KILLED proof of input_LOCxx5HRyE.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) InliningProof [UPPER BOUND(ID), 139 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 56 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 121 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 55 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 41 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 89 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 1065 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 414 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 181 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 85 ms] (56) CpxRNTS (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 1969 ms] (60) CpxRNTS (61) IntTrsBoundProof [UPPER BOUND(ID), 3888 ms] (62) CpxRNTS (63) CompletionProof [UPPER BOUND(ID), 0 ms] (64) CpxTypedWeightedCompleteTrs (65) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (66) CpxRNTS (67) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 87 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 91 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 380 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 390 ms] (162) CdtProblem (163) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 396 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 316 ms] (178) CdtProblem (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtLeafRemovalProof [ComplexityIfPolyImplication, 3 ms] (198) CdtProblem (199) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 455 ms] (200) CdtProblem (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (208) CdtProblem (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (210) 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0') -> true zero(s(x)) -> false p(0') -> 0' p(s(0')) -> 0' p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0', nil) if1(false, x, y) -> cons(0', int(s(0'), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) 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: empty(nil) -> true empty(cons(x, y)) -> false tail(nil) -> nil tail(cons(x, y)) -> y head(cons(x, y)) -> x zero(0) -> true zero(s(x)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(x))) -> s(p(s(x))) intlist(x) -> if_intlist(empty(x), x) if_intlist(true, x) -> nil if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) int(x, y) -> if_int(zero(x), zero(y), x, y) if_int(true, b, x, y) -> if1(b, x, y) if_int(false, b, x, y) -> if2(b, x, y) if1(true, x, y) -> cons(0, nil) if1(false, x, y) -> cons(0, int(s(0), y)) if2(true, x, y) -> nil if2(false, x, y) -> intlist(int(p(x), p(y))) 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: empty(nil) -> true [1] empty(cons(x, y)) -> false [1] tail(nil) -> nil [1] tail(cons(x, y)) -> y [1] head(cons(x, y)) -> x [1] zero(0) -> true [1] zero(s(x)) -> false [1] p(0) -> 0 [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] intlist(x) -> if_intlist(empty(x), x) [1] if_intlist(true, x) -> nil [1] if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1] int(x, y) -> if_int(zero(x), zero(y), x, y) [1] if_int(true, b, x, y) -> if1(b, x, y) [1] if_int(false, b, x, y) -> if2(b, x, y) [1] if1(true, x, y) -> cons(0, nil) [1] if1(false, x, y) -> cons(0, int(s(0), y)) [1] if2(true, x, y) -> nil [1] if2(false, x, y) -> intlist(int(p(x), p(y))) [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: empty(nil) -> true [1] empty(cons(x, y)) -> false [1] tail(nil) -> nil [1] tail(cons(x, y)) -> y [1] head(cons(x, y)) -> x [1] zero(0) -> true [1] zero(s(x)) -> false [1] p(0) -> 0 [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] intlist(x) -> if_intlist(empty(x), x) [1] if_intlist(true, x) -> nil [1] if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1] int(x, y) -> if_int(zero(x), zero(y), x, y) [1] if_int(true, b, x, y) -> if1(b, x, y) [1] if_int(false, b, x, y) -> if2(b, x, y) [1] if1(true, x, y) -> cons(0, nil) [1] if1(false, x, y) -> cons(0, int(s(0), y)) [1] if2(true, x, y) -> nil [1] if2(false, x, y) -> intlist(int(p(x), p(y))) [1] The TRS has the following type information: empty :: nil:cons -> true:false nil :: nil:cons true :: true:false cons :: 0:s -> nil:cons -> nil:cons false :: true:false tail :: nil:cons -> nil:cons head :: nil:cons -> 0:s zero :: 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s intlist :: nil:cons -> nil:cons if_intlist :: true:false -> nil:cons -> nil:cons int :: 0:s -> 0:s -> nil:cons if_int :: true:false -> true:false -> 0:s -> 0:s -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons if2 :: true:false -> 0:s -> 0:s -> nil:cons Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: tail_1 zero_1 int_2 p_1 empty_1 if_int_4 if2_3 if1_3 intlist_1 if_intlist_2 head_1 Due to the following rules being added: head(v0) -> 0 [0] 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: empty(nil) -> true [1] empty(cons(x, y)) -> false [1] tail(nil) -> nil [1] tail(cons(x, y)) -> y [1] head(cons(x, y)) -> x [1] zero(0) -> true [1] zero(s(x)) -> false [1] p(0) -> 0 [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] intlist(x) -> if_intlist(empty(x), x) [1] if_intlist(true, x) -> nil [1] if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1] int(x, y) -> if_int(zero(x), zero(y), x, y) [1] if_int(true, b, x, y) -> if1(b, x, y) [1] if_int(false, b, x, y) -> if2(b, x, y) [1] if1(true, x, y) -> cons(0, nil) [1] if1(false, x, y) -> cons(0, int(s(0), y)) [1] if2(true, x, y) -> nil [1] if2(false, x, y) -> intlist(int(p(x), p(y))) [1] head(v0) -> 0 [0] The TRS has the following type information: empty :: nil:cons -> true:false nil :: nil:cons true :: true:false cons :: 0:s -> nil:cons -> nil:cons false :: true:false tail :: nil:cons -> nil:cons head :: nil:cons -> 0:s zero :: 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s intlist :: nil:cons -> nil:cons if_intlist :: true:false -> nil:cons -> nil:cons int :: 0:s -> 0:s -> nil:cons if_int :: true:false -> true:false -> 0:s -> 0:s -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons if2 :: true:false -> 0:s -> 0:s -> 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: empty(nil) -> true [1] empty(cons(x, y)) -> false [1] tail(nil) -> nil [1] tail(cons(x, y)) -> y [1] head(cons(x, y)) -> x [1] zero(0) -> true [1] zero(s(x)) -> false [1] p(0) -> 0 [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] intlist(nil) -> if_intlist(true, nil) [2] intlist(cons(x', y')) -> if_intlist(false, cons(x', y')) [2] if_intlist(true, x) -> nil [1] if_intlist(false, nil) -> cons(s(head(nil)), intlist(nil)) [2] if_intlist(false, cons(x'', y'')) -> cons(s(head(cons(x'', y''))), intlist(y'')) [2] int(0, 0) -> if_int(true, true, 0, 0) [3] int(0, s(x2)) -> if_int(true, false, 0, s(x2)) [3] int(s(x1), 0) -> if_int(false, true, s(x1), 