KILLED proof of input_7fIA6abg0t.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 18 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 4 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 247 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 84 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 258 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 87 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 250 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 34 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 6181 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 3572 ms] (42) CpxRNTS (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (50) CdtProblem (51) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 111 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 146 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (90) CdtProblem (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (114) CdtProblem (115) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (122) CdtProblem (123) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) 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: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, s(y)) -> 0 minus(s(x), s(y)) -> minus(x, y) plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) mod(s(x), 0) -> 0 mod(x, s(y)) -> help(x, s(y), 0) help(x, s(y), c) -> if(le(c, x), x, s(y), c) if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) if(false, x, s(y), c) -> minus(x, minus(c, s(y))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) minus(x, 0') -> x minus(0', s(y)) -> 0' minus(s(x), s(y)) -> minus(x, y) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) mod(s(x), 0') -> 0' mod(x, s(y)) -> help(x, s(y), 0') help(x, s(y), c) -> if(le(c, x), x, s(y), c) if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) if(false, x, s(y), c) -> minus(x, minus(c, s(y))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) minus(x, 0) -> x minus(0, s(y)) -> 0 minus(s(x), s(y)) -> minus(x, y) plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) mod(s(x), 0) -> 0 mod(x, s(y)) -> help(x, s(y), 0) help(x, s(y), c) -> if(le(c, x), x, s(y), c) if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) if(false, x, s(y), c) -> minus(x, minus(c, s(y))) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] mod(s(x), 0) -> 0 [1] mod(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(le(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) [1] if(false, x, s(y), c) -> minus(x, minus(c, s(y))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] mod(s(x), 0) -> 0 [1] mod(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(le(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) [1] if(false, x, s(y), c) -> minus(x, minus(c, s(y))) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false minus :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s mod :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: mod_2 help_3 if_4 (c) The following functions are completely defined: plus_2 le_2 minus_2 Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] mod(s(x), 0) -> 0 [1] mod(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(le(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) [1] if(false, x, s(y), c) -> minus(x, minus(c, s(y))) [1] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false minus :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s mod :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] mod(s(x), 0) -> 0 [1] mod(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), 0) -> if(true, x, s(y), 0) [2] help(0, s(y), s(x')) -> if(false, 0, s(y), s(x')) [2] help(s(y'), s(y), s(x'')) -> if(le(x'', y'), s(y'), s(y), s(x'')) [2] if(true, x, s(y), c) -> help(x, s(y), s(plus(c, y))) [2] if(false, x, s(y), 0) -> minus(x, 0) [2] if(false, x, s(y), s(x1)) -> minus(x, minus(x1, y)) [2] The TRS has the following type information: le :: 0:s -> 0:s -> true:false 0 :: 0:s true :: true:false s :: 0:s -> 0:s false :: true:false minus :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s mod :: 0:s -> 0:s -> 0:s help :: 0:s -> 0:s -> 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(le(x'', y'), 1 + y', 1 + y, 1 + x'') :|: z' = 1 + y, y >= 0, z = 1 + y', y' >= 0, z'' = 1 + x'', x'' >= 0 help(z, z', z'') -{ 2 }-> if(1, x, 1 + y, 0) :|: z' = 1 + y, z'' = 0, x >= 0, y >= 0, z = x help(z, z', z'') -{ 2 }-> if(0, 0, 1 + y, 1 + x') :|: z' = 1 + y, z'' = 1 + x', y >= 0, x' >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(x, minus(x1, y)) :|: x1 >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y, z1 = 1 + x1, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(x, 0) :|: z1 = 0, z' = x, x >= 0, y >= 0, z'' = 1 + y, z = 0 if(z, z', z'', z1) -{ 2 }-> help(x, 1 + y, 1 + plus(c, y)) :|: z' = x, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z'' = 1 + y le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 mod(z, z') -{ 1 }-> help(x, 1 + y, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 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: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', minus(z1 - 1, z'' - 1)) :|: z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { le } { plus } { help, if } { mod } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', minus(z1 - 1, z'' - 1)) :|: z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {minus}, {le}, {plus}, {help,if}, {mod} ---------------------------------------- (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: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', minus(z1 - 1, z'' - 1)) :|: z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {minus}, {le}, {plus}, {help,if}, {mod} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', minus(z1 - 1, z'' - 1)) :|: z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {minus}, {le}, {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', minus(z1 - 1, z'' - 1)) :|: z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> minus(z', 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {le}, {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [z] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {le}, {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {le}, {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: help(z, z', z'') -{ 2 }-> if(le(z'' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, 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 - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: help(z, z', z'') -{ 3 + z }-> if(s3, 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 le(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: help(z, z', z'') -{ 3 + z }-> if(s3, 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 le(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {plus}, {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: help(z, z', z'') -{ 3 + z }-> if(s3, 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 }-> help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 le(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 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 Function symbols to be analyzed: {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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: help(z, z', z'') -{ 3 + z }-> if(s3, 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 + z'' }-> help(z', 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 le(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s4 :|: s4 >= 0, s4 <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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 CoFloCo for: help after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s3, 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 + z'' }-> help(z', 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 le(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s4 :|: s4 >= 0, s4 <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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'] help: runtime: ?, size: O(n^1) [z] if: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: help after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 3 + z }-> if(s3, 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s3 >= 0, s3 <= 1, z' - 1 >= 0, z - 1 >= 0, z'' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(1, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0 help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 4 }-> s' :|: s' >= 0, s' <= z', z1 = 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 5 + s'' + z'' }-> s1 :|: s'' >= 0, s'' <= z1 - 1, s1 >= 0, s1 <= z', z1 - 1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0 if(z, z', z'', z1) -{ 2 + z'' }-> help(z', 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 0 le(z, z') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 1, z - 1 >= 0, z' - 1 >= 0 le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0 le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 minus(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 mod(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0 mod(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s4 :|: s4 >= 0, s4 <= z + (z' - 1), z >= 0, z' - 1 >= 0 Function symbols to be analyzed: {help,if}, {mod} Previous analysis results are: minus: runtime: O(n^1) [2 + z'], size: O(n^1) [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'] help: runtime: INF, size: O(n^1) [z] if: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (43) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: mod(v0, v1) -> null_mod [0] help(v0, v1, v2) -> null_help [0] if(v0, v1, v2, v3) -> null_if [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] plus(v0, v1) -> null_plus [0] And the following fresh constants: null_mod, null_help, null_if, null_le, null_minus, null_plus ---------------------------------------- (44) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] minus(x, 0) -> x [1] minus(0, s(y)) -> 0 [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] mod(s(x), 0) -> 0 [1] mod(x, s(y)) -> help(x, s(y), 0) [1] help(x, s(y), c) -> if(le(c, x), x, s(y), c) [1] if(true, x, s(y), c) -> help(x, s(y), plus(c, s(y))) [1] if(false, x, s(y), c) -> minus(x, minus(c, s(y))) [1] mod(v0, v1) -> null_mod [0] help(v0, v1, v2) -> null_help [0] if(v0, v1, v2, v3) -> null_if [0] le(v0, v1) -> null_le [0] minus(v0, v1) -> null_minus [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: le :: 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> true:false:null_le 0 :: 0:s:null_mod:null_help:null_if:null_minus:null_plus true :: true:false:null_le s :: 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus false :: true:false:null_le minus :: 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus plus :: 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus mod :: 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus help :: 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus if :: true:false:null_le -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus -> 0:s:null_mod:null_help:null_if:null_minus:null_plus null_mod :: 0:s:null_mod:null_help:null_if:null_minus:null_plus null_help :: 0:s:null_mod:null_help:null_if:null_minus:null_plus null_if :: 0:s:null_mod:null_help:null_if:null_minus:null_plus null_le :: true:false:null_le null_minus :: 0:s:null_mod:null_help:null_if:null_minus:null_plus null_plus :: 0:s:null_mod:null_help:null_if:null_minus:null_plus Rewrite Strategy: INNERMOST ---------------------------------------- (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 null_mod => 0 null_help => 0 null_if => 0 null_le => 0 null_minus => 0 null_plus => 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: help(z, z', z'') -{ 1 }-> if(le(c, x), x, 1 + y, c) :|: z' = 1 + y, c >= 0, x >= 0, y >= 0, z = x, z'' = c help(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 if(z, z', z'', z1) -{ 1 }-> minus(x, minus(c, 1 + y)) :|: z' = x, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z'' = 1 + y if(z, z', z'', z1) -{ 1 }-> help(x, 1 + y, plus(c, 1 + y)) :|: z = 2, z' = x, z1 = c, c >= 0, x >= 0, y >= 0, z'' = 1 + y if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x le(z, z') -{ 1 }-> 2 :|: y >= 0, z = 0, z' = y le(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0 le(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 mod(z, z') -{ 1 }-> help(x, 1 + y, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x mod(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0 mod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) minus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) mod(s(z0), 0) -> 0 mod(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1))) if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1))) Tuples: LE(0, z0) -> c LE(s(z0), 0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(z0, 0) -> c3 MINUS(0, s(z0)) -> c4 MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, 0) -> c6 PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) MOD(s(z0), 0) -> c8 MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1))) S tuples: LE(0, z0) -> c LE(s(z0), 0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(z0, 0) -> c3 MINUS(0, s(z0)) -> c4 MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, 0) -> c6 PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) MOD(s(z0), 0) -> c8 MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1))) K tuples:none Defined Rule Symbols: le_2, minus_2, plus_2, mod_2, help_3, if_4 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, MOD_2, HELP_3, IF_4 Compound Symbols: c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10_2, c11_2, c12_2 ---------------------------------------- (49) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: MOD(z0, s(z1)) -> c9(HELP(z0, s(z1), 0)) Removed 6 trailing nodes: MOD(s(z0), 0) -> c8 PLUS(z0, 0) -> c6 LE(s(z0), 0) -> c1 MINUS(z0, 0) -> c3 LE(0, z0) -> c MINUS(0, s(z0)) -> c4 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) minus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) mod(s(z0), 0) -> 0 mod(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1))) if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c12(MINUS(z0, minus(z2, s(z1))), MINUS(z2, s(z1))) K tuples:none Defined Rule Symbols: le_2, minus_2, plus_2, mod_2, help_3, if_4 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c11_2, c12_2 ---------------------------------------- (51) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) minus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) mod(s(z0), 0) -> 0 mod(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1))) if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1))) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) K tuples:none Defined Rule Symbols: le_2, minus_2, plus_2, mod_2, help_3, if_4 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c11_2, c_1 ---------------------------------------- (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: mod(s(z0), 0) -> 0 mod(z0, s(z1)) -> help(z0, s(z1), 0) help(z0, s(z1), z2) -> if(le(z2, z0), z0, s(z1), z2) if(true, z0, s(z1), z2) -> help(z0, s(z1), plus(z2, s(z1))) if(false, z0, s(z1), z2) -> minus(z0, minus(z2, s(z1))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) K tuples:none Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c11_2, c_1 ---------------------------------------- (55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) We considered the (Usable) Rules: le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(HELP(x_1, x_2, x_3)) = [1] + x_1 + x_2 POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = [1] POL(PLUS(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 POL(true) = [1] ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c11_2, c_1 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HELP(z0, s(z1), z2) -> c10(IF(le(z2, z0), z0, s(z1), z2), LE(z2, z0)) by HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0), LE(0, z0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0)), LE(s(z0), 0)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0), LE(0, z0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0)), LE(s(z0), 0)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0), LE(0, z0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0)), LE(s(z0), 0)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c11_2, c_1, c10_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c11_2, c_1, c10_2, c10_1 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) We considered the (Usable) Rules: plus(z0, s(z1)) -> s(plus(z0, z1)) And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(HELP(x_1, x_2, x_3)) = x_3 + [2]x_1^2 POL(IF(x_1, x_2, x_3, x_4)) = [2]x_2^2 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = 0 POL(le(x_1, x_2)) = 0 POL(minus(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = 0 POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c11_2, c_1, c10_2, c10_1 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) We considered the (Usable) Rules: plus(z0, s(z1)) -> s(plus(z0, z1)) le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [2] POL(HELP(x_1, x_2, x_3)) = [2]x_3 POL(IF(x_1, x_2, x_3, x_4)) = [3] + x_1 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = 0 POL(le(x_1, x_2)) = [1] POL(minus(x_1, x_2)) = [3] POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = [2] POL(true) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c11_2, c_1, c10_2, c10_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, z0, s(z1), z2) -> c11(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) by IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_2, c10_1, c11_2 ---------------------------------------- (67) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: HELP(z0, s(x1), 0) -> c10(IF(true, z0, s(x1), 0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_2, c10_1, c11_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) by IF(false, x0, s(z0), 0) -> c(MINUS(x0, 0)) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z0), 0) -> c(MINUS(x0, 0)) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z0, minus(z2, s(z1)))) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_2, c10_1, c11_2 ---------------------------------------- (71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF(false, x0, s(z0), 0) -> c(MINUS(x0, 0)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_2, c10_1, c11_2 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace HELP(s(z1), s(x1), s(z0)) -> c10(IF(le(z0, z1), s(z1), s(x1), s(z0)), LE(s(z0), s(z1))) by HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(0), s(x1), s(s(z0))) -> c10(IF(false, s(0), s(x1), s(s(z0))), LE(s(s(z0)), s(0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(0), s(x1), s(s(z0))) -> c10(IF(false, s(0), s(x1), s(s(z0))), LE(s(s(z0)), s(0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(0), s(x1), s(s(z0))) -> c10(IF(false, s(0), s(x1), s(s(z0))), LE(s(s(z0)), s(0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_1, c11_2, c10_2 ---------------------------------------- (75) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_1, c11_2, c10_2, c1_1 ---------------------------------------- (77) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(HELP(x_1, x_2, x_3)) = x_2 + x_3 POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_3 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = [1] POL(PLUS(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] POL(true) = [1] ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_1, c11_2, c10_2, c1_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. HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) We considered the (Usable) Rules: le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(HELP(x_1, x_2, x_3)) = x_3 POL(IF(x_1, x_2, x_3, x_4)) = x_1 POL(LE(x_1, x_2)) = 0 POL(MINUS(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c10(x_1, x_2)) = x_1 + x_2 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = 0 POL(le(x_1, x_2)) = [1] POL(minus(x_1, x_2)) = [1] + x_1 + x_2 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] POL(true) = [1] ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_1, c11_2, c10_2, c1_1 ---------------------------------------- (81) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(z1), s(z0)) -> c(MINUS(x0, minus(z0, z1))) by IF(false, x0, s(s(z0)), s(0)) -> c(MINUS(x0, 0)) IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(s(z0)), s(0)) -> c(MINUS(x0, 