KILLED proof of input_IFrZc9OGDO.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) CompletionProof [UPPER BOUND(ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 314 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 130 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 19 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 241 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 93 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 385 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 167 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 13.9 s] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 9706 ms] (54) CpxRNTS (55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 136 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (100) CdtProblem (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 14 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 10 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 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) CdtInstantiationProof [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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (180) CdtProblem (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (196) 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: table -> gen(s(0)) gen(x) -> if1(le(x, 10), x) if1(false, x) -> nil if1(true, x) -> if2(x, x) if2(x, y) -> if3(le(y, 10), x, y) if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) if3(false, x, y) -> gen(s(x)) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(x, y)) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) 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: table -> gen(s(0')) gen(x) -> if1(le(x, 10'), x) if1(false, x) -> nil if1(true, x) -> if2(x, x) if2(x, y) -> if3(le(y, 10'), x, y) if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) if3(false, x, y) -> gen(s(x)) le(0', y) -> true le(s(x), 0') -> false le(s(x), s(y)) -> le(x, y) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) times(0', y) -> 0' times(s(x), y) -> plus(y, times(x, y)) 10' -> s(s(s(s(s(s(s(s(s(s(0')))))))))) 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: table -> gen(s(0)) gen(x) -> if1(le(x, 10), x) if1(false, x) -> nil if1(true, x) -> if2(x, x) if2(x, y) -> if3(le(y, 10), x, y) if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) if3(false, x, y) -> gen(s(x)) le(0, y) -> true le(s(x), 0) -> false le(s(x), s(y)) -> le(x, y) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) times(0, y) -> 0 times(s(x), y) -> plus(y, times(x, y)) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) 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: table -> gen(s(0)) [1] gen(x) -> if1(le(x, 10), x) [1] if1(false, x) -> nil [1] if1(true, x) -> if2(x, x) [1] if2(x, y) -> if3(le(y, 10), x, y) [1] if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) [1] if3(false, x, y) -> gen(s(x)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: 10 => 10' ---------------------------------------- (8) 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: table -> gen(s(0)) [1] gen(x) -> if1(le(x, 10'), x) [1] if1(false, x) -> nil [1] if1(true, x) -> if2(x, x) [1] if2(x, y) -> if3(le(y, 10'), x, y) [1] if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) [1] if3(false, x, y) -> gen(s(x)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] 10' -> s(s(s(s(s(s(s(s(s(s(0)))))))))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: table -> gen(s(0)) [1] gen(x) -> if1(le(x, 10'), x) [1] if1(false, x) -> nil [1] if1(true, x) -> if2(x, x) [1] if2(x, y) -> if3(le(y, 10'), x, y) [1] if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) [1] if3(false, x, y) -> gen(s(x)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] 10' -> s(s(s(s(s(s(s(s(s(s(0)))))))))) [1] The TRS has the following type information: table :: nil:cons gen :: 0:s -> nil:cons s :: 0:s -> 0:s 0 :: 0:s if1 :: false:true -> 0:s -> nil:cons le :: 0:s -> 0:s -> false:true 10' :: 0:s false :: false:true nil :: nil:cons true :: false:true if2 :: 0:s -> 0:s -> nil:cons if3 :: false:true -> 0:s -> 0:s -> nil:cons cons :: entry -> nil:cons -> nil:cons entry :: 0:s -> 0:s -> 0:s -> entry times :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: const ---------------------------------------- (12) 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: table -> gen(s(0)) [1] gen(x) -> if1(le(x, 10'), x) [1] if1(false, x) -> nil [1] if1(true, x) -> if2(x, x) [1] if2(x, y) -> if3(le(y, 10'), x, y) [1] if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) [1] if3(false, x, y) -> gen(s(x)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] 10' -> s(s(s(s(s(s(s(s(s(s(0)))))))))) [1] The TRS has the following type information: table :: nil:cons gen :: 0:s -> nil:cons s :: 0:s -> 0:s 0 :: 0:s if1 :: false:true -> 0:s -> nil:cons le :: 0:s -> 0:s -> false:true 10' :: 0:s false :: false:true nil :: nil:cons true :: false:true if2 :: 0:s -> 0:s -> nil:cons if3 :: false:true -> 0:s -> 0:s -> nil:cons cons :: entry -> nil:cons -> nil:cons entry :: 0:s -> 0:s -> 0:s -> entry times :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s const :: entry 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 false => 0 nil => 0 true => 1 const => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 