0) [3] int(s(x1), s(x3)) -> if_int(false, false, s(x1), s(x3)) [3] if_int(true, b, x, y) -> if1(b, x, y) [1] if_int(false, b, x, y) -> if2(b, x, y) [1] if1(true, x, y) -> cons(0, nil) [1] if1(false, x, y) -> cons(0, int(s(0), y)) [1] if2(true, x, y) -> nil [1] if2(false, 0, 0) -> intlist(int(0, 0)) [3] if2(false, 0, s(0)) -> intlist(int(0, 0)) [3] if2(false, 0, s(s(x5))) -> intlist(int(0, s(p(s(x5))))) [3] if2(false, s(0), 0) -> intlist(int(0, 0)) [3] if2(false, s(0), s(0)) -> intlist(int(0, 0)) [3] if2(false, s(0), s(s(x6))) -> intlist(int(0, s(p(s(x6))))) [3] if2(false, s(s(x4)), 0) -> intlist(int(s(p(s(x4))), 0)) [3] if2(false, s(s(x4)), s(0)) -> intlist(int(s(p(s(x4))), 0)) [3] if2(false, s(s(x4)), s(s(x7))) -> intlist(int(s(p(s(x4))), s(p(s(x7))))) [3] head(v0) -> 0 [0] The TRS has the following type information: empty :: nil:cons -> true:false nil :: nil:cons true :: true:false cons :: 0:s -> nil:cons -> nil:cons false :: true:false tail :: nil:cons -> nil:cons head :: nil:cons -> 0:s zero :: 0:s -> true:false 0 :: 0:s s :: 0:s -> 0:s p :: 0:s -> 0:s intlist :: nil:cons -> nil:cons if_intlist :: true:false -> nil:cons -> nil:cons int :: 0:s -> 0:s -> nil:cons if_int :: true:false -> true:false -> 0:s -> 0:s -> nil:cons if1 :: true:false -> 0:s -> 0:s -> nil:cons if2 :: true:false -> 0:s -> 0:s -> 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 true => 1 false => 0 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x5))) :|: x5 >= 0, z'' = 1 + (1 + x5), z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x6))) :|: z'' = 1 + (1 + x6), x6 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z'' = 0, z' = 1 + (1 + x4), x4 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z' = 1 + (1 + x4), x4 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 1 + p(1 + x7))) :|: z' = 1 + (1 + x4), x4 >= 0, z'' = 1 + (1 + x7), x7 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x if_intlist(z, z') -{ 1 }-> 0 :|: z' = x, z = 1, x >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + head(0)) + intlist(0) :|: z = 0, z' = 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + head(1 + x'' + y'')) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + x2) :|: z' = 1 + x2, z = 0, x2 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + x1, 0) :|: x1 >= 0, z = 1 + x1, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + x1, 1 + x3) :|: x1 >= 0, z' = 1 + x3, z = 1 + x1, x3 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, y) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x5))) :|: x5 >= 0, z'' = 1 + (1 + x5), z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + x6))) :|: z'' = 1 + (1 + x6), x6 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z'' = 0, z' = 1 + (1 + x4), x4 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 0)) :|: z' = 1 + (1 + x4), x4 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + x4), 1 + p(1 + x7))) :|: z' = 1 + (1 + x4), x4 >= 0, z'' = 1 + (1 + x7), x7 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x if_intlist(z, z') -{ 1 }-> 0 :|: z' = x, z = 1, x >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + x2) :|: z' = 1 + x2, z = 0, x2 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + x1, 0) :|: x1 >= 0, z = 1 + x1, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + x1, 1 + x3) :|: x1 >= 0, z' = 1 + x3, z = 1 + x1, x3 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: x >= 0, z = 1 + x ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { empty } { zero } { tail } { head } { if_intlist, intlist } { p } { if_int, int, if2, if1 } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {empty}, {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {empty}, {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: empty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {empty}, {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: ?, size: O(1) [1] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: empty after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: zero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {zero}, {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: zero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: tail after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {tail}, {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: ?, size: O(n^1) [z] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: tail after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: head after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {head}, {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: ?, size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: head after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if_intlist after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 2*z' + z'^2 Computed SIZE bound using KoAT for: intlist after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 11 + 8*z + 4*z^2 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_intlist,intlist}, {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: ?, size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: ?, size: O(n^2) [11 + 8*z + 4*z^2] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if_intlist after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 11 + 5*z' Computed RUNTIME bound using CoFloCo for: intlist after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 13 + 5*z ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 3 }-> 1 + (1 + x) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(y'') :|: z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 if_intlist(z, z') -{ 2 }-> 1 + (1 + 0) + intlist(0) :|: z = 0, z' = 0, v0 >= 0, 0 = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 2 }-> if_intlist(1, 0) :|: z = 0 intlist(z) -{ 2 }-> if_intlist(0, 1 + x' + y') :|: x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0 if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0 intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0 if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0 intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (55) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 1 + p(1 + (z'' - 2)))) :|: z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 0)) :|: z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 3 }-> intlist(int(1 + p(1 + (z' - 2)), 1 + p(1 + (z'' - 2)))) :|: z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0 if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0 intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] p: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (57) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s4)) :|: s4 >= 0, s4 <= 1 + (z'' - 2), z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s5)) :|: s5 >= 0, s5 <= 1 + (z'' - 2), z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s6, 0)) :|: s6 >= 0, s6 <= 1 + (z' - 2), z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s7, 0)) :|: s7 >= 0, s7 <= 1 + (z' - 2), z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 5 + z' + z'' }-> intlist(int(1 + s8, 1 + s9)) :|: s8 >= 0, s8 <= 1 + (z' - 2), s9 >= 0, s9 <= 1 + (z'' - 2), z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0 if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0 intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 2 + z }-> 1 + s3 :|: s3 >= 0, s3 <= 1 + (z - 2), z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] p: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s4)) :|: s4 >= 0, s4 <= 1 + (z'' - 2), z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s5)) :|: s5 >= 0, s5 <= 1 + (z'' - 2), z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s6, 0)) :|: s6 >= 0, s6 <= 1 + (z' - 2), z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s7, 0)) :|: s7 >= 0, s7 <= 1 + (z' - 2), z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 5 + z' + z'' }-> intlist(int(1 + s8, 1 + s9)) :|: s8 >= 0, s8 <= 1 + (z' - 2), s9 >= 0, s9 <= 1 + (z'' - 2), z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0 if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0 intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 2 + z }-> 1 + s3 :|: s3 >= 0, s3 <= 1 + (z - 2), z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] p: runtime: O(n^1) [2 + z], size: O(n^1) [z] if_int: runtime: ?, size: INF int: runtime: ?, size: INF if2: runtime: ?, size: INF if1: runtime: ?, size: INF ---------------------------------------- (61) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if_int after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (62) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 1 :|: z = 0 empty(z) -{ 1 }-> 0 :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: z >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, z'') :|: z' >= 0, z'' >= 0, z = 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 1, z' >= 0, z'' >= 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z = 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z = 0, z'' = 1 + 0, z' = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z'' = 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 3 }-> intlist(int(0, 0)) :|: z' = 1 + 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s4)) :|: s4 >= 0, s4 <= 1 + (z'' - 2), z'' - 2 >= 0, z = 0, z' = 0 if2(z, z', z'') -{ 4 + z'' }-> intlist(int(0, 1 + s5)) :|: s5 >= 0, s5 <= 1 + (z'' - 2), z'' - 2 >= 0, z' = 1 + 0, z = 0 if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s6, 0)) :|: s6 >= 0, s6 <= 1 + (z' - 2), z'' = 0, z' - 2 >= 0, z = 0 if2(z, z', z'') -{ 4 + z' }-> intlist(int(1 + s7, 0)) :|: s7 >= 0, s7 <= 1 + (z' - 2), z' - 2 >= 0, z = 0, z'' = 1 + 0 if2(z, z', z'') -{ 5 + z' + z'' }-> intlist(int(1 + s8, 1 + s9)) :|: s8 >= 0, s8 <= 1 + (z' - 2), s9 >= 0, s9 <= 1 + (z'' - 2), z' - 2 >= 0, z'' - 2 >= 0, z = 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0 if_int(z, z', z'', z1) -{ 1 }-> if1(z', z'', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 1, z' >= 0 if_intlist(z, z') -{ 16 + 5*y'' }-> 1 + (1 + x) + s1 :|: s1 >= 0, s1 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, 1 + x'' + y'' = 1 + x + y, x >= 0, y >= 0 if_intlist(z, z') -{ 15 }-> 1 + (1 + 0) + s'' :|: s'' >= 0, s'' <= 8 * 0 + 4 * (0 * 0) + 11, z = 0, z' = 0, v0 >= 0, 0 = v0 if_intlist(z, z') -{ 15 + 5*y'' }-> 1 + (1 + 0) + s2 :|: s2 >= 0, s2 <= 8 * y'' + 4 * (y'' * y'') + 11, z' = 1 + x'' + y'', y'' >= 0, x'' >= 0, z = 0, v0 >= 0, 1 + x'' + y'' = v0 int(z, z') -{ 3 }-> if_int(1, 1, 0, 0) :|: z = 0, z' = 0 int(z, z') -{ 3 }-> if_int(1, 0, 0, 1 + (z' - 1)) :|: z = 0, z' - 1 >= 0 int(z, z') -{ 3 }-> if_int(0, 1, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0 int(z, z') -{ 3 }-> if_int(0, 0, 1 + (z - 1), 1 + (z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 intlist(z) -{ 13 }-> s :|: s >= 0, s <= 2 * 0 + 4 + 0 * 0, z = 0 intlist(z) -{ 18 + 5*x' + 5*y' }-> s' :|: s' >= 0, s' <= 2 * (1 + x' + y') + 4 + (1 + x' + y') * (1 + x' + y'), x' >= 0, y' >= 0, z = 1 + x' + y' p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 2 + z }-> 1 + s3 :|: s3 >= 0, s3 <= 1 + (z - 2), z - 2 >= 0 tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if_int,int,if2,if1} Previous analysis results are: empty: runtime: O(1) [1], size: O(1) [1] zero: runtime: O(1) [1], size: O(1) [1] tail: runtime: O(1) [1], size: O(n^1) [z] head: runtime: O(1) [1], size: O(n^1) [z] if_intlist: runtime: O(n^1) [11 + 5*z'], size: O(n^2) [4 + 2*z' + z'^2] intlist: runtime: O(n^1) [13 + 5*z], size: O(n^2) [11 + 8*z + 4*z^2] p: runtime: O(n^1) [2 + z], size: O(n^1) [z] if_int: runtime: INF, size: INF int: runtime: ?, size: INF if2: runtime: ?, size: INF if1: runtime: ?, size: INF ---------------------------------------- (63) 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: head(v0) -> null_head [0] zero(v0) -> null_zero [0] p(v0) -> null_p [0] if_intlist(v0, v1) -> null_if_intlist [0] if_int(v0, v1, v2, v3) -> null_if_int [0] if1(v0, v1, v2) -> null_if1 [0] if2(v0, v1, v2) -> null_if2 [0] empty(v0) -> null_empty [0] tail(v0) -> null_tail [0] And the following fresh constants: null_head, null_zero, null_p, null_if_intlist, null_if_int, null_if1, null_if2, null_empty, null_tail ---------------------------------------- (64) 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: empty(nil) -> true [1] empty(cons(x, y)) -> false [1] tail(nil) -> nil [1] tail(cons(x, y)) -> y [1] head(cons(x, y)) -> x [1] zero(0) -> true [1] zero(s(x)) -> false [1] p(0) -> 0 [1] p(s(0)) -> 0 [1] p(s(s(x))) -> s(p(s(x))) [1] intlist(x) -> if_intlist(empty(x), x) [1] if_intlist(true, x) -> nil [1] if_intlist(false, x) -> cons(s(head(x)), intlist(tail(x))) [1] int(x, y) -> if_int(zero(x), zero(y), x, y) [1] if_int(true, b, x, y) -> if1(b, x, y) [1] if_int(false, b, x, y) -> if2(b, x, y) [1] if1(true, x, y) -> cons(0, nil) [1] if1(false, x, y) -> cons(0, int(s(0), y)) [1] if2(true, x, y) -> nil [1] if2(false, x, y) -> intlist(int(p(x), p(y))) [1] head(v0) -> null_head [0] zero(v0) -> null_zero [0] p(v0) -> null_p [0] if_intlist(v0, v1) -> null_if_intlist [0] if_int(v0, v1, v2, v3) -> null_if_int [0] if1(v0, v1, v2) -> null_if1 [0] if2(v0, v1, v2) -> null_if2 [0] empty(v0) -> null_empty [0] tail(v0) -> null_tail [0] The TRS has the following type information: empty :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> true:false:null_zero:null_empty nil :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail true :: true:false:null_zero:null_empty cons :: 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail false :: true:false:null_zero:null_empty tail :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail head :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> 