0)) IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_1, c11_2, c10_2, c1_1 ---------------------------------------- (83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF(false, x0, s(s(z0)), s(0)) -> c(MINUS(x0, 0)) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) K tuples: IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, IF_4, HELP_3 Compound Symbols: c2_1, c5_1, c7_1, c_1, c10_1, c11_2, c10_2, c1_1 ---------------------------------------- (85) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, z0, s(z1), z2) -> c(MINUS(z2, s(z1))) by IF(false, 0, s(x0), s(x1)) -> c(MINUS(s(x1), s(x0))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, 0, s(x0), s(x1)) -> c(MINUS(s(x1), s(x0))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) K tuples: HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, 0, s(x0), s(x1)) -> c(MINUS(s(x1), s(x0))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_1, c11_2, c10_2, c1_1, c_1 ---------------------------------------- (87) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, x0, s(z1), z0) -> c11(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) by IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, 0, s(x0), s(x1)) -> c(MINUS(s(x1), s(x0))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) K tuples: HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, 0, s(x0), s(x1)) -> c(MINUS(s(x1), s(x0))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_1, c10_2, c1_1, c_1, c11_2 ---------------------------------------- (89) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: HELP(0, s(x1), s(z0)) -> c10(IF(false, 0, s(x1), s(z0))) IF(false, 0, s(x0), s(x1)) -> c(MINUS(s(x1), s(x0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c10_1, c1_1, c_1, c11_2 ---------------------------------------- (91) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, x0, s(s(z1)), s(s(z0))) -> c(MINUS(x0, minus(z0, z1))) by IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: LE_2, MINUS_2, PLUS_2, HELP_3, IF_4 Compound Symbols: c2_1, c5_1, c7_1, c10_2, c10_1, c1_1, c_1, c11_2 ---------------------------------------- (93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: MINUS_2, PLUS_2, HELP_3, IF_4, LE_2 Compound Symbols: c5_1, c7_1, c10_2, c10_1, c1_1, c_1, c11_2, c2_1 ---------------------------------------- (95) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: HELP(s(0), s(x1), s(s(z0))) -> c1(LE(s(s(z0)), s(0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0)), LE(s(0), s(z0))) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: MINUS_2, PLUS_2, HELP_3, IF_4, LE_2 Compound Symbols: c5_1, c7_1, c10_2, c10_1, c1_1, c_1, c11_2, c2_1 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: MINUS_2, PLUS_2, HELP_3, IF_4, LE_2 Compound Symbols: c5_1, c7_1, c10_2, c10_1, c1_1, c_1, c11_2, c2_1 ---------------------------------------- (99) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, x0, s(0), s(z0)) -> c(MINUS(x0, z0)) by IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) IF(false, s(0), s(0), s(s(x1))) -> c(MINUS(s(0), s(x1))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) IF(false, s(0), s(0), s(s(x1))) -> c(MINUS(s(0), s(x1))) S tuples: MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: MINUS_2, PLUS_2, HELP_3, IF_4, LE_2 Compound Symbols: c5_1, c7_1, c10_2, c10_1, c1_1, c_1, c11_2, c2_1 ---------------------------------------- (101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(z0), s(z1)) -> c5(MINUS(z0, z1)) by MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) IF(false, s(0), s(0), s(s(x1))) -> c(MINUS(s(0), s(x1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) S tuples: PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: PLUS_2, HELP_3, IF_4, LE_2, MINUS_2 Compound Symbols: c7_1, c10_2, c10_1, c1_1, c_1, c11_2, c2_1, c5_1 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: IF(false, s(0), s(s(z1)), s(s(x1))) -> c(MINUS(s(0), minus(x1, z1))) IF(false, s(0), s(0), s(s(x1))) -> c(MINUS(s(0), s(x1))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) S tuples: PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: PLUS_2, HELP_3, IF_4, LE_2, MINUS_2 Compound Symbols: c7_1, c10_2, c10_1, c1_1, c_1, c11_2, c2_1, c5_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(z1)) -> c7(PLUS(z0, z1)) by PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) K tuples: HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c10_1, c1_1, c_1, c11_2, c2_1, c5_1, c7_1 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(x0), s(x1), s(x2)) -> c10(LE(s(x2), s(x0))) by HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c11_2, c2_1, c10_1, c5_1, c7_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, s(x0), s(x1), s(0)) -> c11(HELP(s(x0), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) by IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0))), PLUS(s(0), s(0))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0))), PLUS(s(0), s(0))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0))), PLUS(s(0), s(0))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c11_2, c2_1, c10_1, c5_1, c7_1 