KILLED proof of input_7uBMKELrPL.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 4 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 260 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 99 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 269 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 193 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 429 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 3893 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 297 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (132) CdtProblem (133) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) 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: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) cond(true, x, y) -> s(0) cond(false, x, y) -> double(log(x, square(s(s(y))))) le(0, v) -> true le(s(u), 0) -> false le(s(u), s(v)) -> le(u, v) double(0) -> 0 double(s(x)) -> s(s(double(x))) square(0) -> 0 square(s(x)) -> s(plus(square(x), double(x))) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) 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: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) cond(true, x, y) -> s(0') cond(false, x, y) -> double(log(x, square(s(s(y))))) le(0', v) -> true le(s(u), 0') -> false le(s(u), s(v)) -> le(u, v) double(0') -> 0' double(s(x)) -> s(s(double(x))) square(0') -> 0' square(s(x)) -> s(plus(square(x), double(x))) plus(n, 0') -> n plus(n, s(m)) -> s(plus(n, m)) 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: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) cond(true, x, y) -> s(0) cond(false, x, y) -> double(log(x, square(s(s(y))))) le(0, v) -> true le(s(u), 0) -> false le(s(u), s(v)) -> le(u, v) double(0) -> 0 double(s(x)) -> s(s(double(x))) square(0) -> 0 square(s(x)) -> s(plus(square(x), double(x))) plus(n, 0) -> n plus(n, s(m)) -> s(plus(n, m)) 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: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) [1] cond(true, x, y) -> s(0) [1] cond(false, x, y) -> double(log(x, square(s(s(y))))) [1] le(0, v) -> true [1] le(s(u), 0) -> false [1] le(s(u), s(v)) -> le(u, v) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] square(0) -> 0 [1] square(s(x)) -> s(plus(square(x), double(x))) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [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: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) [1] cond(true, x, y) -> s(0) [1] cond(false, x, y) -> double(log(x, square(s(s(y))))) [1] le(0, v) -> true [1] le(s(u), 0) -> false [1] le(s(u), s(v)) -> le(u, v) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] square(0) -> 0 [1] square(s(x)) -> s(plus(square(x), double(x))) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] The TRS has the following type information: log :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 cond :: true:false -> s:0 -> s:0 -> s:0 le :: s:0 -> s:0 -> true:false true :: true:false 0 :: s:0 false :: true:false double :: s:0 -> s:0 square :: s:0 -> s:0 plus :: s:0 -> s:0 -> s:0 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: le_2 log_2 square_1 double_1 cond_3 plus_2 Due to the following rules being added: log(v0, v1) -> 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: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) [1] cond(true, x, y) -> s(0) [1] cond(false, x, y) -> double(log(x, square(s(s(y))))) [1] le(0, v) -> true [1] le(s(u), 0) -> false [1] le(s(u), s(v)) -> le(u, v) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] square(0) -> 0 [1] square(s(x)) -> s(plus(square(x), double(x))) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] log(v0, v1) -> 0 [0] The TRS has the following type information: log :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 cond :: true:false -> s:0 -> s:0 -> s:0 le :: s:0 -> s:0 -> true:false true :: true:false 0 :: s:0 false :: true:false double :: s:0 -> s:0 square :: s:0 -> s:0 plus :: s:0 -> s:0 -> s:0 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: log(0, s(s(y))) -> cond(true, 0, y) [2] log(s(u'), s(s(y))) -> cond(le(u', s(y)), s(u'), y) [2] cond(true, x, y) -> s(0) [1] cond(false, x, y) -> double(log(x, s(plus(square(s(y)), double(s(y)))))) [2] le(0, v) -> true [1] le(s(u), 0) -> false [1] le(s(u), s(v)) -> le(u, v) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] square(0) -> 0 [1] square(s(0)) -> s(plus(0, 0)) [3] square(s(s(x'))) -> s(plus(s(plus(square(x'), double(x'))), s(s(double(x'))))) [3] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] log(v0, v1) -> 0 [0] The TRS has the following type information: log :: s:0 -> s:0 -> s:0 s :: s:0 -> s:0 cond :: true:false -> s:0 -> s:0 -> s:0 le :: s:0 -> s:0 -> true:false true :: true:false 0 :: s:0 false :: true:false double :: s:0 -> s:0 square :: s:0 -> s:0 plus :: s:0 -> s:0 -> s:0 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: true => 1 0 => 0 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 2 }-> double(log(x, 1 + plus(square(1 + y), double(1 + y)))) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x le(z, z') -{ 1 }-> le(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 le(z, z') -{ 1 }-> 1 :|: v >= 0, z' = v, z = 0 le(z, z') -{ 1 }-> 0 :|: z = 1 + u, z' = 0, u >= 0 log(z, z') -{ 2 }-> cond(le(u', 1 + y), 1 + u', y) :|: z' = 1 + (1 + y), y >= 0, u' >= 0, z = 1 + u' log(z, z') -{ 2 }-> cond(1, 0, y) :|: z' = 1 + (1 + y), y >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> n :|: n >= 0, z = n, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(n, m) :|: n >= 0, z = n, z' = 1 + m, m >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 3 }-> 1 + plus(1 + plus(square(x'), double(x')), 1 + (1 + double(x'))) :|: x' >= 0, z = 1 + (1 + x') ---------------------------------------- (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: cond(z, z', z'') -{ 2 }-> double(log(z', 1 + plus(square(1 + z''), double(1 + z'')))) :|: z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 3 }-> 1 + plus(1 + plus(square(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { double } { le } { plus } { square } { log, cond } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 2 }-> double(log(z', 1 + plus(square(1 + z''), double(1 + z'')))) :|: z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 3 }-> 1 + plus(1 + plus(square(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0 Function symbols to be analyzed: {double}, {le}, {plus}, {square}, {log,cond} ---------------------------------------- (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: cond(z, z', z'') -{ 2 }-> double(log(z', 1 + plus(square(1 + z''), double(1 + z'')))) :|: z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 3 }-> 1 + plus(1 + plus(square(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0 Function symbols to be analyzed: {double}, {le}, {plus}, {square}, {log,cond} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 2 }-> double(log(z', 1 + plus(square(1 + z''), double(1 + z'')))) :|: z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 3 }-> 1 + plus(1 + plus(square(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0 Function symbols to be analyzed: {double}, {le}, {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: double after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 2 }-> double(log(z', 1 + plus(square(1 + z''), double(1 + z'')))) :|: z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 3 }-> 1 + plus(1 + plus(square(z - 2), double(z - 2)), 1 + (1 + double(z - 2))) :|: z - 2 >= 0 Function symbols to be analyzed: {le}, {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {le}, {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: le after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {le}, {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: ?, size: O(1) [1] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: le after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(le(z - 1, 1 + (z' - 2)), 1 + (z - 1), z' - 2) :|: z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (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: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (33) 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' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {plus}, {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (35) 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' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (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: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z + (z' - 1), z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 4 }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 0, z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: square after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 4*z + 4*z^2 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z + (z' - 1), z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 4 }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 0, z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {square}, {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] square: runtime: ?, size: O(n^2) [1 + 4*z + 4*z^2] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: square after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 5 + 6*z^2 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 4 + z'' }-> double(log(z', 1 + plus(square(1 + z''), s))) :|: s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z + (z' - 1), z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 4 }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 0, z = 1 + 0 square(z) -{ 1 + 2*z }-> 1 + plus(1 + plus(square(z - 2), s''), 1 + (1 + s1)) :|: s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 Function symbols to be analyzed: {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] square: runtime: O(n^2) [5 + 6*z^2], size: O(n^2) [1 + 4*z + 4*z^2] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 16 + s + 13*z'' + 6*z''^2 }-> double(log(z', 1 + s7)) :|: s6 >= 0, s6 <= 4 * ((1 + z'') * (1 + z'')) + 4 * (1 + z'') + 1, s7 >= 0, s7 <= s6 + s, s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z + (z' - 1), z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 34 + s'' + s1 + -22*z + 6*z^2 }-> 1 + s10 :|: s8 >= 0, s8 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s9 >= 0, s9 <= s8 + s'', s10 >= 0, s10 <= 1 + s9 + (1 + (1 + s1)), s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 