0:s:null_head:null_p zero :: 0:s:null_head:null_p -> true:false:null_zero:null_empty 0 :: 0:s:null_head:null_p s :: 0:s:null_head:null_p -> 0:s:null_head:null_p p :: 0:s:null_head:null_p -> 0:s:null_head:null_p intlist :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail if_intlist :: true:false:null_zero:null_empty -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail int :: 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail if_int :: true:false:null_zero:null_empty -> true:false:null_zero:null_empty -> 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail if1 :: true:false:null_zero:null_empty -> 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail if2 :: true:false:null_zero:null_empty -> 0:s:null_head:null_p -> 0:s:null_head:null_p -> nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail null_head :: 0:s:null_head:null_p null_zero :: true:false:null_zero:null_empty null_p :: 0:s:null_head:null_p null_if_intlist :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail null_if_int :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail null_if1 :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail null_if2 :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail null_empty :: true:false:null_zero:null_empty null_tail :: nil:cons:null_if_intlist:null_if_int:null_if1:null_if2:null_tail Rewrite Strategy: INNERMOST ---------------------------------------- (65) 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 true => 2 false => 1 0 => 0 null_head => 0 null_zero => 0 null_p => 0 null_if_intlist => 0 null_if_int => 0 null_if1 => 0 null_if2 => 0 null_empty => 0 null_tail => 0 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: empty(z) -{ 1 }-> 2 :|: z = 0 empty(z) -{ 1 }-> 1 :|: z = 1 + x + y, x >= 0, y >= 0 empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 head(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0 head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 if1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + int(1 + 0, y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if1(z, z', z'') -{ 1 }-> 1 + 0 + 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 if2(z, z', z'') -{ 1 }-> intlist(int(p(x), p(y))) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 if2(z, z', z'') -{ 1 }-> 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 if2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if_int(z, z', z'', z1) -{ 1 }-> if2(b, x, y) :|: b >= 0, z1 = y, z = 1, x >= 0, y >= 0, z' = b, z'' = x if_int(z, z', z'', z1) -{ 1 }-> if1(b, x, y) :|: z = 2, b >= 0, z1 = y, x >= 0, y >= 0, z' = b, z'' = x if_int(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if_intlist(z, z') -{ 1 }-> 0 :|: z = 2, z' = x, x >= 0 if_intlist(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 if_intlist(z, z') -{ 1 }-> 1 + (1 + head(x)) + intlist(tail(x)) :|: z' = x, z = 1, x >= 0 int(z, z') -{ 1 }-> if_int(zero(x), zero(y), x, y) :|: x >= 0, y >= 0, z = x, z' = y intlist(z) -{ 1 }-> if_intlist(empty(x), x) :|: x >= 0, z = x p(z) -{ 1 }-> 0 :|: z = 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) tail(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0 tail(z) -{ 1 }-> 0 :|: z = 0 tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 zero(z) -{ 1 }-> 2 :|: z = 0 zero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (67) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: EMPTY(nil) -> c EMPTY(cons(z0, z1)) -> c1 TAIL(nil) -> c2 TAIL(cons(z0, z1)) -> c3 HEAD(cons(z0, z1)) -> c4 ZERO(0) -> c5 ZERO(s(z0)) -> c6 P(0) -> c7 P(s(0)) -> c8 P(s(s(z0))) -> c9(P(s(z0))) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0)) IF_INTLIST(true, z0) -> c11 IF_INTLIST(false, z0) -> c12(HEAD(z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1)) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(true, z0, z1) -> c18 IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(true, z0, z1) -> c20 IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) S tuples: EMPTY(nil) -> c EMPTY(cons(z0, z1)) -> c1 TAIL(nil) -> c2 TAIL(cons(z0, z1)) -> c3 HEAD(cons(z0, z1)) -> c4 ZERO(0) -> c5 ZERO(s(z0)) -> c6 P(0) -> c7 P(s(0)) -> c8 P(s(s(z0))) -> c9(P(s(z0))) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0)) IF_INTLIST(true, z0) -> c11 IF_INTLIST(false, z0) -> c12(HEAD(z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1)) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(true, z0, z1) -> c18 IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(true, z0, z1) -> c20 IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: EMPTY_1, TAIL_1, HEAD_1, ZERO_1, P_1, INTLIST_1, IF_INTLIST_2, INT_2, IF_INT_4, IF1_3, IF2_3 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8, c9_1, c10_2, c11, c12_1, c13_2, c14_2, c15_2, c16_1, c17_1, c18, c19_1, c20, c21_3, c22_3 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 13 trailing nodes: IF2(true, z0, z1) -> c20 IF_INTLIST(true, z0) -> c11 P(0) -> c7 HEAD(cons(z0, z1)) -> c4 ZERO(0) -> c5 TAIL(cons(z0, z1)) -> c3 IF1(true, z0, z1) -> c18 EMPTY(cons(z0, z1)) -> c1 TAIL(nil) -> c2 P(s(0)) -> c8 ZERO(s(z0)) -> c6 IF_INTLIST(false, z0) -> c12(HEAD(z0)) EMPTY(nil) -> c ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1)) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0), EMPTY(z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0)), TAIL(z0)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z0)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1), ZERO(z1)) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, INTLIST_1, IF_INTLIST_2, INT_2, IF_INT_4, IF1_3, IF2_3 Compound Symbols: c9_1, c10_2, c13_2, c14_2, c15_2, c16_1, c17_1, c19_1, c21_3, c22_3 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c22_3, c10_1, c13_1, c14_1, c15_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0, z1) -> c21(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z0)) by IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c10_1, c13_1, c14_1, c15_1, c21_3 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, z0, z1) -> c22(INTLIST(int(p(z0), p(z1))), INT(p(z0), p(z1)), P(z1)) by IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INTLIST(z0) -> c10(IF_INTLIST(empty(z0), z0)) by INTLIST(nil) -> c10(IF_INTLIST(true, nil)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(nil) -> c10(IF_INTLIST(true, nil)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(nil) -> c10(IF_INTLIST(true, nil)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF_INTLIST_2, INT_2, IF2_3, INTLIST_1 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: INTLIST(nil) -> c10(IF_INTLIST(true, nil)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF_INTLIST_2, INT_2, IF2_3, INTLIST_1 