KILLED proof of input_0tfWGfR65q.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 138 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 72 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 104 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 407 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 1257 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 557 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 4023 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 3308 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 141 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms] (82) CdtProblem (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 4 ms] (108) CdtProblem (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 34 ms] (110) CdtProblem (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2112 ms] (112) CdtProblem (113) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms] (120) CdtProblem (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 5 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 3 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms] (192) 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: double(0) -> 0 double(s(x)) -> s(s(double(x))) del(x, nil) -> nil del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) if(true, x, y, xs) -> xs if(false, x, y, xs) -> cons(y, del(x, xs)) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) first(nil) -> 0 first(cons(x, xs)) -> x doublelist(nil) -> nil doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: double(0') -> 0' double(s(x)) -> s(s(double(x))) del(x, nil) -> nil del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) if(true, x, y, xs) -> xs if(false, x, y, xs) -> cons(y, del(x, xs)) eq(0', 0') -> true eq(0', s(y)) -> false eq(s(x), 0') -> false eq(s(x), s(y)) -> eq(x, y) first(nil) -> 0' first(cons(x, xs)) -> x doublelist(nil) -> nil doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: double(0) -> 0 double(s(x)) -> s(s(double(x))) del(x, nil) -> nil del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) if(true, x, y, xs) -> xs if(false, x, y, xs) -> cons(y, del(x, xs)) eq(0, 0) -> true eq(0, s(y)) -> false eq(s(x), 0) -> false eq(s(x), s(y)) -> eq(x, y) first(nil) -> 0 first(cons(x, xs)) -> x doublelist(nil) -> nil doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) [1] if(true, x, y, xs) -> xs [1] if(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] first(nil) -> 0 [1] first(cons(x, xs)) -> x [1] doublelist(nil) -> nil [1] doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) [1] if(true, x, y, xs) -> xs [1] if(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] first(nil) -> 0 [1] first(cons(x, xs)) -> x [1] doublelist(nil) -> nil [1] doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) [1] The TRS has the following type information: double :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s del :: 0:s -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons if :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false first :: nil:cons -> 0:s doublelist :: nil:cons -> 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: double_1 doublelist_1 (c) The following functions are completely defined: eq_2 del_2 first_1 if_4 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) [1] if(true, x, y, xs) -> xs [1] if(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] first(nil) -> 0 [1] first(cons(x, xs)) -> x [1] doublelist(nil) -> nil [1] doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) [1] The TRS has the following type information: double :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s del :: 0:s -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons if :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false first :: nil:cons -> 0:s doublelist :: nil:cons -> nil:cons 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: double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] del(x, nil) -> nil [1] del(0, cons(0, xs)) -> if(true, 0, 0, xs) [2] del(0, cons(s(y'), xs)) -> if(false, 0, s(y'), xs) [2] del(s(x'), cons(0, xs)) -> if(false, s(x'), 0, xs) [2] del(s(x''), cons(s(y''), xs)) -> if(eq(x'', y''), s(x''), s(y''), xs) [2] if(true, x, y, xs) -> xs [1] if(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] first(nil) -> 0 [1] first(cons(x, xs)) -> x [1] doublelist(nil) -> nil [1] doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(x, cons(x, xs)))) [2] The TRS has the following type information: double :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s del :: 0:s -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons if :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false first :: nil:cons -> 0:s doublelist :: nil:cons -> nil:cons 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 nil => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(x'', y''), 1 + x'', 1 + y'', xs) :|: z = 1 + x'', xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, x'' >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + x', 0, xs) :|: z = 1 + x', xs >= 0, z' = 1 + 0 + xs, x' >= 0 del(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 2 }-> 1 + double(x) + doublelist(del(x, 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> xs :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if(z, z', z'', z1) -{ 1 }-> 1 + y + del(x, xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 2 }-> 1 + double(x) + doublelist(del(x, 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { double } { first } { eq } { del, if } { doublelist } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 2 }-> 1 + double(x) + doublelist(del(x, 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {double}, {first}, {eq}, {del,if}, {doublelist} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 