square(z) -{ 4 }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 0, z = 1 + 0 Function symbols to be analyzed: {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] square: runtime: O(n^2) [5 + 6*z^2], size: O(n^2) [1 + 4*z + 4*z^2] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: log after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: cond after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 16 + s + 13*z'' + 6*z''^2 }-> double(log(z', 1 + s7)) :|: s6 >= 0, s6 <= 4 * ((1 + z'') * (1 + z'')) + 4 * (1 + z'') + 1, s7 >= 0, s7 <= s6 + s, s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z + (z' - 1), z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 34 + s'' + s1 + -22*z + 6*z^2 }-> 1 + s10 :|: s8 >= 0, s8 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s9 >= 0, s9 <= s8 + s'', s10 >= 0, s10 <= 1 + s9 + (1 + (1 + s1)), s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 square(z) -{ 4 }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 0, z = 1 + 0 Function symbols to be analyzed: {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] square: runtime: O(n^2) [5 + 6*z^2], size: O(n^2) [1 + 4*z + 4*z^2] log: runtime: ?, size: INF cond: runtime: ?, size: INF ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: log after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 16 + s + 13*z'' + 6*z''^2 }-> double(log(z', 1 + s7)) :|: s6 >= 0, s6 <= 4 * ((1 + z'') * (1 + z'')) + 4 * (1 + z'') + 1, s7 >= 0, s7 <= s6 + s, s >= 0, s <= 2 * (1 + z''), z' >= 0, z'' >= 0, z = 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 1, z' >= 0, z'' >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 1 + z }-> 1 + (1 + s') :|: s' >= 0, s' <= 2 * (z - 1), z - 1 >= 0 le(z, z') -{ 2 + z' }-> s3 :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z' = 0, z - 1 >= 0 log(z, z') -{ 3 + z' }-> cond(s2, 1 + (z - 1), z' - 2) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 1 >= 0 log(z, z') -{ 2 }-> cond(1, 0, z' - 2) :|: z' - 2 >= 0, z = 0 log(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= z + (z' - 1), z >= 0, z' - 1 >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 34 + s'' + s1 + -22*z + 6*z^2 }-> 1 + s10 :|: s8 >= 0, s8 <= 4 * ((z - 2) * (z - 2)) + 4 * (z - 2) + 1, s9 >= 0, s9 <= s8 + s'', s10 >= 0, s10 <= 1 + s9 + (1 + (1 + s1)), s'' >= 0, s'' <= 2 * (z - 2), s1 >= 0, s1 <= 2 * (z - 2), z - 2 >= 0 square(z) -{ 4 }-> 1 + s4 :|: s4 >= 0, s4 <= 0 + 0, z = 1 + 0 Function symbols to be analyzed: {log,cond} Previous analysis results are: double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z] le: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] square: runtime: O(n^2) [5 + 6*z^2], size: O(n^2) [1 + 4*z + 4*z^2] log: runtime: INF, size: INF cond: runtime: ?, size: INF ---------------------------------------- (49) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: log(v0, v1) -> null_log [0] le(v0, v1) -> null_le [0] double(v0) -> null_double [0] square(v0) -> null_square [0] plus(v0, v1) -> null_plus [0] cond(v0, v1, v2) -> null_cond [0] And the following fresh constants: null_log, null_le, null_double, null_square, null_plus, null_cond ---------------------------------------- (50) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: log(x, s(s(y))) -> cond(le(x, s(s(y))), x, y) [1] cond(true, x, y) -> s(0) [1] cond(false, x, y) -> double(log(x, square(s(s(y))))) [1] le(0, v) -> true [1] le(s(u), 0) -> false [1] le(s(u), s(v)) -> le(u, v) [1] double(0) -> 0 [1] double(s(x)) -> s(s(double(x))) [1] square(0) -> 0 [1] square(s(x)) -> s(plus(square(x), double(x))) [1] plus(n, 0) -> n [1] plus(n, s(m)) -> s(plus(n, m)) [1] log(v0, v1) -> null_log [0] le(v0, v1) -> null_le [0] double(v0) -> null_double [0] square(v0) -> null_square [0] plus(v0, v1) -> null_plus [0] cond(v0, v1, v2) -> null_cond [0] The TRS has the following type information: log :: s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond s :: s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond cond :: true:false:null_le -> s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond le :: s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond -> true:false:null_le true :: true:false:null_le 0 :: s:0:null_log:null_double:null_square:null_plus:null_cond false :: true:false:null_le double :: s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond square :: s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond plus :: s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond -> s:0:null_log:null_double:null_square:null_plus:null_cond null_log :: s:0:null_log:null_double:null_square:null_plus:null_cond null_le :: true:false:null_le null_double :: s:0:null_log:null_double:null_square:null_plus:null_cond null_square :: s:0:null_log:null_double:null_square:null_plus:null_cond null_plus :: s:0:null_log:null_double:null_square:null_plus:null_cond null_cond :: s:0:null_log:null_double:null_square:null_plus:null_cond Rewrite Strategy: INNERMOST ---------------------------------------- (51) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: true => 2 0 => 0 false => 1 null_log => 0 null_le => 0 null_double => 0 null_square => 0 null_plus => 0 null_cond => 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: cond(z, z', z'') -{ 1 }-> double(log(x, square(1 + (1 + y)))) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 cond(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 cond(z, z', z'') -{ 1 }-> 1 + 0 :|: z = 2, z' = x, z'' = y, x >= 0, y >= 0 double(z) -{ 1 }-> 0 :|: z = 0 double(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x le(z, z') -{ 1 }-> le(u, v) :|: v >= 0, z' = 1 + v, z = 1 + u, u >= 0 le(z, z') -{ 1 }-> 2 :|: v >= 0, z' = v, z = 0 le(z, z') -{ 1 }-> 1 :|: z = 1 + u, z' = 0, u >= 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 log(z, z') -{ 1 }-> cond(le(x, 1 + (1 + y)), x, y) :|: z' = 1 + (1 + y), x >= 0, y >= 0, z = x log(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> n :|: n >= 0, z = n, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(n, m) :|: n >= 0, z = n, z' = 1 + m, m >= 0 square(z) -{ 1 }-> 0 :|: z = 0 square(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 square(z) -{ 1 }-> 1 + plus(square(x), double(x)) :|: x >= 0, z = 1 + x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) COND(true, z0, z1) -> c1 COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(0) -> c6 DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(0) -> c8 SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) SQUARE(s(z0)) -> c10(PLUS(square(z0), double(z0)), DOUBLE(z0)) PLUS(z0, 0) -> c11 PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) S tuples: LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) COND(true, z0, z1) -> c1 COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(0, z0) -> c3 LE(s(z0), 0) -> c4 LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(0) -> c6 DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(0) -> c8 SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) SQUARE(s(z0)) -> c10(PLUS(square(z0), double(z0)), DOUBLE(z0)) PLUS(z0, 0) -> c11 PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, COND_3, LE_2, DOUBLE_1, SQUARE_1, PLUS_2 Compound Symbols: c_2, c1, c2_3, c3, c4, c5_1, c6, c7_1, c8, c9_2, c10_2, c11, c12_1 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: SQUARE(0) -> c8 DOUBLE(0) -> c6 COND(true, z0, z1) -> c1 LE(0, z0) -> c3 PLUS(z0, 0) -> c11 LE(s(z0), 0) -> c4 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) SQUARE(s(z0)) -> c10(PLUS(square(z0), double(z0)), DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) S tuples: LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) SQUARE(s(z0)) -> c10(PLUS(square(z0), double(z0)), DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, COND_3, LE_2, DOUBLE_1, SQUARE_1, PLUS_2 Compound Symbols: c_2, c2_3, c5_1, c7_1, c9_2, c10_2, c12_1 ---------------------------------------- (57) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) S tuples: LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, COND_3, LE_2, DOUBLE_1, SQUARE_1, PLUS_2 Compound Symbols: c_2, c2_3, c5_1, c7_1, c9_2, c12_1, c1_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOG(z0, s(s(z1))) -> c(COND(le(z0, s(s(z1))), z0, z1), LE(z0, s(s(z1)))) by LOG(0, s(s(x1))) -> c(COND(true, 0, x1), LE(0, s(s(x1)))) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(0, s(s(x1))) -> c(COND(true, 0, x1), LE(0, s(s(x1)))) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) S tuples: COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(0, s(s(x1))) -> c(COND(true, 0, x1), LE(0, s(s(x1)))) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: COND_3, LE_2, DOUBLE_1, SQUARE_1, PLUS_2, LOG_2 Compound Symbols: c2_3, c5_1, c7_1, c9_2, c12_1, c1_1, c_2 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LOG(0, s(s(x1))) -> c(COND(true, 0, x1), LE(0, s(s(x1)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) S tuples: COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: COND_3, LE_2, DOUBLE_1, SQUARE_1, PLUS_2, LOG_2 Compound Symbols: c2_3, c5_1, c7_1, c9_2, c12_1, c1_1, c_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND(false, z0, z1) -> c2(DOUBLE(log(z0, square(s(s(z1))))), LOG(z0, square(s(s(z1)))), SQUARE(s(s(z1)))) by COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, SQUARE_1, PLUS_2, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c9_2, c12_1, c1_1, c_2, c2_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(z0)) -> c9(PLUS(square(z0), double(z0)), SQUARE(z0)) by SQUARE(s(0)) -> c9(PLUS(square(0), 0), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0)), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(0)) -> c9(PLUS(square(0), 0), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0)), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(0)) -> c9(PLUS(square(0), 0), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0)), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c2_3, c9_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQUARE(s(0)) -> c9(PLUS(square(0), 0), SQUARE(0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0)), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0)), SQUARE(0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c2_3, c9_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c2_3, c9_2, c9_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(z0)) -> c1(PLUS(square(z0), double(z0))) by SQUARE(s(0)) -> c1(PLUS(square(0), 