1 }-> if1(le(x, 10'), x) :|: x >= 0, z = x if1(z, z') -{ 1 }-> if2(x, x) :|: z' = x, z = 1, x >= 0 if1(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 if2(z, z') -{ 1 }-> if3(le(y, 10'), x, y) :|: x >= 0, y >= 0, z = x, z' = y if3(z, z', z'') -{ 1 }-> gen(1 + x) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + x + y + times(x, y)) + if2(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 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 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 1 }-> plus(y, times(x, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (15) 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: table gen_1 if1_2 if2_2 if3_3 (c) The following functions are completely defined: times_2 le_2 10' plus_2 Due to the following rules being added: none And the following fresh constants: const ---------------------------------------- (16) 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: table -> gen(s(0)) [1] gen(x) -> if1(le(x, 10'), x) [1] if1(false, x) -> nil [1] if1(true, x) -> if2(x, x) [1] if2(x, y) -> if3(le(y, 10'), x, y) [1] if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) [1] if3(false, x, y) -> gen(s(x)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(x), y) -> plus(y, times(x, y)) [1] 10' -> s(s(s(s(s(s(s(s(s(s(0)))))))))) [1] The TRS has the following type information: table :: nil:cons gen :: 0:s -> nil:cons s :: 0:s -> 0:s 0 :: 0:s if1 :: false:true -> 0:s -> nil:cons le :: 0:s -> 0:s -> false:true 10' :: 0:s false :: false:true nil :: nil:cons true :: false:true if2 :: 0:s -> 0:s -> nil:cons if3 :: false:true -> 0:s -> 0:s -> nil:cons cons :: entry -> nil:cons -> nil:cons entry :: 0:s -> 0:s -> 0:s -> entry times :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s const :: entry Rewrite Strategy: INNERMOST ---------------------------------------- (17) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (18) 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: table -> gen(s(0)) [1] gen(x) -> if1(le(x, s(s(s(s(s(s(s(s(s(s(0))))))))))), x) [2] if1(false, x) -> nil [1] if1(true, x) -> if2(x, x) [1] if2(x, y) -> if3(le(y, s(s(s(s(s(s(s(s(s(s(0))))))))))), x, y) [2] if3(true, x, y) -> cons(entry(x, y, times(x, y)), if2(x, s(y))) [1] if3(false, x, y) -> gen(s(x)) [1] le(0, y) -> true [1] le(s(x), 0) -> false [1] le(s(x), s(y)) -> le(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(s(0), y) -> plus(y, 0) [2] times(s(s(x')), y) -> plus(y, plus(y, times(x', y))) [2] 10' -> s(s(s(s(s(s(s(s(s(s(0)))))))))) [1] The TRS has the following type information: table :: nil:cons gen :: 0:s -> nil:cons s :: 0:s -> 0:s 0 :: 0:s if1 :: false:true -> 0:s -> nil:cons le :: 0:s -> 0:s -> false:true 10' :: 0:s false :: false:true nil :: nil:cons true :: false:true if2 :: 0:s -> 0:s -> nil:cons if3 :: false:true -> 0:s -> 0:s -> nil:cons cons :: entry -> nil:cons -> nil:cons entry :: 0:s -> 0:s -> 0:s -> entry times :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s const :: entry Rewrite Strategy: INNERMOST ---------------------------------------- (19) 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 false => 0 nil => 0 true => 1 const => 0 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 2 }-> if1(le(x, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), x) :|: x >= 0, z = x if1(z, z') -{ 1 }-> if2(x, x) :|: z' = x, z = 1, x >= 0 if1(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0 if2(z, z') -{ 2 }-> if3(le(y, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), x, y) :|: x >= 0, y >= 0, z = x, z' = y if3(z, z', z'') -{ 1 }-> gen(1 + x) :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + x + y + times(x, y)) + if2(x, 1 + y) :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0 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 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y times(z, z') -{ 2 }-> plus(y, 0) :|: z = 1 + 0, y >= 0, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 2 }-> if1(le(z, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z) :|: z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 2 }-> if3(le(z', 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z, z') :|: z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { 10' } { plus } { times } { if2, gen, if1, if3 } { table } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 2 }-> if1(le(z, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z) :|: z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 2 }-> if3(le(z', 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z, z') :|: z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {le}, {10'}, {plus}, {times}, {if2,gen,if1,if3}, {table} ---------------------------------------- (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: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 2 }-> if1(le(z, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z) :|: z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 2 }-> if3(le(z', 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z, z') :|: z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {le}, {10'}, {plus}, {times}, {if2,gen,if1,if3}, {table} ---------------------------------------- (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: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 2 }-> if1(le(z, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z) :|: z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 2 }-> if3(le(z', 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z, z') :|: z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {le}, {10'}, {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: 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: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 2 }-> if1(le(z, 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z) :|: z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 2 }-> if3(le(z', 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))), z, z') :|: z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {10'}, {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: 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: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {10'}, {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: 10' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 10 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {10'}, {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: ?, size: O(1) [10] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: 10' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] ---------------------------------------- (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: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] ---------------------------------------- (39) 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' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {plus}, {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (41) 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 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', 0) :|: z = 1 + 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= z' + 0, z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + z' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= z' + 0, z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {times}, {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [2*z*z' + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 4*z + 2*z*z' + z' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 1 }-> 1 + (1 + z' + z'' + times(z', z'')) + if2(z', 1 + z'') :|: z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= z' + 0, z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(z', plus(z', times(z - 2, z'))) :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 5 + 4*z' + 2*z'*z'' + z'' }-> 1 + (1 + z' + z'' + s3) + if2(z', 1 + z'') :|: s3 >= 0, s3 <= z'' + 2 * (z'' * z'), z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= z' + 0, z = 1 + 0, z' >= 0 times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s6 :|: s4 >= 0, s4 <= z' + 2 * (z' * (z - 2)), s5 >= 0, s5 <= z' + s4, s6 >= 0, s6 <= z' + s5, z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: gen after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if1 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: if3 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 5 + 4*z' + 2*z'*z'' + z'' }-> 1 + (1 + z' + z'' + s3) + if2(z', 1 + z'') :|: s3 >= 0, s3 <= z'' + 2 * (z'' * z'), z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= z' + 0, z = 1 + 0, z' >= 0 times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s6 :|: s4 >= 0, s4 <= z' + 2 * (z' * (z - 2)), s5 >= 0, s5 <= z' + s4, s6 >= 0, s6 <= z' + s5, z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] if2: runtime: ?, size: INF gen: runtime: ?, size: INF if1: runtime: ?, size: INF if3: runtime: ?, size: INF ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if2 after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: 10' -{ 1 }-> 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) :|: gen(z) -{ 14 }-> if1(s, z) :|: s >= 0, s <= 1, z >= 0 if1(z, z') -{ 1 }-> if2(z', z') :|: z = 1, z' >= 0 if1(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 if2(z, z') -{ 14 }-> if3(s', z, z') :|: s' >= 0, s' <= 1, z >= 0, z' >= 0 if3(z, z', z'') -{ 1 }-> gen(1 + z') :|: z' >= 0, z'' >= 0, z = 0 if3(z, z', z'') -{ 5 + 4*z' + 2*z'*z'' + z'' }-> 1 + (1 + z' + z'' + s3) + if2(z', 1 + z'') :|: s3 >= 0, s3 <= z'' + 2 * (z'' * z'), z = 1, z' >= 0, z'' >= 0 le(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 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 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0 table -{ 1 }-> gen(1 + 0) :|: times(z, z') -{ 3 + z' }-> s2 :|: s2 >= 0, s2 <= z' + 0, z = 1 + 0, z' >= 0 times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s6 :|: s4 >= 0, s4 <= z' + 2 * (z' * (z - 2)), s5 >= 0, s5 <= z' + s4, s6 >= 0, s6 <= z' + s5, z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 Function symbols to be analyzed: {if2,gen,if1,if3}, {table} Previous analysis results are: le: runtime: O(n^1) [2 + z'], size: O(1) [1] 10': runtime: O(1) [1], size: O(1) [10] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] if2: runtime: INF, size: INF gen: runtime: ?, size: INF if1: runtime: ?, size: INF if3: runtime: ?, size: INF ---------------------------------------- (55) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: table -> gen(s(0)) gen(z0) -> if1(le(z0, 10), z0) if1(false, z0) -> nil if1(true, z0) -> if2(z0, z0) if2(z0, z1) -> if3(le(z1, 