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0)))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c11_2, c2_1, c10_1, c5_1, c7_1, c11_1 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(true, s(s(x0)), s(x1), s(s(x2))) -> c11(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) by IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c2_1, c10_1, c5_1, c7_1, c11_2, c11_1 ---------------------------------------- (115) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: HELP(s(z0), s(x1), s(0)) -> c10(IF(true, s(z0), s(x1), s(0))) IF(true, s(x0), s(0), s(0)) -> c11(HELP(s(x0), s(0), s(s(0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c2_1, c5_1, c7_1, c10_1, c11_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(x0), s(s(z1)), s(0)) -> c11(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1 ---------------------------------------- (119) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(PLUS(s(0), s(s(z1)))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(PLUS(s(0), s(s(z1)))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1 ---------------------------------------- (121) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF(true, s(x0), s(s(z1)), s(0)) -> c3(PLUS(s(0), s(s(z1)))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1 ---------------------------------------- (123) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) S tuples: HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c10_2, c1_1, c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1 ---------------------------------------- (125) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(z1)), s(x1), s(s(z0))) -> c10(IF(le(z0, z1), s(s(z1)), s(x1), s(s(z0))), LE(s(s(z0)), s(s(z1)))) by HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) K tuples: HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: HELP_3, IF_4, LE_2, MINUS_2, PLUS_2 Compound Symbols: c1_1, c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2 ---------------------------------------- (127) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(0), s(x1), s(s(z0))) -> c1(IF(false, s(0), s(x1), s(s(z0)))) by HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) K tuples: IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, LE_2, MINUS_2, PLUS_2, HELP_3 Compound Symbols: c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2, c1_1 ---------------------------------------- (129) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(x0)), s(x1), s(s(x2))) -> c(MINUS(s(s(x2)), s(x1))) by IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(s(x1))), s(0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(s(x1))), s(0))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) K tuples: IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(s(x1))), s(0))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, LE_2, MINUS_2, PLUS_2, HELP_3 Compound Symbols: c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2, c1_1 ---------------------------------------- (131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(s(x1))), s(0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) K tuples: IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, LE_2, MINUS_2, PLUS_2, HELP_3 Compound Symbols: c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2, c1_1 ---------------------------------------- (133) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(0), s(x0), s(s(x1))) -> c(MINUS(s(s(x1)), s(x0))) by IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) S tuples: LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) K tuples: HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, LE_2, MINUS_2, PLUS_2, HELP_3 Compound Symbols: c_1, c2_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2, c1_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) by LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, MINUS_2, PLUS_2, HELP_3, LE_2 Compound Symbols: c_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2, c1_1, c2_1 ---------------------------------------- (137) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(x0)), s(0), s(s(x2))) -> c(MINUS(s(s(x0)), s(x2))) by IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) S tuples: MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, MINUS_2, PLUS_2, HELP_3, LE_2 Compound Symbols: c_1, c5_1, c7_1, c10_1, c11_2, c11_1, c3_1, c10_2, c1_1, c2_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(y1)), s(z1), s(s(y0))) -> c10(LE(s(s(y0)), s(s(y1)))) by HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) S tuples: MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, MINUS_2, PLUS_2, HELP_3, LE_2 Compound Symbols: c_1, c5_1, c7_1, c11_2, c11_1, c3_1, c10_2, c1_1, c2_1, c10_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MINUS(s(s(y0)), s(s(y1))) -> c5(MINUS(s(y0), s(y1))) by MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) S tuples: PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, PLUS_2, HELP_3, LE_2, MINUS_2 Compound Symbols: c_1, c7_1, c11_2, c11_1, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1 ---------------------------------------- (143) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(true, s(s(x0)), s(0), s(s(x2))) -> c11(HELP(s(s(x0)), s(0), s(s(s(x2))))) by IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) S tuples: PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, PLUS_2, HELP_3, LE_2, MINUS_2 Compound Symbols: c_1, c7_1, c11_2, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1, c11_1 ---------------------------------------- (145) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(IF(le(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LE(s(s(s(x1))), s(s(x0)))) by HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) S tuples: PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, PLUS_2, HELP_3, LE_2, MINUS_2 Compound Symbols: c_1, c7_1, c11_2, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1, c11_1 ---------------------------------------- (147) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c10(LE(s(s(s(x1))), s(s(x0)))) by HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) S tuples: PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, PLUS_2, HELP_3, LE_2, MINUS_2 Compound Symbols: c_1, c7_1, c11_2, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1, c11_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c7(PLUS(z0, s(y1))) by PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1, c11_1, c7_1 ---------------------------------------- (151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(x0)), s(s(x1)), s(s(x2))) -> c(MINUS(s(s(x2)), s(s(x1)))) by IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1, c11_1, c7_1 ---------------------------------------- (153) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(0), s(s(x0)), s(s(x1))) -> c(MINUS(s(s(x1)), s(s(x0)))) by IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c1_1, c2_1, c10_1, c5_1, c11_1, c7_1 ---------------------------------------- (155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(0), s(s(x1)), s(s(y0))) -> c1(IF(false, s(0), s(s(x1)), s(s(y0)))) by HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c2_1, c10_1, c5_1, c11_1, c7_1, c1_1 ---------------------------------------- (157) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace IF(false, s(s(x0)), s(0), s(s(s(x1)))) -> c(MINUS(s(s(x0)), s(s(x1)))) by IF(false, s(s(s(y0))), s(0), s(s(s(s(y1))))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) IF(false, s(s(s(y0))), s(0), s(s(s(s(y1))))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c2_1, c10_1, c5_1, c11_1, c7_1, c1_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(LE(s(s(y0)), s(s(x0)))) by HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c10(LE(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) IF(false, s(s(s(y0))), s(0), s(s(s(s(y1))))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c10(LE(s(s(s(y0))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c10(LE(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c2_1, c5_1, c11_1, c10_1, c7_1, c1_1 ---------------------------------------- (161) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(LE(s(s(s(s(x1)))), s(s(x0)))) by HELP(s(s(s(y1))), s(0), s(s(s(s(z1))))) -> c10(LE(s(s(s(s(z1)))), s(s(s(y1))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(z0, s(z1)) -> s(plus(z0, z1)) plus(z0, 0) -> z0 minus(0, s(z0)) -> 0 minus(s(z0), s(z1)) -> minus(z0, z1) minus(z0, 0) -> z0 Tuples: IF(false, s(s(x0)), s(s(z1)), s(s(x2))) -> c(MINUS(s(s(x0)), minus(x2, z1))) IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) IF(false, s(s(s(y0))), s(0), s(s(s(s(y1))))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c10(LE(s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(0), s(s(s(s(z1))))) -> c10(LE(s(s(s(s(z1)))), s(s(s(y1))))) S tuples: IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c11(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1)))) HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c10(IF(le(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LE(s(s(y0)), s(s(x0)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) MINUS(s(s(s(y0))), s(s(s(y1)))) -> c5(MINUS(s(s(y0)), s(s(y1)))) IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c11(HELP(s(s(x0)), s(0), s(s(s(s(x1)))))) HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c10(IF(le(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(x0)))) PLUS(z0, s(s(s(y1)))) -> c7(PLUS(z0, s(s(y1)))) K tuples: IF(true, s(x0), s(s(z1)), s(0)) -> c3(HELP(s(x0), s(s(z1)), s(s(plus(s(0), z1))))) IF(false, s(s(z0)), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) IF(false, s(0), s(s(s(y1))), s(s(s(y0)))) -> c(MINUS(s(s(s(y0))), s(s(s(y1))))) HELP(s(0), s(s(s(y0))), s(s(s(y1)))) -> c1(IF(false, s(0), s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c10(LE(s(s(s(y0))), s(s(s(y1))))) HELP(s(s(s(y1))), s(0), s(s(s(s(z1))))) -> c10(LE(s(s(s(s(z1)))), s(s(s(y1))))) Defined Rule Symbols: le_2, plus_2, minus_2 Defined Pair Symbols: IF_4, HELP_3, LE_2, MINUS_2, PLUS_2 Compound Symbols: c_1, c11_2, c3_1, c10_2, c2_1, c5_1, c11_1, c7_1, c1_1, c10_1