KILLED proof of input_T79Ppvynzs.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 1 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), 353 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 164 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 257 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 37 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 1097 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 529 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 1327 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 229 ms] (42) CpxRNTS (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 120 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 58 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms] (82) CdtProblem (83) CdtRewritingProof [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) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem ---------------------------------------- (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: last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 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) reverse(nil) -> nil reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(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: last(nil) -> 0' last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 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) reverse(nil) -> nil reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(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: last(nil) -> 0 last(cons(x, nil)) -> x last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 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) reverse(nil) -> nil reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(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: last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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] reverse(nil) -> nil [1] reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(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: last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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] reverse(nil) -> nil [1] reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) [1] The TRS has the following type information: last :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons del :: 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 s :: 0:s -> 0:s reverse :: 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: reverse_1 (c) The following functions are completely defined: del_2 last_1 eq_2 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: last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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] reverse(nil) -> nil [1] reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) [1] The TRS has the following type information: last :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons del :: 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 s :: 0:s -> 0:s reverse :: 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: last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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] reverse(nil) -> nil [1] reverse(cons(x, nil)) -> cons(last(cons(x, nil)), reverse(del(x, cons(x, nil)))) [2] reverse(cons(x, cons(y1, xs'))) -> cons(last(cons(x, cons(y1, xs'))), reverse(del(last(cons(y1, xs')), cons(x, cons(y1, xs'))))) [2] The TRS has the following type information: last :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons del :: 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 s :: 0:s -> 0:s reverse :: 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: nil => 0 0 => 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 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 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 last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + 0) + reverse(del(x, 1 + x + 0)) :|: x >= 0, z = 1 + x + 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + 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 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { eq } { last } { del, if } { reverse } ---------------------------------------- (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 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {eq}, {last}, {del,if}, {reverse} ---------------------------------------- (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 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {eq}, {last}, {del,if}, {reverse} ---------------------------------------- (21) 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 ---------------------------------------- (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 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {eq}, {last}, {del,if}, {reverse} Previous analysis