10), z0, z1) if3(true, z0, z1) -> cons(entry(z0, z1, times(z0, z1)), if2(z0, s(z1))) if3(false, z0, z1) -> gen(s(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: TABLE -> c(GEN(s(0))) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10), 10') IF1(false, z0) -> c2 IF1(true, z0) -> c3(IF2(z0, z0)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10), 10') IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(0, z0) -> c8 LE(s(z0), 0) -> c9 LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(0, z0) -> c11 PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(0, z0) -> c13 TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 10' -> c15 S tuples: TABLE -> c(GEN(s(0))) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10), 10') IF1(false, z0) -> c2 IF1(true, z0) -> c3(IF2(z0, z0)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10), 10') IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(0, z0) -> c8 LE(s(z0), 0) -> c9 LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(0, z0) -> c11 PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(0, z0) -> c13 TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) 10' -> c15 K tuples:none Defined Rule Symbols: table, gen_1, if1_2, if2_2, if3_3, le_2, plus_2, times_2, 10 Defined Pair Symbols: TABLE, GEN_1, IF1_2, IF2_2, IF3_3, LE_2, PLUS_2, TIMES_2, 10' Compound Symbols: c_1, c1_3, c2, c3_1, c4_3, c5_1, c6_1, c7_1, c8, c9, c10_1, c11, c12_1, c13, c14_2, c15 ---------------------------------------- (57) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: TABLE -> c(GEN(s(0))) Removed 6 trailing nodes: PLUS(0, z0) -> c11 IF1(false, z0) -> c2 LE(0, z0) -> c8 10' -> c15 LE(s(z0), 0) -> c9 TIMES(0, z0) -> c13 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: table -> gen(s(0)) gen(z0) -> if1(le(z0, 10), z0) if1(false, z0) -> nil if1(true, z0) -> if2(z0, z0) if2(z0, z1) -> if3(le(z1, 10), z0, z1) if3(true, z0, z1) -> cons(entry(z0, z1, times(z0, z1)), if2(z0, s(z1))) if3(false, z0, z1) -> gen(s(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10), 10') IF1(true, z0) -> c3(IF2(z0, z0)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10), 10') IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) S tuples: GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10), 10') IF1(true, z0) -> c3(IF2(z0, z0)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10), 10') IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) K tuples:none Defined Rule Symbols: table, gen_1, if1_2, if2_2, if3_3, le_2, plus_2, times_2, 10 Defined Pair Symbols: GEN_1, IF1_2, IF2_2, IF3_3, LE_2, PLUS_2, TIMES_2 Compound Symbols: c1_3, c3_1, c4_3, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: table -> gen(s(0)) gen(z0) -> if1(le(z0, 10), z0) if1(false, z0) -> nil if1(true, z0) -> if2(z0, z0) if2(z0, z1) -> if3(le(z1, 10), z0, z1) if3(true, z0, z1) -> cons(entry(z0, z1, times(z0, z1)), if2(z0, s(z1))) if3(false, z0, z1) -> gen(s(z0)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) K tuples:none Defined Rule Symbols: table, gen_1, if1_2, if2_2, if3_3, le_2, plus_2, times_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF2_2 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c4_2 ---------------------------------------- (61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: table -> gen(s(0)) gen(z0) -> if1(le(z0, 10), z0) if1(false, z0) -> nil if1(true, z0) -> if2(z0, z0) if2(z0, z1) -> if3(le(z1, 10), z0, z1) if3(true, z0, z1) -> cons(entry(z0, z1, times(z0, z1)), if2(z0, s(z1))) if3(false, z0, z1) -> gen(s(z0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF2_2 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c4_2 ---------------------------------------- (63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF3(true, z0, z1) -> c5(TIMES(z0, z1)) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(10) = 0 POL(GEN(x_1)) = [1] + x_1 POL(IF1(x_1, x_2)) = [1] + x_2 POL(IF2(x_1, x_2)) = [1] + x_1 POL(IF3(x_1, x_2, x_3)) = [1] + x_2 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 POL(true) = [1] ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF2_2 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c4_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GEN(z0) -> c1(IF1(le(z0, 10), z0), LE(z0, 10)) by GEN(0) -> c1(IF1(true, 0), LE(0, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) GEN(0) -> c1(IF1(true, 0), LE(0, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) GEN(0) -> c1(IF1(true, 0), LE(0, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c1_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) GEN(0) -> c1(IF1(true, 0)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) GEN(0) -> c1(IF1(true, 0)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c1_2, c1_1 ---------------------------------------- (69) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GEN(0) -> c1(IF1(true, 0)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c1_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(z0, z1) -> c4(IF3(le(z1, 10), z0, z1), LE(z1, 10)) by IF2(x0, 0) -> c4(IF3(true, x0, 0), LE(0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0), LE(0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0), LE(0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF2_2 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c4_2 