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c13_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_INTLIST(false, z0) -> c13(INTLIST(tail(z0))) by IF_INTLIST(false, nil) -> c13(INTLIST(nil)) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, nil) -> c13(INTLIST(nil)) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, nil) -> c13(INTLIST(nil)) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF_INTLIST(false, nil) -> c13(INTLIST(nil)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c14_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(z0, z1) -> c14(IF_INT(zero(z0), zero(z1), z0, z1)) by INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) K tuples:none Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) We considered the (Usable) Rules: zero(0) -> true zero(s(z0)) -> false And the Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF1(x_1, x_2, x_3)) = x_1 POL(IF2(x_1, x_2, x_3)) = x_1 POL(IF_INT(x_1, x_2, x_3, x_4)) = x_2 POL(IF_INTLIST(x_1, x_2)) = 0 POL(INT(x_1, x_2)) = [1] POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 POL(empty(x_1)) = [1] + x_1 POL(false) = [1] POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = [1] ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INT_2, IF2_3, INTLIST_1, IF_INTLIST_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c15_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1 ---------------------------------------- (93) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(z0, z1) -> c15(IF_INT(zero(z0), zero(z1), z0, z1)) by INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1 ---------------------------------------- (95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) We considered the (Usable) Rules: zero(0) -> true zero(s(z0)) -> false And the Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF1(x_1, x_2, x_3)) = x_1 POL(IF2(x_1, x_2, x_3)) = x_1 POL(IF_INT(x_1, x_2, x_3, x_4)) = x_2 POL(IF_INTLIST(x_1, x_2)) = 0 POL(INT(x_1, x_2)) = [1] POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 POL(empty(x_1)) = [1] + x_1 POL(false) = [1] POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = [1] ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, x1) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x0)) by IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(0)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(s(0))) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, 0) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(0)), P(x0)) by IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0)) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(s(0))) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0)) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(s(0))) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0)) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(s(0))) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, s(0)) -> c21(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) by IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(0)) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0))) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(0)) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0))) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(0)) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0))) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, s(s(z0))) -> c21(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) by IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(0)) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(0))) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(0)) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(0))) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(0)) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(0))) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_3, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, s(s(z0)), x1) -> c21(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) by IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(x0)), s(0)) -> c21(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(s(x0)))) IF2(false, s(s(x0)), s(s(z0))) -> c21(INTLIST(int(s(p(s(x0))), s(p(s(z0))))), INT(p(s(s(x0))), p(s(s(z0)))), P(s(s(x0)))) IF2(false, s(s(x0)), 0) -> c21(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, 0, x1) -> c21(INTLIST(int(0, p(x1))), INT(p(0), p(x1))) by IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(z0))) -> c21(INTLIST(int(0, s(p(s(z0))))), INT(p(0), p(s(s(z0))))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c21_2, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, s(0), x1) -> c21(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1))) by IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(0), s(s(z0))) -> c21(INTLIST(int(0, s(p(s(z0))))), INT(p(s(0)), p(s(s(z0))))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, x1) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(x1)), p(x0), p(x1))), INT(p(x0), p(x1)), P(x1)) by IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(0)) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(s(0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, s(s(z0))) -> c22(INTLIST(int(p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) by IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, 0, x1) -> c22(INTLIST(int(0, p(x1))), INT(p(0), p(x1)), P(x1)) by IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0)) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(s(0))) IF2(false, 0, s(s(z0))) -> c22(INTLIST(int(0, s(p(s(z0))))), INT(p(0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0)) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(s(0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0)), P(0)) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0))), P(s(0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, s(0), x1) -> c22(INTLIST(int(0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) by IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(0)) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0))) IF2(false, s(0), s(s(z0))) -> c22(INTLIST(int(0, s(p(s(z0))))), INT(p(s(0)), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(0)) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0))) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0)), P(0)) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0))), P(s(0))) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_3, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_1 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, s(s(z0)), x1) -> c22(INTLIST(int(s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) by IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(0))) IF2(false, s(s(x0)), s(s(z0))) -> c22(INTLIST(int(s(p(s(x0))), s(p(s(z0))))), INT(p(s(s(x0))), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(0)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(0))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(0)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0))), P(s(0))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0)), P(0)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_1 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, 0) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(0))) by IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(s(z0)), 0) -> c22(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, IF2_3, INTLIST_1, IF_INTLIST_2, INT_2 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c22_2, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(false, x0, s(0)) -> c22(INTLIST(int(p(x0), 0)), INT(p(x0), p(s(0)))) by IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(z0)), s(0)) -> c22(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) by INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(x0, s(z0)) -> c14(IF_INT(zero(x0), false, x0, s(z0))) by INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) by INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace INT(x0, s(z0)) -> c15(IF_INT(zero(x0), false, x0, s(z0))) by INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0)), P(x0)) by IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) by IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (153) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) We considered the (Usable) Rules: p(0) -> 0 p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 And the Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(IF1(x_1, x_2, x_3)) = x_3 POL(IF2(x_1, x_2, x_3)) = x_3 POL(IF_INT(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF_INTLIST(x_1, x_2)) = x_1 POL(INT(x_1, x_2)) = x_2 POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1)) = x_1 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 POL(empty(x_1)) = x_1 POL(false) = 0 POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 POL(tail(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) by IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c16_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1 ---------------------------------------- (157) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_INT(true, z0, z1, z2) -> c16(IF1(z0, z1, z2)) by IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, true, 0, 0) -> c16(IF1(true, 0, 0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, true, 0, 0) -> c16(IF1(true, 0, 0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, true, 0, 0) -> c16(IF1(true, 0, 0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1 ---------------------------------------- (159) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16 ---------------------------------------- (161) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_INT(true, true, 0, 0) -> c16 We considered the (Usable) Rules:none And the Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF1(x_1, x_2, x_3)) = [1] + x_2 POL(IF2(x_1, x_2, x_3)) = [1] POL(IF_INT(x_1, x_2, x_3, x_4)) = [1] + x_1 POL(IF_INTLIST(x_1, x_2)) = x_1 POL(INT(x_1, x_2)) = [1] POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16) = 0 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1)) = x_1 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 POL(empty(x_1)) = x_1 POL(false) = 0 POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF_INT(true, true, 0, 0) -> c16 Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16 ---------------------------------------- (163) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) We considered the (Usable) Rules:none And the Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(IF1(x_1, x_2, x_3)) = [1] POL(IF2(x_1, x_2, x_3)) = [1] + x_2 POL(IF_INT(x_1, x_2, x_3, x_4)) = x_1 + x_3 POL(IF_INTLIST(x_1, x_2)) = 0 POL(INT(x_1, x_2)) = [1] POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16) = 0 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1)) = x_1 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 POL(empty(x_1)) = x_1 POL(false) = [1] POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF_INT(true, true, 0, 0) -> c16 IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) by IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF_INT(true, true, 0, 0) -> c16 IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) K tuples: INT(x0, 0) -> c14(IF_INT(zero(x0), true, x0, 0)) INT(x0, 0) -> c15(IF_INT(zero(x0), true, x0, 0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF_INT(true, true, 0, 0) -> c16 IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c16 ---------------------------------------- (167) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF_INT(true, true, 0, 0) -> c16 ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0)), P(x0)) by IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) S tuples: P(s(s(z0))) -> c9(P(s(z0))) IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: P_1, IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3 Compound Symbols: c9_1, c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1 ---------------------------------------- (171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(z0))) -> c9(P(s(z0))) by P(s(s(s(y0)))) -> c9(P(s(s(y0)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(s(s(0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(s(s(0)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0))), P(x0)) by IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (177) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) We considered the (Usable) Rules: p(0) -> 0 p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 And the Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF1(x_1, x_2, x_3)) = x_2 + x_3 POL(IF2(x_1, x_2, x_3)) = x_3 POL(IF_INT(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(IF_INTLIST(x_1, x_2)) = x_1 POL(INT(x_1, x_2)) = x_2 POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1)) = x_1 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = [1] + x_1 POL(empty(x_1)) = x_1 POL(false) = 0 POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = x_1 POL(s(x_1)) = [1] POL(tail(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, s(s(z0))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(x0)) by IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (181) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, s(s(z0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(s(s(z0)))) by IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) by IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (185) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1))) by IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) by IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_2, c21_3, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (189) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, s(0), x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1))) by IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) S tuples: IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF_INT_4, IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, P_1 Compound Symbols: c17_1, c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1 ---------------------------------------- (191) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_INT(false, z0, z1, z2) -> c17(IF2(z0, z1, z2)) by IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, true, s(x0), 0) -> c17(IF2(true, s(x0), 0)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, true, s(x0), 0) -> c17(IF2(true, s(x0), 0)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, true, s(x0), 0) -> c17(IF2(true, s(x0), 0)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1 ---------------------------------------- (193) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(p(0)), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, 0, z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(0), p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, 0) -> c21(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c21(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1)), P(x1)) IF2(false, 0, s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(0), p(x0)), P(x0)) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, 0, 0) -> c22(INTLIST(int(0, 0)), INT(p(0), p(0))) IF2(false, 0, s(0)) -> c22(INTLIST(int(0, 0)), INT(p(0), p(s(0)))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17 ---------------------------------------- (195) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(P(x1)) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, s(s(x1))) -> c(P(s(s(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1))), INT(p(0), p(x1))) IF2(false, 0, x1) -> c21(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1))), INT(p(0), p(x1))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(P(x1)) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, s(s(x1))) -> c(P(s(s(x1)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1 ---------------------------------------- (197) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: IF2(false, 0, x1) -> c(P(x1)) IF2(false, 0, s(s(x1))) -> c(P(s(s(x1)))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1 ---------------------------------------- (199) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_INT(false, true, s(x0), 0) -> c17 We considered the (Usable) Rules: zero(0) -> true zero(s(z0)) -> false And the Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF1(x_1, x_2, x_3)) = x_1 + x_2 POL(IF2(x_1, x_2, x_3)) = x_1 POL(IF_INT(x_1, x_2, x_3, x_4)) = x_2 POL(IF_INTLIST(x_1, x_2)) = 0 POL(INT(x_1, x_2)) = [1] POL(INTLIST(x_1)) = 0 POL(P(x_1)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17) = 0 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c21(x_1)) = x_1 POL(c21(x_1, x_2)) = x_1 + x_2 POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c22(x_1)) = x_1 POL(c22(x_1, x_2)) = x_1 + x_2 POL(c22(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c9(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 POL(empty(x_1)) = [1] + x_1 POL(false) = [1] POL(head(x_1)) = [1] + x_1 POL(if1(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if2(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(if_int(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(if_intlist(x_1, x_2)) = [1] + x_1 + x_2 POL(int(x_1, x_2)) = [1] POL(intlist(x_1)) = [1] POL(nil) = [1] POL(p(x_1)) = 0 POL(s(x_1)) = 0 POL(tail(x_1)) = [1] + x_1 POL(true) = [1] POL(zero(x_1)) = [1] ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF_INT(false, true, s(x0), 0) -> c17 Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1 ---------------------------------------- (201) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, 0) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0)), P(x0)) by IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF_INT(false, true, s(x0), 0) -> c17 IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF_INT(false, true, s(x0), 0) -> c17 Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c17, c_1 ---------------------------------------- (203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF_INT(false, true, s(x0), 0) -> c17 ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1 ---------------------------------------- (205) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) by IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_1, c21_2, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1 ---------------------------------------- (207) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, x0, 0) -> c21(INT(p(x0), p(0))) by IF2(false, z0, 0) -> c21(INT(p(z0), 0)) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INT(p(z0), 0)) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INT(p(z0), 0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1 ---------------------------------------- (209) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) by IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), 0)) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: empty(nil) -> true empty(cons(z0, z1)) -> false tail(nil) -> nil tail(cons(z0, z1)) -> z1 head(cons(z0, z1)) -> z0 zero(0) -> true zero(s(z0)) -> false p(0) -> 0 p(s(0)) -> 0 p(s(s(z0))) -> s(p(s(z0))) intlist(z0) -> if_intlist(empty(z0), z0) if_intlist(true, z0) -> nil if_intlist(false, z0) -> cons(s(head(z0)), intlist(tail(z0))) int(z0, z1) -> if_int(zero(z0), zero(z1), z0, z1) if_int(true, z0, z1, z2) -> if1(z0, z1, z2) if_int(false, z0, z1, z2) -> if2(z0, z1, z2) if1(true, z0, z1) -> cons(0, nil) if1(false, z0, z1) -> cons(0, int(s(0), z1)) if2(true, z0, z1) -> nil if2(false, z0, z1) -> intlist(int(p(z0), p(z1))) Tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(0)), P(s(s(z0)))) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, 0) -> c14(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c14(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, 0) -> c15(IF_INT(true, true, 0, 0)) INT(s(z0), 0) -> c15(IF_INT(false, true, s(z0), 0)) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) IF2(false, 0, z0) -> c(INT(0, p(z0))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INT(p(z0), 0)) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), 0)) S tuples: IF1(false, z0, z1) -> c19(INT(s(0), z1)) INTLIST(cons(z0, z1)) -> c10(IF_INTLIST(false, cons(z0, z1))) IF_INTLIST(false, cons(z0, z1)) -> c13(INTLIST(z1)) INT(0, x1) -> c14(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c14(IF_INT(false, zero(x1), s(z0), x1)) INT(0, x1) -> c15(IF_INT(true, zero(x1), 