results are: eq: runtime: ?, size: O(1) [1] ---------------------------------------- (23) 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' ---------------------------------------- (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 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {last}, {del,if}, {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (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') -{ 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {last}, {del,if}, {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: last 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') -{ 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {last}, {del,if}, {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: last after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (30) 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 2 }-> 1 + last(1 + x + (1 + y1 + xs')) + reverse(del(last(1 + y1 + xs'), 1 + x + (1 + y1 + xs'))) :|: y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') reverse(z) -{ 2 }-> 1 + last(1 + (z - 1) + 0) + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: z - 1 >= 0 Function symbols to be analyzed: {del,if}, {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], 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') -{ 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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 last(z) -{ 4 + xs + y }-> s'' :|: s'' >= 0, s'' <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 4 + z }-> 1 + s1 + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: s1 >= 0, s1 <= 1 + (z - 1) + 0, z - 1 >= 0 reverse(z) -{ 9 + x + 2*xs' + 2*y1 }-> 1 + s2 + reverse(del(s3, 1 + x + (1 + y1 + xs'))) :|: s2 >= 0, s2 <= 1 + x + (1 + y1 + xs'), s3 >= 0, s3 <= 1 + y1 + xs', y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') Function symbols to be analyzed: {del,if}, {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (33) 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 ---------------------------------------- (34) 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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 last(z) -{ 4 + xs + y }-> s'' :|: s'' >= 0, s'' <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 4 + z }-> 1 + s1 + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: s1 >= 0, s1 <= 1 + (z - 1) + 0, z - 1 >= 0 reverse(z) -{ 9 + x + 2*xs' + 2*y1 }-> 1 + s2 + reverse(del(s3, 1 + x + (1 + y1 + xs'))) :|: s2 >= 0, s2 <= 1 + x + (1 + y1 + xs'), s3 >= 0, s3 <= 1 + y1 + xs', y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') Function symbols to be analyzed: {del,if}, {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] del: runtime: ?, size: O(n^1) [z'] if: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (35) 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 ---------------------------------------- (36) 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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 last(z) -{ 4 + xs + y }-> s'' :|: s'' >= 0, s'' <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 4 + z }-> 1 + s1 + reverse(del(z - 1, 1 + (z - 1) + 0)) :|: s1 >= 0, s1 <= 1 + (z - 1) + 0, z - 1 >= 0 reverse(z) -{ 9 + x + 2*xs' + 2*y1 }-> 1 + s2 + reverse(del(s3, 1 + x + (1 + y1 + xs'))) :|: s2 >= 0, s2 <= 1 + x + (1 + y1 + xs'), s3 >= 0, s3 <= 1 + y1 + xs', y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') Function symbols to be analyzed: {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] ---------------------------------------- (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') -{ 35 + 17*z' }-> s4 :|: s4 >= 0, s4 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s5 :|: s5 >= 0, s5 <= 1 + y' + xs + 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s6 :|: s6 >= 0, s6 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y'' }-> s7 :|: s7 >= 0, s7 <= 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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'' + s8 :|: s8 >= 0, s8 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s'' :|: s'' >= 0, s'' <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 53 + 18*z }-> 1 + s1 + reverse(s9) :|: s9 >= 0, s9 <= 1 + (z - 1) + 0, s1 >= 0, s1 <= 1 + (z - 1) + 0, z - 1 >= 0 reverse(z) -{ 92 + 18*x + 19*xs' + 19*y1 }-> 1 + s2 + reverse(s10) :|: s10 >= 0, s10 <= 1 + x + (1 + y1 + xs'), s2 >= 0, s2 <= 1 + x + (1 + y1 + xs'), s3 >= 0, s3 <= 1 + y1 + xs', y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') Function symbols to be analyzed: {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: reverse after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s4 :|: s4 >= 0, s4 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s5 :|: s5 >= 0, s5 <= 1 + y' + xs + 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s6 :|: s6 >= 0, s6 