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF2_2 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c4_2, c4_1 ---------------------------------------- (75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(x0, 0) -> c4(IF3(true, x0, 0)) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(10) = [1] POL(GEN(x_1)) = x_1 POL(IF1(x_1, x_2)) = x_2 POL(IF2(x_1, x_2)) = x_2 POL(IF3(x_1, x_2, x_3)) = 0 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_2 POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 POL(true) = [1] ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF2_2 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c4_2, c4_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GEN(x0) -> c1(IF1(le(x0, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0), LE(x0, 10)) by GEN(0) -> c1(IF1(true, 0), LE(0, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(0) -> c1(IF1(true, 0), LE(0, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) GEN(0) -> c1(IF1(true, 0), LE(0, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c4_1, c1_2, c1_1 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) GEN(0) -> c1(IF1(true, 0)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) GEN(0) -> c1(IF1(true, 0)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c4_1, c1_2, c1_1 ---------------------------------------- (81) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: GEN(0) -> c1(IF1(true, 0)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c4_1, c1_2, c1_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GEN(x0) -> c1(LE(x0, 10)) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(10) = [3] POL(GEN(x_1)) = [1] + x_1 POL(IF1(x_1, x_2)) = [2] POL(IF2(x_1, x_2)) = [2] POL(IF3(x_1, x_2, x_3)) = [2] POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [2] POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [3] POL(le(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = [3] + [3]x_1 POL(s(x_1)) = [1] POL(times(x_1, x_2)) = [3] + [3]x_1 + [3]x_2 POL(true) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c4_1, c1_2, c1_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(x0, x1) -> c4(IF3(le(x1, s(s(s(s(s(s(s(s(s(s(0))))))))))), x0, x1), LE(x1, 10)) by IF2(x0, 0) -> c4(IF3(true, x0, 0), LE(0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0), LE(0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0), LE(0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c1_2, c1_1, c4_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c1_2, c1_1, c4_2 ---------------------------------------- (89) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF2(x0, 0) -> c4(IF3(true, x0, 0)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c1_2, c1_1, c4_2 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(x0, x1) -> c4(LE(x1, 10)) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(10) = [1] POL(GEN(x_1)) = [1] + x_1 POL(IF1(x_1, x_2)) = [1] POL(IF2(x_1, x_2)) = [1] POL(IF3(x_1, x_2, x_3)) = [1] + x_3 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 POL(true) = [1] ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, x1) -> c4(LE(x1, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c1_2, c1_1, c4_2 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), 10)) by GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, x1) -> c4(LE(x1, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c1_1, c4_2, c1_2 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace GEN(x0) -> c1(LE(x0, 10)) by GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF1(true, z0) -> c3(IF2(z0, z0)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, x1) -> c4(LE(x1, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF1_2, IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1 Compound Symbols: c3_1, c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c4_2, c1_2, c1_1 ---------------------------------------- (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, z0) -> c3(IF2(z0, z0)) by IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, 0) -> c4(IF3(true, x0, 0)) GEN(x0) -> c1(LE(x0, 10)) IF2(x0, x1) -> c4(LE(x1, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1, IF1_2 Compound Symbols: c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c4_2, c1_2, c1_1, c3_1 ---------------------------------------- (99) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: IF2(x0, 0) -> c4(IF3(true, x0, 0)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) K tuples: IF3(true, z0, z1) -> c5(TIMES(z0, z1)) IF2(x0, x1) -> c4(LE(x1, 10)) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1, IF1_2 Compound Symbols: c5_1, c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c4_1, c1_2, c1_1, c3_1 ---------------------------------------- (101) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, z0, z1) -> c5(TIMES(z0, z1)) by IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1, IF1_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c4_2, c4_1, c1_2, c1_1, c3_1, c5_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(x0, s(z0)) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), x0, s(z0)), LE(s(z0), 10)) by IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1, IF1_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c4_1, c1_2, c1_1, c3_1, c5_1, c4_2 