2 }-> 1 + double(x) + doublelist(del(x, 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {double}, {first}, {eq}, {del,if}, {doublelist} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 2 }-> 1 + double(x) + doublelist(del(x, 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {double}, {first}, {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 2 }-> 1 + double(x) + doublelist(del(x, 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {first}, {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {first}, {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: first after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {first}, {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: first after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: eq after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {eq}, {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: ?, size: O(1) [1] ---------------------------------------- (35) 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' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 2 }-> if(eq(z - 1, y''), 1 + (z - 1), 1 + y'', xs) :|: xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y'' }-> if(s'', 1 + (z - 1), 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: del after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z'' + z1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y'' }-> if(s'', 1 + (z - 1), 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {del,if}, {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] del: runtime: ?, size: O(n^1) [z'] if: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: del after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 49 + 17*z' Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 50 + 17*z1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 5 + y'' }-> if(s'', 1 + (z - 1), 1 + y'', xs) :|: s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 2 }-> if(1, 0, 0, z' - 1) :|: z' - 1 >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 0, 1 + y', xs) :|: xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 3 + x }-> 1 + s' + doublelist(del(x, 1 + x + xs)) :|: s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 1 }-> 1 + z'' + del(z', z1) :|: z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s2 :|: s2 >= 0, s2 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s3 :|: s3 >= 0, s3 <= 1 + y' + xs + 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s4 :|: s4 >= 0, s4 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y'' }-> s5 :|: s5 >= 0, s5 <= 1 + y'' + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 69 + 18*x + 17*xs }-> 1 + s' + doublelist(s7) :|: s7 >= 0, s7 <= 1 + x + xs, s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s6 :|: s6 >= 0, s6 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: doublelist after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s2 :|: s2 >= 0, s2 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s3 :|: s3 >= 0, s3 <= 1 + y' + xs + 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s4 :|: s4 >= 0, s4 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y'' }-> s5 :|: s5 >= 0, s5 <= 1 + y'' + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 69 + 18*x + 17*xs }-> 1 + s' + doublelist(s7) :|: s7 >= 0, s7 <= 1 + x + xs, s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s6 :|: s6 >= 0, s6 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] doublelist: runtime: ?, size: INF ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: doublelist after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s2 :|: s2 >= 0, s2 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s3 :|: s3 >= 0, s3 <= 1 + y' + xs + 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s4 :|: s4 >= 0, s4 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y'' }-> s5 :|: s5 >= 0, s5 <= 1 + y'' + xs + 1, s'' >= 0, s'' <= 1, xs >= 0, z' = 1 + (1 + y'') + xs, y'' >= 0, z - 1 >= 0 del(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0 doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 69 + 18*x + 17*xs }-> 1 + s' + doublelist(s7) :|: s7 >= 0, s7 <= 1 + x + xs, s' >= 0, s' <= 2 * x, z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 3 + z' }-> s1 :|: s1 >= 0, s1 <= 1, z - 1 >= 0, z' - 1 >= 0 eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> z1 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0 if(z, z', z'', z1) -{ 50 + 17*z1 }-> 1 + z'' + s6 :|: s6 >= 0, s6 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 Function symbols to be analyzed: {doublelist} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] first: runtime: O(1) [1], size: O(n^1) [z] eq: runtime: O(n^1) [3 + z'], size: O(1) [1] del: runtime: O(n^1) [49 + 17*z'], size: O(n^1) [z'] if: runtime: O(n^1) [50 + 17*z1], size: O(n^1) [1 + z'' + z1] doublelist: runtime: INF, size: INF ---------------------------------------- (49) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (50) 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: double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] del(x, nil) -> nil [1] del(x, cons(y, xs)) -> if(eq(x, y), x, y, xs) [1] if(true, x, y, xs) -> xs [1] if(false, x, y, xs) -> cons(y, del(x, xs)) [1] eq(0, 0) -> true [1] eq(0, s(y)) -> false [1] eq(s(x), 0) -> false [1] eq(s(x), s(y)) -> eq(x, y) [1] first(nil) -> 0 [1] first(cons(x, xs)) -> x [1] doublelist(nil) -> nil [1] doublelist(cons(x, xs)) -> cons(double(x), doublelist(del(first(cons(x, xs)), cons(x, xs)))) [1] The TRS has the following type information: double :: 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s del :: 0:s -> nil:cons -> nil:cons nil :: nil:cons cons :: 