KILLED proof of input_gkvxaJXNQV.trs # AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 150 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 81 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 2 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 316 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 293 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 115 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 7511 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 2216 ms] (42) CpxRNTS (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 169 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 34 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 96 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 62 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms] (88) CdtProblem (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms] (96) CdtProblem (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 42 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 26 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 91 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 78 ms] (124) CdtProblem (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms] (132) CdtProblem (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 73 ms] (136) CdtProblem (137) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 74 ms] (144) CdtProblem (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 74 ms] (148) CdtProblem (149) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 50 ms] (172) CdtProblem (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms] (174) CdtProblem (175) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 83 ms] (176) CdtProblem (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 78 ms] (182) CdtProblem (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 91 ms] (186) CdtProblem (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (196) CdtProblem (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (198) 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: even(0) -> true even(s(0)) -> false even(s(s(x))) -> even(x) half(0) -> 0 half(s(s(x))) -> s(half(x)) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(0, y) -> 0 times(s(x), y) -> if_times(even(s(x)), s(x), y) if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) if_times(false, s(x), y) -> plus(y, times(x, 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: even(0') -> true even(s(0')) -> false even(s(s(x))) -> even(x) half(0') -> 0' half(s(s(x))) -> s(half(x)) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) times(0', y) -> 0' times(s(x), y) -> if_times(even(s(x)), s(x), y) if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) if_times(false, s(x), y) -> plus(y, times(x, 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: even(0) -> true even(s(0)) -> false even(s(s(x))) -> even(x) half(0) -> 0 half(s(s(x))) -> s(half(x)) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(0, y) -> 0 times(s(x), y) -> if_times(even(s(x)), s(x), y) if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) if_times(false, s(x), y) -> plus(y, times(x, 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: even(0) -> true [1] even(s(0)) -> false [1] even(s(s(x))) -> even(x) [1] half(0) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> if_times(even(s(x)), s(x), y) [1] if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) [1] if_times(false, s(x), y) -> plus(y, times(x, 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: even(0) -> true [1] even(s(0)) -> false [1] even(s(s(x))) -> even(x) [1] half(0) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> if_times(even(s(x)), s(x), y) [1] if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) [1] if_times(false, s(x), y) -> plus(y, times(x, y)) [1] The TRS has the following type information: even :: 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false half :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s if_times :: true:false -> 0:s -> 0:s -> 0:s 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: even_1 times_2 half_1 if_times_3 plus_2 Due to the following rules being added: half(v0) -> 0 [0] if_times(v0, v1, v2) -> 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: even(0) -> true [1] even(s(0)) -> false [1] even(s(s(x))) -> even(x) [1] half(0) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> if_times(even(s(x)), s(x), y) [1] if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) [1] if_times(false, s(x), y) -> plus(y, times(x, y)) [1] half(v0) -> 0 [0] if_times(v0, v1, v2) -> 0 [0] The TRS has the following type information: even :: 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false half :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s if_times :: true:false -> 0:s -> 0:s -> 0:s 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: even(0) -> true [1] even(s(0)) -> false [1] even(s(s(x))) -> even(x) [1] half(0) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(0), y) -> if_times(false, s(0), y) [2] times(s(s(x')), y) -> if_times(even(x'), s(s(x')), y) [2] if_times(true, s(s(x'')), y) -> plus(times(s(half(x'')), y), times(s(half(x'')), y)) [3] if_times(true, s(s(x'')), y) -> plus(times(s(half(x'')), y), times(0, y)) [2] if_times(true, s(s(x1)), y) -> plus(times(0, y), times(s(half(x1)), y)) [2] if_times(true, s(x), y) -> plus(times(0, y), times(0, y)) [1] if_times(false, s(0), y) -> plus(y, 0) [2] if_times(false, s(s(x2)), y) -> plus(y, if_times(even(s(x2)), s(x2), y)) [2] half(v0) -> 0 [0] if_times(v0, v1, v2) -> 0 [0] The TRS has the following type information: even :: 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false half :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s if_times :: true:false -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(x) :|: x >= 0, z = 1 + (1 + x) even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) if_times(z, z', z'') -{ 2 }-> plus(y, if_times(even(1 + x2), 1 + x2, y)) :|: z'' = y, z' = 1 + (1 + x2), y >= 0, z = 0, x2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, y), times(0, y)) :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(0, y), times(1 + half(x1), y)) :|: x1 >= 0, z' = 1 + (1 + x1), z'' = y, z = 1, y >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(1 + half(x''), y), times(0, y)) :|: z' = 1 + (1 + x''), z'' = y, z = 1, y >= 0, x'' >= 0 if_times(z, z', z'') -{ 3 }-> plus(times(1 + half(x''), y), times(1 + half(x''), y)) :|: z' = 1 + (1 + x''), z'' = y, z = 1, y >= 0, x'' >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 2 }-> if_times(even(x'), 1 + (1 + x'), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y times(z, z') -{ 2 }-> if_times(0, 1 + 0, y) :|: z = 1 + 0, y >= 0, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(0, z''), times(1 + half(z' - 2), z'')) :|: z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(1 + half(z' - 2), z''), times(0, z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 3 }-> plus(times(1 + half(z' - 2), z''), times(1 + half(z' - 2), z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { half } { plus } { even } { if_times, times } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(0, z''), times(1 + half(z' - 2), z'')) :|: z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(1 + half(z' - 2), z''), times(0, z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 3 }-> plus(times(1 + half(z' - 2), z''), times(1 + half(z' - 2), z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {half}, {plus}, {even}, {if_times,times} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(0, z''), times(1 + half(z' - 2), z'')) :|: z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(1 + half(z' - 2), z''), times(0, z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 3 }-> plus(times(1 + half(z' - 2), z''), times(1 + half(z' - 2), z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {half}, {plus}, {even}, {if_times,times} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(0, z''), times(1 + half(z' - 2), z'')) :|: z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(1 + half(z' - 2), z''), times(0, z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 3 }-> plus(times(1 + half(z' - 2), z''), times(1 + half(z' - 2), z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {half}, {plus}, {even}, {if_times,times} Previous analysis results are: half: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(0, z''), times(1 + half(z' - 2), z'')) :|: z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 2 }-> plus(times(1 + half(z' - 2), z''), times(0, z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 3 }-> plus(times(1 + half(z' - 2), z''), times(1 + half(z' - 2), z'')) :|: z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {even}, {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {even}, {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {even}, {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {even}, {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= z'' + 0, z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {even}, {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: even after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= z'' + 0, z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {even}, {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] even: runtime: ?, size: O(1) [1] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: even after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(z - 2) :|: z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= z'' + 0, z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 2 }-> plus(z'', if_times(even(1 + (z' - 2)), 1 + (z' - 2), z'')) :|: z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(even(z - 2), 1 + (1 + (z - 2)), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] even: runtime: O(n^1) [2 + z], size: O(1) [1] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 + z }-> s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= z'' + 0, z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 3 + z' }-> plus(z'', if_times(s7, 1 + (z' - 2), z'')) :|: s7 >= 0, s7 <= 1, z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 2 + z }-> if_times(s6, 1 + (1 + (z - 2)), z') :|: s6 >= 0, s6 <= 1, z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] even: runtime: O(n^1) [2 + z], size: O(1) [1] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: if_times after applying outer abstraction to obtain an ITS, resulting in: EXP with polynomial bound: ? Computed SIZE bound using CoFloCo for: times after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 + z }-> s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= z'' + 0, z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 3 + z' }-> plus(z'', if_times(s7, 1 + (z' - 2), z'')) :|: s7 >= 0, s7 <= 1, z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 2 + z }-> if_times(s6, 1 + (1 + (z - 2)), z') :|: s6 >= 0, s6 <= 1, z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] even: runtime: O(n^1) [2 + z], size: O(1) [1] if_times: runtime: ?, size: EXP times: runtime: ?, size: INF ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if_times after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 + z }-> s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0 even(z) -{ 1 }-> 1 :|: z = 0 even(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: z >= 0 half(z) -{ z }-> 1 + s :|: s >= 0, s <= z - 2, z - 2 >= 0 if_times(z, z', z'') -{ 3 + z'' }-> s4 :|: s4 >= 0, s4 <= z'' + 0, z'' >= 0, z' = 1 + 0, z = 0 if_times(z, z', z'') -{ 3 + z' }-> plus(z'', if_times(s7, 1 + (z' - 2), z'')) :|: s7 >= 0, s7 <= 1, z'' >= 0, z = 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(0, z''), times(0, z'')) :|: z = 1, z' - 1 >= 0, z'' >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(0, z''), times(1 + s2, z'')) :|: s2 >= 0, s2 <= z' - 2, z' - 2 >= 0, z = 1, z'' >= 0 if_times(z, z', z'') -{ 1 + 2*z' }-> plus(times(1 + s', z''), times(1 + s'', z'')) :|: s' >= 0, s' <= z' - 2, s'' >= 0, s'' <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 1 + z' }-> plus(times(1 + s1, z''), times(0, z'')) :|: s1 >= 0, s1 <= z' - 2, z = 1, z'' >= 0, z' - 2 >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 2 + z }-> if_times(s6, 1 + (1 + (z - 2)), z') :|: s6 >= 0, s6 <= 1, z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> if_times(0, 1 + 0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if_times,times} Previous analysis results are: half: runtime: O(n^1) [1 + z], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] even: runtime: O(n^1) [2 + z], size: O(1) [1] if_times: runtime: INF, size: EXP times: runtime: ?, size: INF ---------------------------------------- (43) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: half(v0) -> null_half [0] if_times(v0, v1, v2) -> null_if_times [0] even(v0) -> null_even [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] And the following fresh constants: null_half, null_if_times, null_even, null_plus, null_times ---------------------------------------- (44) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: even(0) -> true [1] even(s(0)) -> false [1] even(s(s(x))) -> even(x) [1] half(0) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> if_times(even(s(x)), s(x), y) [1] if_times(true, s(x), y) -> plus(times(half(s(x)), y), times(half(s(x)), y)) [1] if_times(false, s(x), y) -> plus(y, times(x, y)) [1] half(v0) -> null_half [0] if_times(v0, v1, v2) -> null_if_times [0] even(v0) -> null_even [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] The TRS has the following type information: even :: 0:s:null_half:null_if_times:null_plus:null_times -> true:false:null_even 0 :: 0:s:null_half:null_if_times:null_plus:null_times true :: true:false:null_even s :: 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times false :: true:false:null_even half :: 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times plus :: 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times times :: 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times if_times :: true:false:null_even -> 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times -> 0:s:null_half:null_if_times:null_plus:null_times null_half :: 0:s:null_half:null_if_times:null_plus:null_times null_if_times :: 0:s:null_half:null_if_times:null_plus:null_times null_even :: true:false:null_even null_plus :: 0:s:null_half:null_if_times:null_plus:null_times null_times :: 0:s:null_half:null_if_times:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 null_half => 0 null_if_times => 0 null_even => 0 null_plus => 0 null_times => 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: even(z) -{ 1 }-> even(x) :|: x >= 0, z = 1 + (1 + x) even(z) -{ 1 }-> 