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y'' }-> s7 :|: s7 >= 0, s7 <= 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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'' + s8 :|: s8 >= 0, s8 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s'' :|: s'' >= 0, s'' <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 53 + 18*z }-> 1 + s1 + reverse(s9) :|: s9 >= 0, s9 <= 1 + (z - 1) + 0, s1 >= 0, s1 <= 1 + (z - 1) + 0, z - 1 >= 0 reverse(z) -{ 92 + 18*x + 19*xs' + 19*y1 }-> 1 + s2 + reverse(s10) :|: s10 >= 0, s10 <= 1 + x + (1 + y1 + xs'), s2 >= 0, s2 <= 1 + x + (1 + y1 + xs'), s3 >= 0, s3 <= 1 + y1 + xs', y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') Function symbols to be analyzed: {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] reverse: runtime: ?, size: INF ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: reverse after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: del(z, z') -{ 35 + 17*z' }-> s4 :|: s4 >= 0, s4 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z = 0 del(z, z') -{ 52 + 17*xs }-> s5 :|: s5 >= 0, s5 <= 1 + y' + xs + 1, xs >= 0, z' = 1 + (1 + y') + xs, y' >= 0, z = 0 del(z, z') -{ 35 + 17*z' }-> s6 :|: s6 >= 0, s6 <= 0 + (z' - 1) + 1, z' - 1 >= 0, z - 1 >= 0 del(z, z') -{ 55 + 17*xs + y'' }-> s7 :|: s7 >= 0, s7 <= 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 eq(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 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 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'' + s8 :|: s8 >= 0, s8 <= z1, z1 >= 0, z' >= 0, z'' >= 0, z = 0 last(z) -{ 4 + xs + y }-> s'' :|: s'' >= 0, s'' <= 1 + y + xs, xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 last(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 53 + 18*z }-> 1 + s1 + reverse(s9) :|: s9 >= 0, s9 <= 1 + (z - 1) + 0, s1 >= 0, s1 <= 1 + (z - 1) + 0, z - 1 >= 0 reverse(z) -{ 92 + 18*x + 19*xs' + 19*y1 }-> 1 + s2 + reverse(s10) :|: s10 >= 0, s10 <= 1 + x + (1 + y1 + xs'), s2 >= 0, s2 <= 1 + x + (1 + y1 + xs'), s3 >= 0, s3 <= 1 + y1 + xs', y1 >= 0, x >= 0, xs' >= 0, z = 1 + x + (1 + y1 + xs') Function symbols to be analyzed: {reverse} Previous analysis results are: eq: runtime: O(n^1) [3 + z'], size: O(1) [1] last: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] reverse: runtime: INF, size: INF ---------------------------------------- (43) 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 ---------------------------------------- (44) 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: last(nil) -> 0 [1] last(cons(x, nil)) -> x [1] last(cons(x, cons(y, xs))) -> last(cons(y, xs)) [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] reverse(nil) -> nil [1] reverse(cons(x, xs)) -> cons(last(cons(x, xs)), reverse(del(last(cons(x, xs)), cons(x, xs)))) [1] The TRS has the following type information: last :: nil:cons -> 0:s nil :: nil:cons 0 :: 0:s cons :: 0:s -> nil:cons -> nil:cons del :: 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 s :: 0:s -> 0:s reverse :: nil:cons -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 0 => 0 true => 1 false => 0 ---------------------------------------- (46) 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 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 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 last(z) -{ 1 }-> x :|: x >= 0, z = 1 + x + 0 last(z) -{ 1 }-> last(1 + y + xs) :|: xs >= 0, x >= 0, y >= 0, z = 1 + x + (1 + y + xs) last(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 1 }-> 0 :|: z = 0 reverse(z) -{ 1 }-> 1 + last(1 + x + xs) + reverse(del(last(1 + x + xs), 1 + x + xs)) :|: z = 1 + x + xs, xs >= 0, x >= 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: last(nil) -> 0 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) 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) reverse(nil) -> nil reverse(cons(z0, z1)) -> cons(last(cons(z0, z1)), reverse(del(last(cons(z0, z1)), cons(z0, z1)))) Tuples: LAST(nil) -> c LAST(cons(z0, nil)) -> c1 LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, nil) -> c3 DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(true, z0, z1, z2) -> c5 IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(0, 0) -> c7 EQ(0, s(z0)) -> c8 EQ(s(z0), 0) -> c9 EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(nil) -> c11 REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: LAST(nil) -> c LAST(cons(z0, nil)) -> c1 LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, nil) -> c3 DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(true, z0, z1, z2) -> c5 IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(0, 0) -> c7 EQ(0, s(z0)) -> c8 EQ(s(z0), 0) -> c9 EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(nil) -> c11 REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: last_1, del_2, if_4, eq_2, reverse_1 