0:s -> nil:cons -> nil:cons if :: true:false -> 0:s -> 0:s -> nil:cons -> nil:cons eq :: 0:s -> 0:s -> true:false true :: true:false false :: true:false first :: nil:cons -> 0:s doublelist :: nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (51) 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 nil => 0 true => 1 false => 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 1 }-> if(eq(x, y), x, y, xs) :|: xs >= 0, z' = 1 + y + xs, x >= 0, y >= 0, z = x del(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x doublelist(z) -{ 1 }-> 0 :|: z = 0 doublelist(z) -{ 1 }-> 1 + double(x) + doublelist(del(first(1 + x + xs), 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 eq(z, z') -{ 1 }-> eq(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x eq(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0 eq(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 eq(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 first(z) -{ 1 }-> x :|: z = 1 + x + xs, xs >= 0, x >= 0 first(z) -{ 1 }-> 0 :|: z = 0 if(z, z', z'', z1) -{ 1 }-> xs :|: xs >= 0, z' = x, z'' = y, z = 1, x >= 0, y >= 0, z1 = xs if(z, z', z'', z1) -{ 1 }-> 1 + y + del(x, xs) :|: xs >= 0, z' = x, z'' = y, x >= 0, y >= 0, z = 0, z1 = xs Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) first(nil) -> 0 first(cons(z0, z1)) -> z0 doublelist(nil) -> nil doublelist(cons(z0, z1)) -> cons(double(z0), doublelist(del(first(cons(z0, z1)), cons(z0, z1)))) Tuples: DOUBLE(0) -> c DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, nil) -> c2 DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(true, z0, z1, z2) -> c4 IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(0, 0) -> c6 EQ(0, s(z0)) -> c7 EQ(s(z0), 0) -> c8 EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) FIRST(nil) -> c10 FIRST(cons(z0, z1)) -> c11 DOUBLELIST(nil) -> c12 DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1)), FIRST(cons(z0, z1))) S tuples: DOUBLE(0) -> c DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, nil) -> c2 DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(true, z0, z1, z2) -> c4 IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(0, 0) -> c6 EQ(0, s(z0)) -> c7 EQ(s(z0), 0) -> c8 EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) FIRST(nil) -> c10 FIRST(cons(z0, z1)) -> c11 DOUBLELIST(nil) -> c12 DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1)), FIRST(cons(z0, z1))) K tuples:none Defined Rule Symbols: double_1, del_2, if_4, eq_2, first_1, doublelist_1 Defined Pair Symbols: DOUBLE_1, DEL_2, IF_4, EQ_2, FIRST_1, DOUBLELIST_1 Compound Symbols: c, c1_1, c2, c3_2, c4, c5_1, c6, c7, c8, c9_1, c10, c11, c12, c13_1, c14_3 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: DOUBLELIST(nil) -> c12 EQ(s(z0), 0) -> c8 EQ(0, 0) -> c6 EQ(0, s(z0)) -> c7 DEL(z0, nil) -> c2 FIRST(cons(z0, z1)) -> c11 DOUBLE(0) -> c IF(true, z0, z1, z2) -> c4 FIRST(nil) -> c10 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) first(nil) -> 0 first(cons(z0, z1)) -> z0 doublelist(nil) -> nil doublelist(cons(z0, z1)) -> cons(double(z0), doublelist(del(first(cons(z0, z1)), cons(z0, z1)))) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1)), FIRST(cons(z0, z1))) S tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1)), FIRST(cons(z0, z1))) K tuples:none Defined Rule Symbols: double_1, del_2, if_4, eq_2, first_1, doublelist_1 Defined Pair Symbols: DOUBLE_1, DEL_2, IF_4, EQ_2, DOUBLELIST_1 Compound Symbols: c1_1, c3_2, c5_1, c9_1, c13_1, c14_3 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) del(z0, nil) -> nil del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) first(nil) -> 0 first(cons(z0, z1)) -> z0 doublelist(nil) -> nil doublelist(cons(z0, z1)) -> cons(double(z0), doublelist(del(first(cons(z0, z1)), cons(z0, z1)))) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) S tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) K tuples:none Defined Rule Symbols: double_1, del_2, if_4, eq_2, first_1, doublelist_1 Defined Pair Symbols: DOUBLE_1, DEL_2, IF_4, EQ_2, DOUBLELIST_1 Compound Symbols: c1_1, c3_2, c5_1, c9_1, c13_1, c14_2 ---------------------------------------- (59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: double(0) -> 0 double(s(z0)) -> s(s(double(z0))) first(nil) -> 0 doublelist(nil) -> nil doublelist(cons(z0, z1)) -> cons(double(z0), doublelist(del(first(cons(z0, z1)), cons(z0, z1)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) S tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) K tuples:none Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, DEL_2, IF_4, EQ_2, DOUBLELIST_1 Compound Symbols: c1_1, c3_2, c5_1, c9_1, c13_1, c14_2 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) We considered the (Usable) Rules:none And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = 0 POL(DOUBLELIST(x_1)) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = 0 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(del(x_1, x_2)) = [1] + x_1 POL(eq(x_1, x_2)) = [1] + x_1 + x_2 POL(false) = [1] POL(first(x_1)) = [1] + x_1 POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) S tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, DEL_2, IF_4, EQ_2, DOUBLELIST_1 Compound Symbols: c1_1, c3_2, c5_1, c9_1, c13_1, c14_2 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLE(s(z0)) -> c1(DOUBLE(z0)) We considered the (Usable) Rules: del(z0, nil) -> nil if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = [1] + x_1 POL(DOUBLELIST(x_1)) = [1] + x_1 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = 0 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = [1] + x_1 + x_2 POL(false) = 0 POL(first(x_1)) = [1] + x_1 POL(if(x_1, x_2, x_3, x_4)) = x_3 + x_4 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) S