0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(0)) -> c1(PLUS(square(0), 0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(0)) -> c1(PLUS(square(0), 0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c2_3, c9_2, c9_1 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQUARE(s(0)) -> c1(PLUS(square(0), 0)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c2_3, c9_2, c9_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOG(s(z0), s(s(x1))) -> c(COND(le(z0, s(x1)), s(z0), x1), LE(s(z0), s(s(x1)))) by LOG(s(0), s(s(x1))) -> c(COND(true, s(0), x1), LE(s(0), s(s(x1)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(0), s(s(x1))) -> c(COND(true, s(0), x1), LE(s(0), s(s(x1)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(0), s(s(x1))) -> c(COND(true, s(0), x1), LE(s(0), s(s(x1)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, COND_3, LOG_2 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c2_3, c9_2, c9_1, c_2, c_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) K tuples:none Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, COND_3, LOG_2 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c2_3, c9_2, c9_1, c_2, c_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(COND(x_1, x_2, x_3)) = [1] + x_2 POL(DOUBLE(x_1)) = 0 POL(LE(x_1, x_2)) = 0 POL(LOG(x_1, x_2)) = [1] POL(PLUS(x_1, x_2)) = 0 POL(SQUARE(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(cond(x_1, x_2, x_3)) = [1] + x_2 + x_3 POL(double(x_1)) = 0 POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_2 POL(log(x_1, x_2)) = [1] + x_1 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(square(x_1)) = 0 POL(true) = [1] ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, COND_3, LOG_2 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c2_3, c9_2, c9_1, c_2, c_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace COND(false, x0, x1) -> c2(DOUBLE(log(x0, s(plus(square(s(x1)), double(s(x1)))))), LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) by COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c9_2, c9_1, c_2, c_1, c2_3, c2_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(s(z0))) -> c9(PLUS(square(s(z0)), s(s(double(z0)))), SQUARE(s(z0))) by SQUARE(s(s(0))) -> c9(PLUS(square(s(0)), s(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(0))) -> c9(PLUS(square(s(0)), s(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(0))) -> c9(PLUS(square(s(0)), s(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c9_2, c9_1, c_2, c_1, c2_3, c2_2 ---------------------------------------- (85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c9_2, c9_1, c_2, c_1, c2_3, c2_2, c3_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), double(s(z0))), SQUARE(s(z0))) by SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c9(PLUS(s(plus(square(0), 0)), double(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c9(PLUS(s(plus(0, double(0))), double(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c9(PLUS(s(plus(square(0), 0)), double(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c9(PLUS(s(plus(0, double(0))), double(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c9(PLUS(s(plus(square(0), 0)), double(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c9(PLUS(s(plus(0, double(0))), double(s(0))), SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c9_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1 ---------------------------------------- (89) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(0)) -> c9(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c9_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(0)) -> c9(PLUS(0, double(0))) by SQUARE(s(0)) -> c9(PLUS(0, 0)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(0)) -> c9(PLUS(0, 0)) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(0)) -> c9(PLUS(0, 0)) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1, c9_1 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQUARE(s(0)) -> c9(PLUS(0, 0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(s(z0))) -> c1(PLUS(square(s(z0)), s(s(double(z0))))) by SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(0)) -> c1(PLUS(0, double(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(0)) -> c1(PLUS(0, double(0))) by SQUARE(s(0)) -> c1(PLUS(0, 0)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, 0)) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(0)) -> c1(PLUS(0, 0)) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: SQUARE(s(0)) -> c1(PLUS(0, 0)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), double(s(z0)))) by SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(square(s(z0)), s(s(double(z0))))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) by COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_3, c2_2, c9_2, c3_1, c4_1 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND(false, x0, z0) -> c2(DOUBLE(log(x0, s(plus(s(plus(square(z0), double(z0))), double(s(z0)))))), LOG(x0, square(s(s(z0)))), SQUARE(s(s(z0)))) by COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c2_2, c9_2, c3_1, c4_1, c2_3 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND(false, x0, x1) -> c2(LOG(x0, square(s(s(x1)))), SQUARE(s(s(x1)))) by COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c9(PLUS(square(s(s(z0))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) by SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c3(PLUS(square(s(0)), s(s(0)))) by SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0)))), SQUARE(s(s(z0)))) by SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0)))), SQUARE(s(s(z0)))) by SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), double(s(0)))) by SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (119) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), double(s(0)))) by SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (121) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c1(PLUS(square(s(0)), s(s(0)))) by SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (123) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(square(s(z1)), s(s(double(z1))))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) by COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (125) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND(false, z0, z1) -> c2(DOUBLE(log(z0, s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) by COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_2, c2_3 ---------------------------------------- (127) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c1(PLUS(square(s(s(z0))), s(s(s(s(double(z0))))))) by SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_2, c2_3 ---------------------------------------- (129) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace COND(false, z0, z1) -> c2(LOG(z0, s(plus(square(s(z1)), double(s(z1))))), SQUARE(s(s(z1)))) by COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (131) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: LOG(s(0), s(s(x1))) -> c(LE(s(0), s(s(x1)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) S tuples: LE(s(z0), s(z1)) -> c5(LE(z0, z1)) DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LE_2, DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3 Compound Symbols: c5_1, c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2 ---------------------------------------- (133) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c5(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) S tuples: DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) K tuples: LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3, LE_2 Compound Symbols: c7_1, c12_1, c1_1, c_2, c_1, c9_2, c3_1, c4_1, c2_3, c2_2, c5_1 ---------------------------------------- (135) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOG(s(x0), s(s(x1))) -> c(LE(s(x0), s(s(x1)))) by LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) S tuples: DOUBLE(s(z0)) -> c7(DOUBLE(z0)) PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: DOUBLE_1, PLUS_2, SQUARE_1, LOG_2, COND_3, LE_2 Compound Symbols: c7_1, c12_1, c1_1, c_2, c9_2, c3_1, c4_1, c2_3, c2_2, c5_1, c_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace DOUBLE(s(z0)) -> c7(DOUBLE(z0)) by DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) S tuples: PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: PLUS_2, SQUARE_1, LOG_2, COND_3, LE_2, DOUBLE_1 Compound Symbols: c12_1, c1_1, c_2, c9_2, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1 ---------------------------------------- (139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(z1)) -> c12(PLUS(z0, z1)) by PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) S tuples: SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: SQUARE_1, LOG_2, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c1_1, c_2, c9_2, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (141) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), double(s(0)))) by SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) S tuples: SQUARE(s(z0)) -> c1(DOUBLE(z0)) LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: SQUARE_1, LOG_2, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c1_1, c_2, c9_2, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace SQUARE(s(z0)) -> c1(DOUBLE(z0)) by SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c_2, c9_2, c3_1, c4_1, c1_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: SQUARE(s(s(0))) -> c3(SQUARE(s(0))) SQUARE(s(s(0))) -> c4(SQUARE(s(0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c_2, c9_2, c1_1, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (147) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), double(s(s(z0))))) by SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c_2, c9_2, c1_1, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (149) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), double(s(0)))) by SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c_2, c9_2, c1_1, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (151) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), double(s(s(z0))))) by SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, LE_2, DOUBLE_1, PLUS_2 Compound Symbols: c_2, c9_2, c1_1, c3_1, c4_1, c2_3, c2_2, c5_1, c_1, c7_1, c12_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c5(LE(s(y0), s(y1))) by LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c3_1, c4_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1 ---------------------------------------- (155) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) by SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c3_1, c4_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), double(0))), s(s(0)))) by SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c4_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1 ---------------------------------------- (159) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) by SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c4_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1 ---------------------------------------- (161) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0))))), SQUARE(s(s(z0)))) by SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c4_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(double(0))))) by SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c4_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1 ---------------------------------------- (165) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(double(0))))) by SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1, c4_1 ---------------------------------------- (167) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), double(0))), s(s(0)))) by SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1, c4_1 ---------------------------------------- (169) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(square(s(x1)), s(s(double(x1))))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) by COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c2_3, c2_2, c_1, c7_1, c12_1, c5_1, c3_1, c4_1 ---------------------------------------- (171) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND(false, s(s(x0)), x1) -> c2(DOUBLE(log(s(s(x0)), s(plus(s(plus(square(x1), double(x1))), double(s(x1)))))), LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) by COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c2_2, c_1, c7_1, c12_1, c5_1, c3_1, c4_1, c2_3 ---------------------------------------- (173) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), double(s(z0)))), s(s(s(s(double(z0))))))) by SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0))))))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0))))))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0))))))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, COND_3, DOUBLE_1, PLUS_2, LE_2 Compound Symbols: c_2, c9_2, c1_1, c2_2, c_1, c7_1, c12_1, c5_1, c3_1, c4_1, c2_3 ---------------------------------------- (175) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace COND(false, s(s(x0)), x1) -> c2(LOG(s(s(x0)), s(plus(square(s(x1)), double(s(x1))))), SQUARE(s(s(x1)))) by COND(false, s(s(z0)), z1) -> c2(LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: log(z0, s(s(z1))) -> cond(le(z0, s(s(z1))), z0, z1) cond(true, z0, z1) -> s(0) cond(false, z0, z1) -> double(log(z0, square(s(s(z1))))) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) double(0) -> 0 double(s(z0)) -> s(s(double(z0))) square(0) -> 0 square(s(z0)) -> s(plus(square(z0), double(z0))) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) Tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0))))))) COND(false, s(s(z0)), z1) -> c2(LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) S tuples: LOG(s(s(z0)), s(s(z1))) -> c(COND(le(z0, z1), s(s(z0)), z1), LE(s(s(z0)), s(s(z1)))) SQUARE(s(s(z0))) -> c9(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0)))), SQUARE(s(z0))) SQUARE(s(s(z0))) -> c1(PLUS(s(plus(square(z0), double(z0))), s(s(double(z0))))) DOUBLE(s(s(y0))) -> c7(DOUBLE(s(y0))) PLUS(z0, s(s(y1))) -> c12(PLUS(z0, s(y1))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(double(0))))) SQUARE(s(s(s(y0)))) -> c1(DOUBLE(s(s(y0)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(double(s(z0)))))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(0, double(0))), s(s(double(0))))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(double(s(z0)))))) LE(s(s(s(y0))), s(s(s(y1)))) -> c5(LE(s(s(y0)), s(s(y1)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c3(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(s(z0)))) -> c9(PLUS(s(plus(s(plus(square(z0), double(z0))), double(s(z0)))), s(s(s(s(double(z0)))))), SQUARE(s(s(z0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(square(0), 0)), s(s(0)))) SQUARE(s(s(0))) -> c4(PLUS(s(plus(0, double(0))), s(s(0)))) SQUARE(s(s(0))) -> c1(PLUS(s(plus(square(0), 0)), s(s(0)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1))))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) COND(false, s(s(z0)), z1) -> c2(DOUBLE(log(s(s(z0)), s(plus(s(plus(square(z1), double(z1))), double(s(z1)))))), LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) SQUARE(s(s(s(z0)))) -> c1(PLUS(s(plus(square(s(z0)), s(s(double(z0))))), s(s(s(s(double(z0))))))) COND(false, s(s(z0)), z1) -> c2(LOG(s(s(z0)), s(plus(square(s(z1)), s(s(double(z1)))))), SQUARE(s(s(z1)))) K tuples: LOG(s(s(x0)), s(s(z1))) -> c(LE(s(s(x0)), s(s(z1)))) Defined Rule Symbols: log_2, cond_3, le_2, double_1, square_1, plus_2 Defined Pair Symbols: LOG_2, SQUARE_1, DOUBLE_1, PLUS_2, LE_2, COND_3 Compound Symbols: c_2, c9_2, c1_1, c_1, c7_1, c12_1, c5_1, c3_1, c4_1, c2_3, c2_2