Defined Pair Symbols: LAST_1, DEL_2, IF_4, EQ_2, REVERSE_1 Compound Symbols: c, c1, c2_1, c3, c4_2, c5, c6_1, c7, c8, c9, c10_1, c11, c12_1, c13_3 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: EQ(0, s(z0)) -> c8 EQ(0, 0) -> c7 LAST(cons(z0, nil)) -> c1 DEL(z0, nil) -> c3 REVERSE(nil) -> c11 LAST(nil) -> c EQ(s(z0), 0) -> c9 IF(true, z0, z1, z2) -> c5 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: last(nil) -> 0 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) 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) reverse(nil) -> nil reverse(cons(z0, z1)) -> cons(last(cons(z0, z1)), reverse(del(last(cons(z0, z1)), cons(z0, z1)))) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: last_1, del_2, if_4, eq_2, reverse_1 Defined Pair Symbols: LAST_1, DEL_2, IF_4, EQ_2, REVERSE_1 Compound Symbols: c2_1, c4_2, c6_1, c10_1, c12_1, c13_3 ---------------------------------------- (51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: last(nil) -> 0 reverse(nil) -> nil reverse(cons(z0, z1)) -> cons(last(cons(z0, z1)), reverse(del(last(cons(z0, z1)), cons(z0, z1)))) ---------------------------------------- (52) 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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples:none Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, DEL_2, IF_4, EQ_2, REVERSE_1 Compound Symbols: c2_1, c4_2, c6_1, c10_1, c12_1, c13_3 ---------------------------------------- (53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) We considered the (Usable) Rules:none And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(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(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = 0 POL(LAST(x_1)) = x_1 POL(REVERSE(x_1)) = [1] POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = [1] POL(eq(x_1, x_2)) = [1] + x_1 + x_2 POL(false) = 0 POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(last(x_1)) = [1] POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (54) 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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, DEL_2, IF_4, EQ_2, REVERSE_1 Compound Symbols: c2_1, c4_2, c6_1, c10_1, c12_1, c13_3 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(z0, cons(z1, z2)) -> c4(IF(eq(z0, z1), z0, z1, z2), EQ(z0, z1)) by DEL(0, cons(0, x2)) -> c4(IF(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) ---------------------------------------- (56) 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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) DEL(0, cons(0, x2)) -> c4(IF(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) DEL(0, cons(0, x2)) -> c4(IF(true, 0, 0, x2), EQ(0, 0)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c13_3, c4_2 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: DEL(0, cons(0, x2)) -> c4(IF(true, 0, 0, x2), EQ(0, 0)) ---------------------------------------- (58) 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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2), EQ(0, s(z0))) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2), EQ(s(z0), 0)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c13_3, c4_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c13_3, c4_2, c4_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REVERSE(cons(z0, z1)) -> c13(REVERSE(del(last(cons(z0, z1)), cons(z0, z1))), DEL(last(cons(z0, z1)), cons(z0, z1)), LAST(cons(z0, z1))) by REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_2, c4_1, c13_3 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_2, c4_1, c13_3, c13_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace DEL(s(z0), cons(s(z1), x2)) -> c4(IF(eq(z0, z1), s(z0), s(z1), x2), EQ(s(z0), s(z1))) by DEL(s(0), cons(s(0), x2)) -> c4(IF(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4(IF(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4(IF(true, s(0), s(0), x2), EQ(s(0), s(0))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_3, c13_2, c4_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_3, c13_2, c4_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REVERSE(cons(z1, z2)) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z1), last(cons(z1, z2)), z1, z2)), DEL(last(cons(z1, z2)), cons(z1, z2)), LAST(cons(z1, z2))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil)), LAST(cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_3, c13_2, c4_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_3, c13_2, c4_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(z0, cons(z1, z2)))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c13(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c13(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c13(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2))), LAST(cons(x0, cons(x1, x2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_2, c4_2, c13_3 ---------------------------------------- (75) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_2, c4_2, c13_3, c_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) We considered the (Usable) Rules:none And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_3 POL(LAST(x_1)) = x_1 POL(REVERSE(x_1)) = [1] POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = 0 POL(eq(x_1, x_2)) = 0 POL(false) = [1] POL(if(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c13_2, c4_2, c13_3, c_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace REVERSE(cons(z0, nil)) -> c13(REVERSE(del(z0, cons(z0, nil))), DEL(last(cons(z0, nil)), cons(z0, nil))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c4_2, c13_3, c13_2, c_1, c13_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) We considered the (Usable) Rules:none And the Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(DEL(x_1, x_2)) = 0 POL(EQ(x_1, x_2)) = 0 POL(IF(x_1, x_2, x_3, x_4)) = x_3 POL(LAST(x_1)) = x_1 POL(REVERSE(x_1)) = [1] POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(cons(x_1, x_2)) = 0 POL(del(x_1, x_2)) = 0 POL(eq(x_1, x_2)) = 0 POL(false) = 0 POL(if(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4 POL(last(x_1)) = 0 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = [1] ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c4_2, c13_3, c13_2, c_1, c13_1 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c4_2, c13_3, c13_2, c_1, c13_1 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c4_2, c13_2, c13_3, c_1, c13_1 ---------------------------------------- (87) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c4_2, c13_2, c13_3, c_1, c13_1 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, IF_4, EQ_2, REVERSE_1, DEL_2 Compound Symbols: c2_1, c6_1, c10_1, c12_1, c4_1, c4_2, c13_3, c_1, c13_2, c13_1 ---------------------------------------- (91) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, z0, z1, z2) -> c6(DEL(z0, z2)) by IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, EQ_2, REVERSE_1, DEL_2, IF_4 Compound Symbols: c2_1, c10_1, c12_1, c4_1, c4_2, c13_3, c_1, c13_2, c13_1, c6_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z1, cons(x1, x2))) -> c13(REVERSE(if(eq(last(cons(x1, x2)), z1), last(cons(x1, x2)), z1, cons(x1, x2))), DEL(last(cons(z1, cons(x1, x2))), cons(z1, cons(x1, x2))), LAST(cons(z1, cons(x1, x2)))) by REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, EQ_2, REVERSE_1, DEL_2, IF_4 Compound Symbols: c2_1, c10_1, c12_1, c4_1, c4_2, c13_3, c_1, c13_2, c13_1, c6_1 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(x0, cons(z0, nil))) -> c13(REVERSE(del(z0, cons(x0, cons(z0, nil)))), DEL(last(cons(x0, cons(z0, nil))), cons(x0, cons(z0, nil))), LAST(cons(x0, cons(z0, nil)))) by REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) ---------------------------------------- (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 last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, EQ_2, REVERSE_1, DEL_2, IF_4 Compound Symbols: c2_1, c10_1, c12_1, c4_1, c4_2, c13_3, c_1, c13_2, c13_1, c6_1 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(x0, cons(z0, cons(z1, z2)))) -> c13(REVERSE(del(last(cons(z1, z2)), cons(x0, cons(z0, cons(z1, z2))))), DEL(last(cons(x0, cons(z0, cons(z1, z2)))), cons(x0, cons(z0, cons(z1, z2)))), LAST(cons(x0, cons(z0, cons(z1, z2))))) by REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, EQ_2, REVERSE_1, DEL_2, IF_4 Compound Symbols: c2_1, c10_1, c12_1, c4_1, c4_2, c_1, c13_2, c13_1, c13_3, c6_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) by REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, EQ_2, REVERSE_1, DEL_2, IF_4 Compound Symbols: c2_1, c10_1, c12_1, c4_1, c4_2, c_1, c13_2, c13_1, c13_3, c6_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(last(cons(z0, nil)), cons(z0, nil))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) S tuples: LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: LAST_1, EQ_2, REVERSE_1, DEL_2, IF_4 Compound Symbols: c2_1, c10_1, c12_1, c4_1, c4_2, c_1, c13_1, c13_3, c13_2, c6_1 ---------------------------------------- (103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LAST(cons(z0, cons(z1, z2))) -> c2(LAST(cons(z1, z2))) by LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) S tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil))), LAST(cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: EQ_2, REVERSE_1, DEL_2, IF_4, LAST_1 Compound Symbols: c10_1, c12_1, c4_1, c4_2, c_1, c13_1, c13_3, c13_2, c6_1, c2_1 ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) S tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: EQ_2, REVERSE_1, DEL_2, IF_4, LAST_1 Compound Symbols: c10_1, c12_1, c4_1, c4_2, c_1, c13_1, c13_3, c13_2, c6_1, c2_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) by REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) S tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: EQ_2, REVERSE_1, DEL_2, IF_4, LAST_1 Compound Symbols: c10_1, c12_1, c4_1, c4_2, c_1, c13_3, c13_2, c6_1, c2_1, c13_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z0, cons(z1, z2))), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) S tuples: EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: EQ_2, REVERSE_1, DEL_2, IF_4, LAST_1 Compound Symbols: c10_1, c12_1, c4_1, c4_2, c_1, c13_3, c13_2, c6_1, c2_1, c13_1 ---------------------------------------- (111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(z0), s(z1)) -> c10(EQ(z0, z1)) by EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2), EQ(s(0), s(s(z0)))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2), EQ(s(s(z0)), s(0))) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) DEL(s(0), cons(s(0), x2)) -> c4(EQ(s(0), s(0))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c13_3, c13_2, c6_1, c2_1, c13_1, c10_1 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c13_3, c13_2, c6_1, c2_1, c13_1, c10_1, c4 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z0, cons(z1, z2))), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) by REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c13_2, c6_1, c13_3, c2_1, c13_1, c10_1, c4 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(last(cons(z0, nil)), z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c13_2, c6_1, c13_3, c2_1, c13_1, c10_1, c4 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), last(cons(z0, nil)), z0, nil)), DEL(z0, cons(z0, nil))) by REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c6_1, c13_3, c13_2, c2_1, c13_1, c10_1, c4 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z1, cons(z2, z3))), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) by REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c6_1, c13_3, c13_2, c2_1, c13_1, c10_1, c4 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(last(cons(z1, nil)), cons(z0, cons(z1, nil)))) by REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) K tuples: REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c12_1, c4_1, c4_2, c_1, c6_1, c13_3, c13_2, c2_1, c13_1, c10_1, c4 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REVERSE(cons(z0, z1)) -> c12(LAST(cons(z0, z1))) by REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(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)) -> c4(EQ(s(x0), s(x1))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, REVERSE_1, IF_4, LAST_1, EQ_2 Compound Symbols: c4_1, c4_2, c_1, c6_1, c13_3, c13_2, c2_1, c13_1, c10_1, c4, c12_1 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(x0), cons(s(x1), x2)) -> c4(EQ(s(x0), s(x1))) by DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, REVERSE_1, IF_4, LAST_1, EQ_2 Compound Symbols: c4_1, c4_2, c_1, c6_1, c13_3, c13_2, c2_1, c13_1, c10_1, c4, c12_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REVERSE(cons(x0, cons(x1, x2))) -> c(LAST(cons(x0, cons(x1, x2)))) by REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_1, c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4, c12_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, 0, s(x0), x1) -> c6(DEL(0, x1)) by IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) S tuples: DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_1, c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4, c12_1 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(0, cons(s(z0), x2)) -> c4(IF(false, 0, s(z0), x2)) by DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) S tuples: DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_1, c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4, c12_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(x0), 0, x1) -> c6(DEL(s(x0), x1)) by IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) S tuples: DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_1, c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4, c12_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(z0), cons(0, x2)) -> c4(IF(false, s(z0), 0, x2)) by DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), 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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4_1, c4, c12_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(0), s(s(x0)), x1) -> c6(DEL(s(0), x1)) by IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4_1, c4, c12_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(x0)), s(0), x1) -> c6(DEL(s(s(x0)), x1)) by IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), 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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, IF_4, REVERSE_1, LAST_1, EQ_2 Compound Symbols: c4_2, c6_1, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4_1, c4, c12_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c6(DEL(s(s(x0)), x2)) by IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, REVERSE_1, LAST_1, EQ_2, IF_4 Compound Symbols: c4_2, c13_3, c_1, c13_2, c2_1, c13_1, c10_1, c4_1, c4, c12_1, c6_1 ---------------------------------------- (145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LAST(cons(z0, cons(z1, cons(y1, y2)))) -> c2(LAST(cons(z1, cons(y1, y2)))) by LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, REVERSE_1, EQ_2, IF_4, LAST_1 Compound Symbols: c4_2, c13_3, c_1, c13_2, c13_1, c10_1, c4_1, c4, c12_1, c6_1, c2_1 ---------------------------------------- (147) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REVERSE(cons(z0, nil)) -> c13(DEL(z0, cons(z0, nil))) by REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, REVERSE_1, EQ_2, IF_4, LAST_1 Compound Symbols: c4_2, c13_3, c_1, c13_2, c10_1, c4_1, c4, c12_1, c6_1, c2_1, c13_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EQ(s(s(y0)), s(s(y1))) -> c10(EQ(s(y0), s(y1))) by EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) S tuples: DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: DEL_2, REVERSE_1, IF_4, LAST_1, EQ_2 Compound Symbols: c4_2, c13_3, c_1, c13_2, c4_1, c4, c12_1, c6_1, c2_1, c13_1, c10_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(s(z0)), cons(s(s(z1)), x2)) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), x2), EQ(s(s(z0)), s(s(z1)))) by DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4_1, c4, c12_1, c6_1, c2_1, c13_1, c10_1, c4_2 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(0), cons(s(s(z0)), x2)) -> c4(IF(false, s(0), s(s(z0)), x2)) by DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4_1, c4, c12_1, c6_1, c2_1, c13_1, c10_1, c4_2 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(s(z0)), cons(s(0), x2)) -> c4(IF(false, s(s(z0)), s(0), x2)) by DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4, c12_1, c4_1, c6_1, c2_1, c13_1, c10_1, c4_2 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(IF(eq(s(y0), s(y1)), s(s(s(y0))), s(s(s(y1))), z2), EQ(s(s(s(y0))), s(s(s(y1))))) by DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4, c12_1, c4_1, c6_1, c2_1, c13_1, c10_1, c4_2 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REVERSE(cons(z0, cons(y1, cons(y2, y3)))) -> c12(LAST(cons(z0, cons(y1, cons(y2, y3))))) by REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4, c4_1, c6_1, c2_1, c13_1, c10_1, c4_2, c12_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DEL(s(s(y0)), cons(s(s(y1)), z2)) -> c4(EQ(s(s(y0)), s(s(y1)))) by DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(y1))))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(y1))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4, c6_1, c4_1, c2_1, c13_1, c10_1, c4_2, c12_1 ---------------------------------------- (163) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace REVERSE(cons(z0, cons(z1, cons(y2, y3)))) -> c(LAST(cons(z0, cons(z1, cons(y2, y3))))) by REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(y1))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(y1))))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4, c6_1, c4_1, c2_1, c13_1, c10_1, c4_2, c12_1 ---------------------------------------- (165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, 0, s(z0), cons(s(y0), y1)) -> c6(DEL(0, cons(s(y0), y1))) by IF(false, 0, s(z0), cons(s(z1), cons(s(y1), y2))) -> c6(DEL(0, cons(s(z1), cons(s(y1), y2)))) ---------------------------------------- (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) del(z0, cons(z1, z2)) -> if(eq(z0, z1), z0, z1, z2) del(z0, nil) -> nil last(cons(z0, nil)) -> z0 last(cons(z0, cons(z1, z2))) -> last(cons(z1, z2)) if(true, z0, z1, z2) -> z2 if(false, z0, z1, z2) -> cons(z1, del(z0, z2)) Tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, cons(z1, z2))) -> c(DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) REVERSE(cons(s(s(y0)), nil)) -> c13(DEL(s(s(y0)), cons(s(s(y0)), nil))) REVERSE(cons(s(0), nil)) -> c13(DEL(s(0), cons(s(0), nil))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(y1))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) IF(false, 0, s(z0), cons(s(z1), cons(s(y1), y2))) -> c6(DEL(0, cons(s(z1), cons(s(y1), y2)))) S tuples: REVERSE(cons(z0, cons(z1, z2))) -> c13(REVERSE(if(eq(last(cons(z1, z2)), z0), last(cons(z1, z2)), z0, cons(z1, z2))), DEL(last(cons(z1, z2)), cons(z0, cons(z1, z2))), LAST(cons(z0, cons(z1, z2)))) REVERSE(cons(z0, nil)) -> c13(REVERSE(if(eq(z0, z0), z0, z0, nil)), DEL(z0, cons(z0, nil))) DEL(s(0), cons(s(0), x2)) -> c4 