tuples: DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, DEL_2, IF_4, EQ_2, DOUBLELIST_1 Compound Symbols: c1_1, c3_2, c5_1, c9_1, c13_1, c14_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(z0, cons(z1, z2)) -> c3(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by DEL(0, cons(0, x2)) -> c3(IF(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(0, cons(0, x2)) -> c3(IF(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(0, cons(0, x2)) -> c3(IF(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c14_2, c3_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: DEL(0, cons(0, x2)) -> c3(IF(true, 0, 0, x2), EQ(0, 0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c14_2, c3_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c14_2, c3_2, c3_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(first(cons(z0, z1)), cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) by DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c3_2, c3_1, c14_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(z0), cons(s(z1), x2)) -> c3(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by DEL(s(0), cons(s(0), x2)) -> c3(IF(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(0), cons(s(0), x2)) -> c3(IF(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(0), cons(s(0), x2)) -> c3(IF(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c3_1, c14_2, c3_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c3_1, c14_2, c3_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DOUBLELIST(cons(z1, z2)) -> c14(DOUBLELIST(if(eq(first(cons(z1, z2)), z1), first(cons(z1, z2)), z1, z2)), DEL(first(cons(z1, z2)), cons(z1, z2))) by DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c3_1, c14_2, c3_2 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(del(z0, cons(z0, z1))), DEL(first(cons(z0, z1)), cons(z0, z1))) by DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c3_1, c3_2, c14_2, c14_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) We considered the (Usable) Rules:none And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = [2] POL(DOUBLELIST(x_1)) = [2] POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = 0 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = [3] + [3]x_1 POL(eq(x_1, x_2)) = 0 POL(false) = [2] POL(first(x_1)) = 0 POL(if(x_1, x_2, x_3, x_4)) = [2] + [2]x_1 + x_4 POL(nil) = [3] POL(s(x_1)) = 0 POL(true) = [3] ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) S tuples: IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, IF_4, EQ_2, DOUBLELIST_1, DEL_2 Compound Symbols: c1_1, c5_1, c9_1, c13_1, c3_1, c3_2, c14_2, c14_1 ---------------------------------------- (83) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, z0, z1, z2) -> c5(DEL(z0, z2)) by IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c14_2, c14_1, c5_1 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) by DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c14_2, c14_1, c5_1 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(first(cons(z0, z1)), cons(z0, z1))) by DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c14_2, c14_1, c5_1 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(z0, z2)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z2)), DEL(first(cons(z0, z2)), cons(z0, z2))) by DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c14_1, c5_1, c14_2 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) by DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_2, c14_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(first(cons(z0, z1)), z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) by DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_2, c14_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), first(cons(z0, z1)), z0, z1)), DEL(z0, cons(z0, z1))) by DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: eq(0, 0) -> true eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil first(cons(z0, z1)) -> z0 if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, del_2, first_1, if_4 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_2, c14_1 ---------------------------------------- (97) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: first(cons(z0, z1)) -> z0 ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_2, c14_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c3(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) by DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(IF(true, s(s(0)), s(s(0)), x2), EQ(s(s(0)), s(s(0)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(IF(true, s(s(0)), s(s(0)), x2), EQ(s(s(0)), s(s(0)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(IF(true, s(s(0)), s(s(0)), x2), EQ(s(s(0)), s(s(0)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_2, c14_1 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_2, c14_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DOUBLELIST(cons(z0, z1)) -> c14(DOUBLELIST(if(eq(z0, z0), z0, z0, z1)), DEL(z0, cons(z0, z1))) by DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1)), DEL(0, cons(0, x1))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(x0, cons(x0, x1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1)), DEL(0, cons(0, x1))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1)), DEL(0, cons(0, x1))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(x0, cons(x0, x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(x0, x1)) -> c14(DEL(x0, cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) We considered the (Usable) Rules:none And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = [1] POL(DOUBLELIST(x_1)) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_2 + x_3 