2 :|: z = 0 even(z) -{ 1 }-> 1 :|: z = 1 + 0 even(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) if_times(z, z', z'') -{ 1 }-> plus(y, times(x, y)) :|: z' = 1 + x, z'' = y, z = 1, x >= 0, y >= 0 if_times(z, z', z'') -{ 1 }-> plus(times(half(1 + x), y), times(half(1 + x), y)) :|: z = 2, z' = 1 + x, z'' = y, x >= 0, y >= 0 if_times(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> if_times(even(1 + x), 1 + x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(0) -> c EVEN(s(0)) -> c1 EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(0) -> c3 HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(0, z0) -> c5 PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(0, z0) -> c7 TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) S tuples: EVEN(0) -> c EVEN(s(0)) -> c1 EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(0) -> c3 HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(0, z0) -> c5 PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(0, z0) -> c7 TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) K tuples:none Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, TIMES_2, IF_TIMES_3 Compound Symbols: c, c1, c2_1, c3, c4_1, c5, c6_1, c7, c8_2, c9_3, c10_3, c11_2 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: EVEN(0) -> c PLUS(0, z0) -> c5 HALF(0) -> c3 EVEN(s(0)) -> c1 TIMES(0, z0) -> c7 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) K tuples:none Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, TIMES_2, IF_TIMES_3 Compound Symbols: c2_1, c4_1, c6_1, c8_2, c9_3, c10_3, c11_2 ---------------------------------------- (51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] + x_1 POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, TIMES_2, IF_TIMES_3 Compound Symbols: c2_1, c4_1, c6_1, c8_2, c9_3, c10_3, c11_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(z0), z1) -> c8(IF_TIMES(even(s(z0)), s(z0), z1), EVEN(s(z0))) by TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1), EVEN(s(0))) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1), EVEN(s(0))) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1), EVEN(s(0))) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c9_3, c10_3, c11_2, c8_2 ---------------------------------------- (55) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c9_3, c10_3, c11_2, c8_2, c8_1 ---------------------------------------- (57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) And the Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c9_3, c10_3, c11_2, c8_2, c8_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(true, s(z0), z1) -> c9(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) by IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c10_3, c11_2, c8_2, c8_1, c9_3 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(true, s(z0), z1) -> c10(PLUS(times(half(s(z0)), z1), times(half(s(z0)), z1)), TIMES(half(s(z0)), z1), HALF(s(z0))) by IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_2, c8_1, c9_3, c10_3 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace TIMES(s(s(z0)), x1) -> c8(IF_TIMES(even(z0), s(s(z0)), x1), EVEN(s(s(z0)))) by TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c9_3, c10_3, c8_2 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) We considered the (Usable) Rules:none And the Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] + x_1 POL(false) = [1] POL(half(x_1)) = 0 POL(if_times(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c9_3, c10_3, c8_2 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) And the Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_1 + x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [3] + [3]x_2 + [3]x_3 POL(plus(x_1, x_2)) = [3] POL(s(x_1)) = [2] + x_1 POL(times(x_1, x_2)) = 0 POL(true) = [1] ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c9_3, c10_3, c8_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c9_3, c10_3, c8_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c10_3, c8_2, c9_3 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(half(s(s(z0))), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c10_3, c8_2, c9_3 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(half(s(s(z0))), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) S tuples: EVEN(s(s(z0))) -> c2(EVEN(z0)) HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: EVEN_1, HALF_1, PLUS_2, IF_TIMES_3, TIMES_2 Compound Symbols: c2_1, c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3 ---------------------------------------- (77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EVEN(s(s(z0))) -> c2(EVEN(z0)) by EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1), EVEN(s(s(0)))) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1), EVEN(s(s(s(0))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) by IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) by IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_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. IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_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. IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 + x_3 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, IF_TIMES_3, TIMES_2, EVEN_1 Compound Symbols: c4_1, c6_1, c11_2, c8_1, c8_2, c9_3, c10_3, c2_1 ---------------------------------------- (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(false, s(z0), z1) -> c11(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) by IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0)), TIMES(0, x0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0)), TIMES(0, x0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0)), TIMES(0, x0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_2, c11_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) by IF_TIMES(false, s(s(s(s(x0)))), z1) -> c11(PLUS(z1, if_times(even(s(s(s(x0)))), s(s(s(x0))), z1)), TIMES(s(s(s(x0))), z1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(false, s(s(s(s(x0)))), z1) -> c11(PLUS(z1, if_times(even(s(s(s(x0)))), s(s(s(x0))), z1)), TIMES(s(s(s(x0))), z1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_2, c11_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) by IF_TIMES(false, s(s(s(0))), z1) -> c11(PLUS(z1, if_times(even(s(s(0))), s(s(0)), z1)), TIMES(s(s(0)), z1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(false, s(s(s(s(x0)))), z1) -> c11(PLUS(z1, if_times(even(s(s(s(x0)))), s(s(s(x0))), z1)), TIMES(s(s(s(x0))), z1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), z1) -> c11(PLUS(z1, if_times(even(s(s(0))), s(s(0)), z1)), TIMES(s(s(0)), z1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (105) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) by IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), z1) -> c11(PLUS(z1, if_times(even(s(s(s(x0)))), s(s(s(x0))), z1)), TIMES(s(s(s(x0))), z1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), z1) -> c11(PLUS(z1, if_times(even(s(s(0))), s(s(0)), z1)), TIMES(s(s(0)), z1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_2, c11_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(false, s(s(s(s(x0)))), z1) -> c11(PLUS(z1, if_times(even(s(s(s(x0)))), s(s(s(x0))), z1)), TIMES(s(s(s(x0))), z1)) by IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), z1) -> c11(PLUS(z1, if_times(even(s(s(0))), s(s(0)), z1)), TIMES(s(s(0)), z1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF_TIMES(false, s(s(s(0))), z1) -> c11(PLUS(z1, if_times(even(s(s(0))), s(s(0)), z1)), TIMES(s(s(0)), z1)) by IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(z0)), x1) -> c9(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (113) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(x0)), z1) -> c9(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (115) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(times(half(s(s(x0))), z1), if_times(even(s(half(x0))), s(half(x0)), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(0)), x1) -> c9(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) by IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c9_3, c10_3, c2_1, c11_1, c11_2 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(z0)))), x1) -> c9(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) And the Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_1 + x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: HALF(s(s(z0))) -> c4(HALF(z0)) PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: HALF_1, PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1 Compound Symbols: c4_1, c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2 ---------------------------------------- (125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HALF(s(s(z0))) -> c4(HALF(z0)) by HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0), HALF(s(s(0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (129) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(half(s(s(s(s(z0))))), x1), times(s(s(half(z0))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (131) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (133) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (137) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(z0)), x1) -> c10(PLUS(times(s(half(z0)), x1), times(s(half(z0)), x1)), TIMES(half(s(s(z0))), x1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0), HALF(s(s(0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(x0)), z1) -> c10(PLUS(if_times(even(s(half(x0))), s(half(x0)), z1), times(half(s(s(x0))), z1)), TIMES(half(s(s(x0))), z1), HALF(s(s(x0)))) by IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (143) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, IF_TIMES_3, EVEN_1, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c10_3, c2_1, c9_3, c11_1, c11_2, c4_1, c10_2, c9_2 ---------------------------------------- (145) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(z0)))), x1) -> c10(PLUS(times(s(s(half(z0))), x1), times(half(s(s(s(s(z0))))), x1)), TIMES(half(s(s(s(s(z0))))), x1), HALF(s(s(s(s(z0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (147) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (149) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(half(s(s(z0))), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (151) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (153) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) S tuples: PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: PLUS_2, TIMES_2, EVEN_1, IF_TIMES_3, HALF_1 Compound Symbols: c6_1, c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), z1) -> c6(PLUS(z0, z1)) by PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, if_times(even(0), s(s(0)), x0)), TIMES(s(s(0)), x0)) by IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c9_3, c11_1, c11_2, c10_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c9_3, c10_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (165) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(times(s(half(z0)), z1), times(s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c9_3, c10_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c9_3, c10_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c10_3, c9_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (171) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) And the Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_1 + x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c10_3, c9_3, c4_1, c10_2, c9_2, c6_1 ---------------------------------------- (173) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c9_3, c4_1, c10_2, c9_2, c10_3, c6_1 ---------------------------------------- (175) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = 0 ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c9_3, c4_1, c10_2, c9_2, c10_3, c6_1 ---------------------------------------- (177) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) by IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c10_2, c9_2, c10_3, c9_3, c6_1 ---------------------------------------- (179) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(half(s(s(0))), x1), times(s(0), x1)), TIMES(half(s(s(0))), x1)) by IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c10_2, c9_2, c10_3, c9_3, c6_1 ---------------------------------------- (181) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = [1] + x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = 0 POL(false) = [1] POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = 0 ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c10_2, c9_2, c10_3, c9_3, c6_1 ---------------------------------------- (183) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(0)), x1) -> c10(PLUS(times(s(0), x1), times(half(s(s(0))), x1)), TIMES(half(s(s(0))), x1)) by IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1 ---------------------------------------- (185) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) We considered the (Usable) Rules: half(s(s(z0))) -> s(half(z0)) half(0) -> 0 even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) And the Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(EVEN(x_1)) = 0 POL(HALF(x_1)) = 0 POL(IF_TIMES(x_1, x_2, x_3)) = x_1 + x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c8(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2)) = x_1 + x_2 POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(even(x_1)) = [1] POL(false) = 0 POL(half(x_1)) = x_1 POL(if_times(x_1, x_2, x_3)) = [1] + x_3 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = x_2 POL(true) = [1] ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1 ---------------------------------------- (187) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) by IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(0), z0)) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(0), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1 ---------------------------------------- (189) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c(PLUS(times(half(s(s(0))), z0), times(s(0), z0))) IF_TIMES(true, s(s(0)), z0) -> c(TIMES(s(0), z0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1, c_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace TIMES(s(s(x0)), x1) -> c8(EVEN(s(s(x0)))) by TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c(PLUS(times(half(s(s(0))), z0), times(s(0), z0))) IF_TIMES(true, s(s(0)), z0) -> c(TIMES(s(0), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1, c_1 ---------------------------------------- (193) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF_TIMES(true, s(s(z0)), z1) -> c10(PLUS(if_times(even(s(half(z0))), s(half(z0)), z1), times(half(s(s(z0))), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) by IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c(PLUS(times(half(s(s(0))), z0), times(s(0), z0))) IF_TIMES(true, s(s(0)), z0) -> c(TIMES(s(0), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0), HALF(s(s(0)))) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1, c_1 ---------------------------------------- (195) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c(PLUS(times(half(s(s(0))), z0), times(s(0), z0))) IF_TIMES(true, s(s(0)), z0) -> c(TIMES(s(0), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0)) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, EVEN_1, IF_TIMES_3, HALF_1, PLUS_2 Compound Symbols: c8_1, c8_2, c2_1, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1, c_1 ---------------------------------------- (197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace EVEN(s(s(s(s(y0))))) -> c2(EVEN(s(s(y0)))) by EVEN(s(s(s(s(s(s(y0))))))) -> c2(EVEN(s(s(s(s(y0)))))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: even(0) -> true even(s(0)) -> false even(s(s(z0))) -> even(z0) half(0) -> 0 half(s(s(z0))) -> s(half(z0)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> if_times(even(s(z0)), s(z0), z1) if_times(true, s(z0), z1) -> plus(times(half(s(z0)), z1), times(half(s(z0)), z1)) if_times(false, s(z0), z1) -> plus(z1, times(z0, z1)) Tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), z0) -> c11(PLUS(z0, 0)) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, if_times(even(s(x0)), s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(half(s(s(s(s(x0))))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) IF_TIMES(false, s(s(s(0))), z0) -> c11(PLUS(z0, if_times(true, s(s(0)), z0)), TIMES(s(s(0)), z0)) IF_TIMES(true, s(s(z0)), z1) -> c9(PLUS(times(s(half(z0)), z1), if_times(even(s(half(z0))), s(half(z0)), z1)), TIMES(s(half(z0)), z1), HALF(s(s(z0)))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(s(half(z0))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c(PLUS(times(half(s(s(0))), z0), times(s(0), z0))) IF_TIMES(true, s(s(0)), z0) -> c(TIMES(s(0), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0)) EVEN(s(s(s(s(s(s(y0))))))) -> c2(EVEN(s(s(s(s(y0)))))) S tuples: TIMES(s(s(s(s(z0)))), x1) -> c8(IF_TIMES(even(z0), s(s(s(s(z0)))), x1), EVEN(s(s(s(s(z0)))))) TIMES(s(s(0)), x1) -> c8(IF_TIMES(true, s(s(0)), x1)) HALF(s(s(s(s(y0))))) -> c4(HALF(s(s(y0)))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(half(s(s(0))), x0)) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(half(s(s(0))), x0)) PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) EVEN(s(s(s(s(s(s(y0))))))) -> c2(EVEN(s(s(s(s(y0)))))) K tuples: TIMES(s(0), x1) -> c8(IF_TIMES(false, s(0), x1)) TIMES(s(s(s(0))), x1) -> c8(IF_TIMES(false, s(s(s(0))), x1)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(false, s(s(s(s(x0)))), x1) -> c11(PLUS(x1, times(s(s(s(x0))), x1)), TIMES(s(s(s(x0))), x1)) IF_TIMES(false, s(s(s(0))), x0) -> c11(PLUS(x0, times(s(s(0)), x0)), TIMES(s(s(0)), x0)) IF_TIMES(false, s(0), x0) -> c11(PLUS(x0, times(0, x0))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c9(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), times(s(s(half(z0))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(s(s(half(z0))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(half(s(s(s(s(x0))))), x1), if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(half(s(s(0))), x0), if_times(even(s(half(0))), s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c9(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c9(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(times(s(half(s(s(x0)))), x1), times(s(half(s(s(x0)))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(times(s(half(0)), x0), times(s(half(0)), x0)), TIMES(s(half(0)), x0)) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c9(PLUS(if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1), times(half(s(s(s(s(z0))))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(s(s(z0)))), z1) -> c10(PLUS(times(half(s(s(s(s(z0))))), z1), if_times(even(s(half(s(s(z0))))), s(half(s(s(z0)))), z1)), TIMES(s(half(s(s(z0)))), z1), HALF(s(s(s(s(z0)))))) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(half(s(s(0))), z0), times(s(0), z0)), TIMES(s(half(0)), z0)) IF_TIMES(true, s(s(0)), z0) -> c10(PLUS(times(s(0), z0), times(half(s(s(0))), z0)), TIMES(s(half(0)), z0)) TIMES(s(s(s(s(y0)))), z1) -> c8(EVEN(s(s(s(s(y0)))))) IF_TIMES(true, s(s(s(s(x0)))), x1) -> c10(PLUS(if_times(even(s(half(s(s(x0))))), s(half(s(s(x0)))), x1), times(half(s(s(s(s(x0))))), x1)), TIMES(s(half(s(s(x0)))), x1), HALF(s(s(s(s(x0)))))) IF_TIMES(true, s(s(0)), x0) -> c10(PLUS(if_times(even(s(half(0))), s(half(0)), x0), times(half(s(s(0))), x0)), TIMES(s(half(0)), x0)) Defined Rule Symbols: even_1, half_1, plus_2, times_2, if_times_3 Defined Pair Symbols: TIMES_2, IF_TIMES_3, HALF_1, PLUS_2, EVEN_1 Compound Symbols: c8_1, c8_2, c11_1, c11_2, c4_1, c9_2, c10_2, c10_3, c9_3, c6_1, c_1, c2_1