REVERSE(cons(z0, cons(z1, cons(z2, z3)))) -> c13(REVERSE(del(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3))))), DEL(last(cons(z2, z3)), cons(z0, cons(z1, cons(z2, z3)))), LAST(cons(z0, cons(z1, cons(z2, z3))))) REVERSE(cons(z0, cons(z1, nil))) -> c13(REVERSE(del(z1, cons(z0, cons(z1, nil)))), DEL(z1, cons(z0, cons(z1, nil)))) DEL(0, cons(s(z0), cons(s(y1), y2))) -> c4(IF(false, 0, s(z0), cons(s(y1), y2))) IF(false, s(z0), 0, cons(0, y1)) -> c6(DEL(s(z0), cons(0, y1))) IF(false, s(s(y0)), 0, cons(s(s(y1)), y2)) -> c6(DEL(s(s(y0)), cons(s(s(y1)), y2))) IF(false, s(0), 0, cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(s(y0)), 0, cons(s(0), y1)) -> c6(DEL(s(s(y0)), cons(s(0), y1))) IF(false, s(0), 0, cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) DEL(s(z0), cons(0, cons(0, y1))) -> c4(IF(false, s(z0), 0, cons(0, y1))) DEL(s(s(y0)), cons(0, cons(s(s(y1)), y2))) -> c4(IF(false, s(s(y0)), 0, cons(s(s(y1)), y2))) DEL(s(0), cons(0, cons(s(s(y0)), y1))) -> c4(IF(false, s(0), 0, cons(s(s(y0)), y1))) DEL(s(s(y0)), cons(0, cons(s(0), y1))) -> c4(IF(false, s(s(y0)), 0, cons(s(0), y1))) DEL(s(0), cons(0, cons(s(0), y0))) -> c4(IF(false, s(0), 0, cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(s(s(y0)), y1)) -> c6(DEL(s(0), cons(s(s(y0)), y1))) IF(false, s(0), s(s(z0)), cons(s(0), y0)) -> c6(DEL(s(0), cons(s(0), y0))) IF(false, s(0), s(s(z0)), cons(0, cons(0, y1))) -> c6(DEL(s(0), cons(0, cons(0, y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(s(y0)), y1))) -> c6(DEL(s(0), cons(0, cons(s(s(y0)), y1)))) IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y0))) -> c6(DEL(s(0), cons(0, cons(s(0), y0)))) IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(0), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(0), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) IF(false, s(s(z0)), s(s(z1)), cons(s(s(y1)), y2)) -> c6(DEL(s(s(z0)), cons(s(s(y1)), y2))) IF(false, s(s(z0)), s(s(z1)), cons(s(0), y1)) -> c6(DEL(s(s(z0)), cons(s(0), y1))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(0, y1))) -> c6(DEL(s(s(z0)), cons(0, cons(0, y1)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(s(y1)), y2))) -> c6(DEL(s(s(z0)), cons(0, cons(s(s(y1)), y2)))) IF(false, s(s(z0)), s(s(z1)), cons(0, cons(s(0), y1))) -> c6(DEL(s(s(z0)), cons(0, cons(s(0), y1)))) LAST(cons(z0, cons(z1, cons(z2, cons(y2, y3))))) -> c2(LAST(cons(z1, cons(z2, cons(y2, y3))))) EQ(s(s(s(y0))), s(s(s(y1)))) -> c10(EQ(s(s(y0)), s(s(y1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(s(y3)), y4))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(s(y3)), y4)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(s(0), y3))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(s(0), y3)), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(0, y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(0, y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(s(y3)), y4)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(s(y3)), y4))), EQ(s(s(z0)), s(s(z1)))) DEL(s(s(z0)), cons(s(s(z1)), cons(0, cons(s(0), y3)))) -> c4(IF(eq(z0, z1), s(s(z0)), s(s(z1)), cons(0, cons(s(0), y3))), EQ(s(s(z0)), s(s(z1)))) DEL(s(0), cons(s(s(z0)), cons(s(s(y1)), y2))) -> c4(IF(false, s(0), s(s(z0)), cons(s(s(y1)), y2))) DEL(s(0), cons(s(s(z0)), cons(s(0), y1))) -> c4(IF(false, s(0), s(s(z0)), cons(s(0), y1))) DEL(s(0), cons(s(s(z0)), cons(0, cons(0, y1)))) -> c4(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)))) -> c4(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)))) -> c4(IF(false, s(0), s(s(z0)), cons(0, cons(s(0), y1)))) DEL(s(s(z0)), cons(s(0), cons(s(s(y1)), y2))) -> c4(IF(false, s(s(z0)), s(0), cons(s(s(y1)), y2))) DEL(s(s(z0)), cons(s(0), cons(s(0), y1))) -> c4(IF(false, s(s(z0)), s(0), cons(s(0), y1))) DEL(s(s(z0)), cons(s(0), cons(0, cons(0, y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(0, y1)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(s(y1)), y2)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(s(y1)), y2)))) DEL(s(s(z0)), cons(s(0), cons(0, cons(s(0), y1)))) -> c4(IF(false, s(s(z0)), s(0), cons(0, cons(s(0), y1)))) DEL(s(s(s(z0))), cons(s(s(s(z1))), z2)) -> c4(IF(eq(z0, z1), s(s(s(z0))), s(s(s(z1))), z2), EQ(s(s(s(z0))), s(s(s(z1))))) DEL(s(s(s(y0))), cons(s(s(s(y1))), z2)) -> c4(EQ(s(s(s(y0))), s(s(s(y1))))) IF(false, 0, s(z0), cons(s(z1), cons(s(y1), y2))) -> c6(DEL(0, cons(s(z1), cons(s(y1), y2)))) K tuples: REVERSE(cons(x0, cons(x1, x2))) -> c(DEL(last(cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) REVERSE(cons(x0, nil)) -> c13(DEL(last(cons(x0, nil)), cons(x0, nil))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c12(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) REVERSE(cons(z0, cons(z1, cons(z2, cons(y3, y4))))) -> c(LAST(cons(z0, cons(z1, cons(z2, cons(y3, y4)))))) Defined Rule Symbols: eq_2, del_2, last_1, if_4 Defined Pair Symbols: REVERSE_1, DEL_2, IF_4, LAST_1, EQ_2 Compound Symbols: c13_3, c_1, c13_2, c4, c4_1, c6_1, c2_1, c13_1, c10_1, c4_2, c12_1