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(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) = [1] POL(if(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) We considered the (Usable) Rules: del(z0, nil) -> nil eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) eq(0, 0) -> true if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = 0 POL(DOUBLELIST(x_1)) = [1] + x_1 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_2 + x_3 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(if(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) We considered the (Usable) Rules: del(z0, nil) -> nil if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(DEL(x_1, x_2)) = x_1*x_2 POL(DOUBLE(x_1)) = [2] POL(DOUBLELIST(x_1)) = [1] + x_1^2 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_2*x_4 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 + x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(if(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (113) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(s(s(z0))), cons(s(s(s(z1))), x2)) -> c3(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), x2), EQ(s(s(s(z0))), s(s(s(z1))))) by DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(IF(true, s(s(s(0))), s(s(s(0))), x2), EQ(s(s(s(0))), s(s(s(0))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(IF(true, s(s(s(0))), s(s(s(0))), x2), EQ(s(s(s(0))), s(s(s(0))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(IF(true, s(s(s(0))), s(s(s(0))), x2), EQ(s(s(s(0))), s(s(s(0))))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DOUBLELIST(cons(s(z0), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(z0), s(z0), x1)), DEL(s(z0), cons(s(z0), x1))) by DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) We considered the (Usable) Rules:none And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = [1] POL(DOUBLELIST(x_1)) = [1] POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_2 + x_3 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(del(x_1, x_2)) = [1] + x_1 + x_2 POL(eq(x_1, x_2)) = 0 POL(false) = [1] POL(if(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) We considered the (Usable) Rules: del(z0, nil) -> nil eq(0, s(z0)) -> false eq(s(z0), 0) -> false eq(s(z0), s(z1)) -> eq(z0, z1) if(true, z0, z1, z2) -> z2 eq(0, 0) -> true if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) And the Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(DOUBLE(x_1)) = 0 POL(DOUBLELIST(x_1)) = [1] + x_1 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_2 + x_3 POL(c1(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_2 POL(del(x_1, x_2)) = x_2 POL(eq(x_1, x_2)) = [1] POL(false) = [1] POL(if(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) by DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) by DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (129) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c5(DEL(s(s(x0)), x2)) by IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLE(s(z0)) -> c1(DOUBLE(z0)) EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLE(s(z0)) -> c1(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLE_1, EQ_2, DOUBLELIST_1, DEL_2, IF_4 Compound Symbols: c1_1, c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(z0)) -> c1(DOUBLE(z0)) by DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) S tuples: EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: EQ_2, DOUBLELIST_1, DEL_2, IF_4, DOUBLE_1 Compound Symbols: c9_1, c13_1, c3_1, c3_2, c5_1, c14_1, c14_2, c1_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c9(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c3(EQ(s(0), s(0))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLELIST_1, DEL_2, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c13_1, c3_1, c3_2, c5_1, c14_1, c14_2, c1_1, c9_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 K tuples: DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLELIST_1, DEL_2, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c13_1, c3_1, c5_1, c14_1, c3_2, c14_2, c1_1, c9_1, c3 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLELIST(cons(z0, z1)) -> c13(DOUBLE(z0)) by DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, IF_4, DOUBLELIST_1, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c5_1, c14_1, c3_2, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(x0), cons(s(x1), x2)) -> c3(EQ(s(x0), s(x1))) by DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c3(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, IF_4, DOUBLELIST_1, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c5_1, c14_1, c3_2, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, 0, s(x0), x1) -> c5(DEL(0, x1)) by IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, IF_4, DOUBLELIST_1, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c5_1, c14_1, c3_2, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(0, cons(s(z0), x2)) -> c3(IF(false, 0, s(z0), x2)) by DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, IF_4, DOUBLELIST_1, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c5_1, c14_1, c3_2, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(x0), 0, x1) -> c5(DEL(s(x0), x1)) by IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) S tuples: DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, IF_4, DOUBLELIST_1, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c5_1, c14_1, c3_2, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(z0), cons(0, x2)) -> c3(IF(false, s(z0), 0, x2)) by DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) S tuples: IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: IF_4, DOUBLELIST_1, DEL_2, DOUBLE_1, EQ_2 Compound Symbols: c5_1, c14_1, c3_2, c3_1, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(0), s(s(x0)), x1) -> c5(DEL(s(0), x1)) by IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) S tuples: IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: IF_4, DOUBLELIST_1, DEL_2, DOUBLE_1, EQ_2 Compound Symbols: c5_1, c14_1, c3_2, c3_1, c14_2, c1_1, c9_1, c3, c13_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(x0)), s(0), x1) -> c5(DEL(s(s(x0)), x1)) by IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLELIST_1, DEL_2, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c14_1, c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) by DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DOUBLELIST(cons(z0, z1)) -> c14(DEL(z0, cons(z0, z1))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DOUBLELIST_1, DEL_2, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c14_1, c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLELIST(cons(x0, x1)) -> c14(DEL(x0, cons(x0, x1))) by DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c14_1, c5_1, c1_1, c9_1, c3, c13_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLELIST(cons(s(x0), x1)) -> c14(DEL(s(x0), cons(s(x0), x1))) by DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c14_1, c5_1, c1_1, c9_1, c3, c13_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLELIST(cons(0, z2)) -> c14(DOUBLELIST(z2)) by DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1, c14_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(0)), s(s(s(x0))), x1) -> c5(DEL(s(s(0)), x1)) by IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1, c14_1 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(s(x0))), s(s(0)), x1) -> c5(DEL(s(s(s(x0))), x1)) by IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1, c14_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(s(0))), s(s(s(s(x0)))), x1) -> c5(DEL(s(s(s(0))), x1)) by IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1, c14_1 ---------------------------------------- (167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(s(s(x0)))), s(s(s(0))), x1) -> c5(DEL(s(s(s(s(x0)))), x1)) by IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c5_1, c1_1, c9_1, c3, c13_1, c14_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(s(s(x0)))), s(s(s(s(x1)))), x2) -> c5(DEL(s(s(s(s(x0)))), x2)) by IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, DOUBLE_1, EQ_2, IF_4 Compound Symbols: c3_2, c3_1, c14_2, c1_1, c9_1, c3, c13_1, c5_1, c14_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(s(y0))) -> c1(DOUBLE(s(y0))) by DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, EQ_2, IF_4, DOUBLE_1 Compound Symbols: c3_2, c3_1, c14_2, c9_1, c3, c13_1, c5_1, c14_1, c1_1 ---------------------------------------- (173) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c9(EQ(s(y0), s(y1))) by EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) K tuples: DOUBLELIST(cons(x0, x1)) -> c14(DEL(first(cons(x0, x1)), cons(x0, x1))) DOUBLELIST(cons(0, x1)) -> c14(DOUBLELIST(if(true, 0, 0, x1))) DOUBLELIST(cons(s(0), x1)) -> c14(DOUBLELIST(if(true, s(0), s(0), x1)), DEL(s(0), cons(s(0), x1))) DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c3, c13_1, c5_1, c14_1, c1_1, c9_1 ---------------------------------------- (175) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: DEL(s(s(0)), cons(s(s(0)), x2)) -> c3(EQ(s(s(0)), s(s(0)))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) S tuples: DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2), EQ(s(s(0)), s(s(s(z0))))) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2), EQ(s(s(s(z0))), s(s(0)))) DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) K tuples: DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c3, c13_1, c5_1, c14_1, c1_1, c9_1 ---------------------------------------- (177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) S tuples: DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) K tuples: DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c3_2, c14_2, c3, c13_1, c5_1, c14_1, c1_1, c9_1 ---------------------------------------- (179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) by DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c3(EQ(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) K tuples: DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c3_2, c14_2, c3, c13_1, c5_1, c14_1, c1_1, c9_1 ---------------------------------------- (181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(0), cons(s(s(z0)), x2)) -> c3(IF(false, s(0), s(s(z0)), x2)) by DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c3_2, c14_2, c3, c13_1, c5_1, c14_1, c1_1, c9_1 ---------------------------------------- (183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(s(z0)), cons(s(0), x2)) -> c3(IF(false, s(s(z0)), s(0), x2)) by DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(0), x2)) -> c3 DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c3_2, c14_2, c3, c13_1, c5_1, c14_1, c1_1, c9_1 ---------------------------------------- (185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLELIST(cons(s(s(y0)), z1)) -> c13(DOUBLE(s(s(y0)))) by DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(x0)), cons(s(s(x1)), x2)) -> c3(EQ(s(s(x0)), s(s(x1)))) DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_1, c3_2, c14_2, c3, c5_1, c14_1, c1_1, c9_1, c13_1 ---------------------------------------- (187) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c3(EQ(s(s(y0)), s(s(y1)))) by DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c3(EQ(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c3, c5_1, c14_1, c1_1, c9_1, c13_1 ---------------------------------------- (189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: IF(false, s(s(0)), s(s(s(z0))), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) DOUBLELIST(cons(s(s(0)), z1)) -> c14(DEL(s(s(0)), cons(s(s(0)), z1))) IF(false, s(s(0)), 0, cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) IF(false, s(s(0)), s(0), cons(s(s(0)), y0)) -> c5(DEL(s(s(0)), cons(s(s(0)), y0))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c3, c5_1, c14_1, c1_1, c9_1, c13_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, 0, s(z0), cons(s(y0), y1)) -> c5(DEL(0, cons(s(y0), y1))) by IF(false, 0, s(z0), cons(s(z1), cons(s(y1), y2))) -> c5(DEL(0, cons(s(z1), cons(s(y1), y2)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: 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, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil Tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DOUBLELIST(cons(s(0), z0)) -> c14(DOUBLELIST(z0), DEL(s(0), cons(s(0), z0))) DEL(s(0), cons(s(0), x2)) -> c3 DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLELIST(cons(0, cons(s(s(y0)), y1))) -> c14(DOUBLELIST(cons(s(s(y0)), y1))) DOUBLELIST(cons(0, cons(0, y0))) -> c14(DOUBLELIST(cons(0, y0))) DOUBLELIST(cons(0, cons(s(0), y0))) -> c14(DOUBLELIST(cons(s(0), y0))) DOUBLELIST(cons(0, cons(s(s(0)), y0))) -> c14(DOUBLELIST(cons(s(s(0)), y0))) DOUBLELIST(cons(0, cons(s(s(s(s(y0)))), y1))) -> c14(DOUBLELIST(cons(s(s(s(s(y0)))), y1))) DOUBLELIST(cons(0, cons(s(s(s(y0))), y1))) -> c14(DOUBLELIST(cons(s(s(s(y0))), y1))) DOUBLELIST(cons(0, cons(s(s(s(0))), y0))) -> c14(DOUBLELIST(cons(s(s(s(0))), y0))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) IF(false, 0, s(z0), cons(s(z1), cons(s(y1), y2))) -> c5(DEL(0, cons(s(z1), cons(s(y1), y2)))) S tuples: DEL(s(s(s(0))), cons(s(s(s(s(z0)))), x2)) -> c3(IF(false, s(s(s(0))), s(s(s(s(z0)))), x2), EQ(s(s(s(0))), s(s(s(s(z0)))))) DEL(s(s(s(s(z0)))), cons(s(s(s(0))), x2)) -> c3(IF(false, s(s(s(s(z0)))), s(s(s(0))), x2), EQ(s(s(s(s(z0)))), s(s(s(0))))) DEL(s(s(s(s(z0)))), cons(s(s(s(s(z1)))), x2)) -> c3(IF(eq(z0, z1), s(s(s(s(z0)))), s(s(s(s(z1)))), x2), EQ(s(s(s(s(z0)))), s(s(s(s(z1)))))) DEL(s(s(s(x0))), cons(s(s(s(x1))), x2)) -> c3(EQ(s(s(s(x0))), s(s(s(x1))))) DEL(s(s(s(0))), cons(s(s(s(0))), x2)) -> c3(EQ(s(s(s(0))), s(s(s(0))))) DOUBLELIST(cons(s(s(z0)), x1)) -> c14(DOUBLELIST(if(eq(z0, z0), s(s(z0)), s(s(z0)), x1)), DEL(s(s(z0)), cons(s(s(z0)), x1))) DEL(s(0), cons(s(0), x2)) -> c3 IF(false, s(z0), 0, cons(0, y1)) -> c5(DEL(s(z0), cons(0, y1))) IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c5(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c5(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c3(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), 0, cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(0)), y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c3(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(s(0)), cons(0, cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), 0, cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), 0, cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), 0, cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), 0, cons(s(s(s(0))), y0))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c3(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c3(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c3(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c5(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c5(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c5(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c5(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c5(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(y0))), cons(s(s(0)), y1))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c5(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(y0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c5(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c5(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(y0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(s(y0))), y1)) -> c5(DEL(s(s(0)), cons(s(s(s(y0))), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(0)), cons(s(s(y1)), y2))) IF(false, s(s(0)), s(s(s(z0))), cons(s(0), y1)) -> c5(DEL(s(s(0)), cons(s(0), y1))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(0)), cons(0, cons(0, y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(s(y0))), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(s(y0))), y1)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(s(0)), y0))) -> c5(DEL(s(s(0)), cons(0, cons(s(s(0)), y0)))) IF(false, s(s(0)), s(s(s(z0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(0)), cons(0, cons(s(0), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(z0))), cons(s(s(0)), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(z0))), s(s(0)), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(z0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(0)), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(z0))), s(s(0)), cons(s(0), y1)) -> c5(DEL(s(s(s(z0))), cons(s(0), y1))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(0, y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(y0)))), s(s(0)), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(y0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(0)), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(z0))), s(s(0)), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(z0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(0)), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(y1)), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(s(y0)))), y1)) -> c5(DEL(s(s(s(0))), cons(s(s(s(s(y0)))), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(0))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(s(s(0))), y0)) -> c5(DEL(s(s(s(0))), cons(s(s(s(0))), y0))