0, x1)) INT(s(z0), x1) -> c15(IF_INT(false, zero(x1), s(z0), x1)) IF2(false, x0, s(0)) -> c21(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0))), P(x0)) IF2(false, s(s(z0)), s(0)) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), p(s(0))), P(s(s(z0)))) IF2(false, x0, s(0)) -> c21(INT(p(x0), p(s(0)))) IF2(false, s(0), s(0)) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, x0, s(s(x1))) -> c21(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(x0)) IF2(false, x0, s(s(0))) -> c21(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0)))), P(x0)) IF2(false, x0, s(s(s(z0)))) -> c21(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(x0)) IF2(false, s(s(z0)), s(s(x1))) -> c21(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(z0)))) IF2(false, x0, s(s(x1))) -> c21(INT(p(x0), p(s(s(x1))))) IF2(false, s(0), s(s(x1))) -> c21(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1))))) IF2(false, s(s(x0)), x1) -> c21(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(s(s(z0))), x1) -> c21(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(s(s(s(z0))))) IF2(false, s(s(x0)), x1) -> c21(INT(p(s(s(x0))), p(x1)), P(s(s(x0)))) IF2(false, s(0), x0) -> c21(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0))) IF2(false, s(0), x0) -> c21(INT(p(s(0)), p(x0))) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(s(z0)))), p(x0), s(p(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(p(s(0))), zero(p(x1)), 0, p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(x1)), s(p(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, s(s(z0))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(z0)))), p(x0), p(s(s(z0))))), INT(p(x0), p(s(s(z0)))), P(s(s(z0)))) IF2(false, s(0), x1) -> c22(INTLIST(if_int(zero(0), zero(p(x1)), p(s(0)), p(x1))), INT(p(s(0)), p(x1)), P(x1)) IF2(false, s(s(z0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(x1)), p(s(s(z0))), p(x1))), INT(p(s(s(z0))), p(x1)), P(x1)) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(0)), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(p(s(0))), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(0))), INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), p(s(0)))), INT(p(x0), p(s(0)))) IF2(false, x0, s(s(x1))) -> c22(INTLIST(if_int(zero(p(x0)), zero(s(p(s(x1)))), p(x0), s(p(s(x1))))), INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(s(z0)))) -> c22(INTLIST(int(p(x0), s(s(p(s(z0)))))), INT(p(x0), p(s(s(s(z0))))), P(s(s(s(z0))))) IF2(false, s(0), s(s(x1))) -> c22(INTLIST(int(0, s(p(s(x1))))), INT(p(s(0)), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(s(z0)), s(s(x1))) -> c22(INTLIST(int(s(p(s(z0))), s(p(s(x1))))), INT(p(s(s(z0))), p(s(s(x1)))), P(s(s(x1)))) IF2(false, x0, s(s(x1))) -> c22(INT(p(x0), p(s(s(x1)))), P(s(s(x1)))) IF2(false, s(0), x0) -> c22(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0))), INT(p(s(0)), p(x0)), P(x0)) IF2(false, s(0), x0) -> c22(INT(p(s(0)), p(x0))) IF2(false, s(0), 0) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(0))) IF2(false, s(0), s(0)) -> c22(INTLIST(int(0, 0)), INT(p(s(0)), p(s(0)))) IF2(false, s(s(x0)), x1) -> c22(INTLIST(if_int(zero(s(p(s(x0)))), zero(p(x1)), s(p(s(x0))), p(x1))), INT(p(s(s(x0))), p(x1)), P(x1)) IF2(false, s(s(0)), x1) -> c22(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1)), P(x1)) IF2(false, s(s(s(z0))), x1) -> c22(INTLIST(int(s(s(p(s(z0)))), p(x1))), INT(p(s(s(s(z0)))), p(x1)), P(x1)) IF2(false, s(s(x0)), x1) -> c22(INT(p(s(s(x0))), p(x1))) IF2(false, s(s(x0)), s(0)) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(s(0)))) IF2(false, s(s(x0)), 0) -> c22(INTLIST(int(s(p(s(x0))), 0)), INT(p(s(s(x0))), p(0))) IF2(false, x0, 0) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(0))) IF2(false, x0, 0) -> c22(INT(p(x0), p(0))) IF2(false, x0, s(0)) -> c22(INTLIST(if_int(zero(p(x0)), zero(0), p(x0), 0)), INT(p(x0), p(s(0)))) IF2(false, x0, s(0)) -> c22(INT(p(x0), p(s(0)))) INT(0, s(x1)) -> c14(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c14(IF_INT(false, false, s(z0), s(x1))) INT(0, s(x1)) -> c15(IF_INT(true, false, 0, s(x1))) INT(s(z0), s(x1)) -> c15(IF_INT(false, false, s(z0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(0)), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(s(z1)))), p(z0), s(p(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF_INT(true, y0, 0, x0) -> c16(IF1(y0, 0, x0)) IF_INT(true, false, 0, s(x0)) -> c16(IF1(false, 0, s(x0))) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(p(s(s(z0)))), zero(p(z1)), s(p(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(0))), INT(p(z0), 0), P(z0)) P(s(s(s(y0)))) -> c9(P(s(s(y0)))) IF2(false, s(s(0)), x1) -> c21(INTLIST(int(s(0), p(x1))), INT(p(s(s(0))), p(x1))) IF2(false, x0, s(s(0))) -> c22(INTLIST(int(p(x0), s(0))), INT(p(x0), p(s(s(0))))) IF2(false, z0, s(s(z1))) -> c21(INTLIST(if_int(zero(p(z0)), zero(s(p(s(z1)))), p(z0), p(s(s(z1))))), INT(p(z0), s(p(s(z1)))), P(z0)) IF2(false, s(s(z0)), z1) -> c21(INTLIST(if_int(zero(s(p(s(z0)))), zero(p(z1)), p(s(s(z0))), p(z1))), INT(s(p(s(z0))), p(z1)), P(s(s(z0)))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(p(s(0))), zero(p(z0)), 0, p(z0))), INT(0, p(z0))) IF2(false, s(0), z0) -> c21(INTLIST(if_int(zero(0), zero(p(z0)), p(s(0)), p(z0))), INT(0, p(z0))) IF_INT(false, y0, s(x0), x1) -> c17(IF2(y0, s(x0), x1)) IF_INT(false, false, s(x0), s(x1)) -> c17(IF2(false, s(x0), s(x1))) IF2(false, z0, 0) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, s(s(z0)), 0) -> c21(INTLIST(int(s(p(s(z0))), 0)), INT(p(s(s(z0))), 0), P(s(s(z0)))) IF2(false, z0, 0) -> c21(INT(p(z0), 0)) IF2(false, s(0), 0) -> c21(INTLIST(int(0, 0)), INT(p(s(0)), 0)) K tuples: IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(p(s(0))), p(z0), 0)), INT(p(z0), 0), P(z0)) IF2(false, 0, x0) -> c21(INT(p(0), p(x0))) IF2(false, 0, x0) -> c22(INT(p(0), p(x0))) IF2(false, z0, s(0)) -> c21(INTLIST(if_int(zero(p(z0)), zero(0), p(z0), p(s(0)))), INT(p(z0), 0), P(z0)) IF2(false, 0, 0) -> c(INTLIST(int(0, 0))) IF2(false, 0, 0) -> c(INT(p(0), p(0))) IF2(false, 0, s(0)) -> c(INTLIST(int(0, 0))) IF2(false, 0, s(0)) -> c(INT(p(0), p(s(0)))) IF2(false, 0, s(s(x1))) -> c(INTLIST(int(0, s(p(s(x1)))))) IF2(false, 0, s(s(x1))) -> c(INT(p(0), p(s(s(x1))))) IF2(false, 0, x0) -> c(INTLIST(if_int(zero(0), zero(p(x0)), 0, p(x0)))) IF2(false, 0, x0) -> c(INT(p(0), p(x0))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(p(0)), zero(p(x1)), 0, p(x1)))) IF2(false, 0, x1) -> c(INTLIST(if_int(zero(0), zero(p(x1)), p(0), p(x1)))) Defined Rule Symbols: empty_1, tail_1, head_1, zero_1, p_1, intlist_1, if_intlist_2, int_2, if_int_4, if1_3, if2_3 Defined Pair Symbols: IF1_3, INTLIST_1, IF_INTLIST_2, INT_2, IF2_3, IF_INT_4, P_1 Compound Symbols: c19_1, c10_1, c13_1, c14_1, c15_1, c21_3, c21_2, c21_1, c22_3, c22_2, c22_1, c16_1, c9_1, c17_1, c_1