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(s(0), y1)) -> c5(DEL(s(s(s(0))), cons(s(0), y1))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(0, y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y0)))), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(s(y0)))), y1)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y0))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(s(s(0))), y0)))) IF(false, s(s(s(0))), s(s(s(s(z0)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(0))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(0))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(0)), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(0)), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(y1)), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(y1)), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(0))), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(0))), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(s(y1)))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(s(y1)))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(s(s(y1))), y2)) -> c5(DEL(s(s(s(s(z0)))), cons(s(s(s(y1))), y2))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(s(0), y1)) -> c5(DEL(s(s(s(s(z0)))), cons(s(0), y1))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(0, y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(0, y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(0)), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(0)), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(y1)), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(0))), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(0))), y1)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(s(y1)))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(s(y1)))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(s(s(y1))), y2))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(s(s(y1))), y2)))) IF(false, s(s(s(s(z0)))), s(s(s(s(z1)))), cons(0, cons(s(0), y1))) -> c5(DEL(s(s(s(s(z0)))), cons(0, cons(s(0), y1)))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c9(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(0)), cons(s(s(s(z0))), x2)) -> c3(IF(false, s(s(0)), s(s(s(z0))), x2)) DEL(s(s(s(z0))), cons(s(s(0)), x2)) -> c3(IF(false, s(s(s(z0))), s(s(0)), x2)) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c3(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c3(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(0, y1)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y1)), y2)))) DEL(s(0), cons(s(s(z0)), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(0)), cons(s(0), cons(s(s(s(y0))), y1))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(s(y0))), y1))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(0)), y1))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(0)), y1))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c3(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(0)), cons(s(0), cons(s(s(0)), y0))) -> c3(IF(false, s(s(0)), s(0), cons(s(s(0)), y0))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(s(y0)))), y1))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(s(y0)))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(0))), y1))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(0))), y1))) DEL(s(s(s(s(y0)))), cons(s(0), cons(s(s(s(s(y1)))), y2))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(s(s(s(s(y1)))), y2))) DEL(s(s(s(y0))), cons(s(0), cons(s(s(s(y1))), y2))) -> c3(IF(false, s(s(s(y0))), s(0), cons(s(s(s(y1))), y2))) DEL(s(s(s(0))), cons(s(0), cons(s(s(s(0))), y0))) -> c3(IF(false, s(s(s(0))), s(0), cons(s(s(s(0))), y0))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c3(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(s(y0))), y1)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(s(y0))), y1)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(0)), y1)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(0)), y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(0)), cons(s(0), cons(0, cons(s(s(0)), y0)))) -> c3(IF(false, s(s(0)), s(0), cons(0, cons(s(s(0)), y0)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(s(y0)))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(0))), y1)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(0))), y1)))) DEL(s(s(s(s(y0)))), cons(s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) -> c3(IF(false, s(s(s(s(y0)))), s(0), cons(0, cons(s(s(s(s(y1)))), y2)))) DEL(s(s(s(y0))), cons(s(0), cons(0, cons(s(s(s(y1))), y2)))) -> c3(IF(false, s(s(s(y0))), s(0), cons(0, cons(s(s(s(y1))), y2)))) DEL(s(s(s(0))), cons(s(0), cons(0, cons(s(s(s(0))), y0)))) -> c3(IF(false, s(s(s(0))), s(0), cons(0, cons(s(s(s(0))), y0)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c3(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: DEL(0, cons(s(z0), cons(s(y1), y2))) -> c3(IF(false, 0, s(z0), cons(s(y1), y2))) DOUBLELIST(cons(s(s(y0)), z1)) -> c14(DEL(s(s(y0)), cons(s(s(y0)), z1))) DOUBLELIST(cons(s(s(s(s(y0)))), z1)) -> c14(DEL(s(s(s(s(y0)))), cons(s(s(s(s(y0)))), z1))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c14(DEL(s(s(s(y0))), cons(s(s(s(y0))), z1))) DOUBLELIST(cons(s(s(s(0))), z1)) -> c14(DEL(s(s(s(0))), cons(s(s(s(0))), z1))) DOUBLELIST(cons(s(0), z1)) -> c14(DEL(s(0), cons(s(0), z1))) DOUBLE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) DOUBLELIST(cons(s(s(s(y0))), z1)) -> c13(DOUBLE(s(s(s(y0))))) IF(false, 0, s(z0), cons(s(z1), cons(s(y1), y2))) -> c5(DEL(0, cons(s(z1), cons(s(y1), y2)))) Defined Rule Symbols: eq_2, if_4, del_2 Defined Pair Symbols: DEL_2, DOUBLELIST_1, IF_4, DOUBLE_1, EQ_2 Compound Symbols: c3_2, c3_1, c14_2, c3, c5_1, c14_1, c1_1, c9_1, c13_1