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(x0, x1) -> c4(LE(x1, 10)) by IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), 0) -> false le(s(z0), s(z1)) -> le(z0, z1) 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) Defined Rule Symbols: times_2, plus_2, le_2, 10 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c1_1, c3_1, c5_1, c4_2, c4_1 ---------------------------------------- (107) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: 10 -> s(s(s(s(s(s(s(s(s(s(0)))))))))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_2, c1_1, c3_1, c5_1, c4_2, c4_1 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GEN(s(z0)) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(s(0)))))))))), s(z0)), LE(s(z0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_2, c4_1, c1_2 ---------------------------------------- (111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) We considered the (Usable) Rules:none And the Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(GEN(x_1)) = [1] POL(IF1(x_1, x_2)) = x_2 POL(IF2(x_1, x_2)) = x_2 POL(IF3(x_1, x_2, x_3)) = [1] POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_2, c4_1, c1_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(z0, s(z1)) -> c4(IF3(le(z1, s(s(s(s(s(s(s(s(s(0)))))))))), z0, s(z1)), LE(s(z1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) 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: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(GEN(x_1)) = [1] POL(IF1(x_1, x_2)) = [1] + x_2 POL(IF2(x_1, x_2)) = [1] POL(IF3(x_1, x_2, x_3)) = x_1 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 POL(true) = [1] ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace GEN(s(s(z0))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(s(0))))))))), s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2 ---------------------------------------- (119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) 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: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(GEN(x_1)) = [1] POL(IF1(x_1, x_2)) = [1] + x_2 POL(IF2(x_1, x_2)) = [1] POL(IF3(x_1, x_2, x_3)) = x_1 POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 POL(true) = [1] ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2 ---------------------------------------- (121) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF2(x0, s(s(z0))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(s(0))))))))), x0, s(s(z0))), LE(s(s(z0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2 ---------------------------------------- (123) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) We considered the (Usable) Rules:none And the Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(GEN(x_1)) = [1] POL(IF1(x_1, x_2)) = [1] + x_2 POL(IF2(x_1, x_2)) = [1] POL(IF3(x_1, x_2, x_3)) = [1] POL(LE(x_1, x_2)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = [1] + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c14(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c6(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(false) = [1] POL(le(x_1, x_2)) = [1] POL(plus(x_1, x_2)) = x_2 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = [1] + x_2 POL(true) = [1] ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c6_1, c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2 ---------------------------------------- (125) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, z0, z1) -> c6(IF2(z0, s(z1))) by IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) S tuples: IF3(false, z0, z1) -> c7(GEN(s(z0))) LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: IF3_3, LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF2_2 Compound Symbols: c7_1, c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2, c6_1 ---------------------------------------- (127) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(false, z0, z1) -> c7(GEN(s(z0))) by IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) S tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, GEN_1, IF1_2, IF3_3, IF2_2 Compound Symbols: c10_1, c12_1, c14_2, c1_1, c3_1, c5_1, c4_1, c1_2, c4_2, c6_1, c7_1 ---------------------------------------- (129) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace GEN(z0) -> c1(LE(z0, s(s(s(s(s(s(s(s(s(s(0)))))))))))) by GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) S tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF1_2, IF3_3, IF2_2, GEN_1 Compound Symbols: c10_1, c12_1, c14_2, c3_1, c5_1, c4_1, c1_2, c1_1, c4_2, c6_1, c7_1 ---------------------------------------- (131) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF1(true, s(x0)) -> c3(IF2(s(x0), s(x0))) by IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) S tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF3_3, IF2_2, GEN_1, IF1_2 Compound Symbols: c10_1, c12_1, c14_2, c5_1, c4_1, c1_2, c1_1, c4_2, c6_1, c7_1, c3_1 ---------------------------------------- (133) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(x1)) -> c5(TIMES(x0, s(x1))) by IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) S tuples: LE(s(z0), s(z1)) -> c10(LE(z0, z1)) PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF2_2, GEN_1, IF3_3, IF1_2 Compound Symbols: c10_1, c12_1, c14_2, c4_1, c1_2, c1_1, c4_2, c6_1, c7_1, c3_1, c5_1 ---------------------------------------- (135) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c10(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) S tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(0)) -> c1(IF1(true, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0)), LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF2_2, GEN_1, IF3_3, IF1_2, LE_2 Compound Symbols: c12_1, c14_2, c4_1, c1_2, c1_1, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) S tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF2_2, GEN_1, IF3_3, IF1_2, LE_2 Compound Symbols: c12_1, c14_2, c4_1, c1_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(z0, z1) -> c4(LE(z1, s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) K tuples: IF2(x0, x1) -> c4(LE(x1, 10)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2 Compound Symbols: c12_1, c14_2, c1_1, c4_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) K tuples: GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2 Compound Symbols: c12_1, c14_2, c1_1, c4_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), z1) -> c12(PLUS(z0, z1)) by PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) K tuples: GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c4_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1, c12_1 ---------------------------------------- (145) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(x1)) -> c4(LE(s(x1), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) K tuples: GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c4_1, c6_1, c7_1, c3_1, c5_1, c10_1, c12_1 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF2(s(0), s(0)) -> c4(LE(s(0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) K tuples: GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c4_1, c6_1, c7_1, c3_1, c5_1, c10_1, c12_1 ---------------------------------------- (149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by GEN(s(s(y0))) -> c1(LE(s(s(y0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c4_1, c6_1, c7_1, c3_1, c5_1, c10_1, c12_1 ---------------------------------------- (151) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(z0)))) -> c4(IF3(le(z0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c4_1, c6_1, c7_1, c3_1, c5_1, c10_1, c12_1 ---------------------------------------- (153) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(x1))) -> c4(LE(s(s(x1)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1, c4_1, c12_1 ---------------------------------------- (155) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(s(s(x1)))) -> c6(IF2(x0, s(s(s(s(x1)))))) by IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1, c4_1, c12_1 ---------------------------------------- (157) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(x0, s(s(s(0)))) -> c4(IF3(le(0, s(s(s(s(s(s(s(0)))))))), x0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c7_1, c3_1, c5_1, c10_1, c4_1, c12_1 ---------------------------------------- (159) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(false, x0, s(s(s(x1)))) -> c7(GEN(s(x0))) by IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c3_1, c5_1, c10_1, c4_1, c12_1, c7_1 ---------------------------------------- (161) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(s(s(x1)))) -> c5(TIMES(x0, s(s(s(x1))))) by IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c3_1, c5_1, c10_1, c4_1, c12_1, c7_1 ---------------------------------------- (163) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF2(x0, s(s(s(s(x1))))) -> c4(IF3(le(s(x1), s(s(s(s(s(s(s(0)))))))), x0, s(s(s(s(x1))))), LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c3_1, c5_1, c10_1, c4_1, c12_1, c7_1 ---------------------------------------- (165) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(0)) -> c4(IF3(true, x0, s(0))) by IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c3_1, c5_1, c10_1, c4_1, c12_1, c7_1 ---------------------------------------- (167) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(0)) -> c6(IF2(x0, s(s(0)))) by IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF2_2, IF3_3, IF1_2, LE_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c4_2, c6_1, c3_1, c5_1, c10_1, c4_1, c12_1, c7_1 ---------------------------------------- (169) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(0))) -> c4(IF3(true, x0, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF3_3, IF1_2, LE_2, IF2_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c6_1, c3_1, c5_1, c10_1, c4_1, c12_1, c4_2, c7_1 ---------------------------------------- (171) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(s(0))) -> c6(IF2(x0, s(s(s(0))))) by IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, IF3_3, LE_2, IF2_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c3_1, c5_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1 ---------------------------------------- (173) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(0)) -> c5(TIMES(x0, s(0))) by IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, IF3_3, LE_2, IF2_2, PLUS_2 Compound Symbols: c14_2, c1_1, c1_2, c3_1, c5_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1 ---------------------------------------- (175) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF3(true, x0, s(s(0))) -> c5(TIMES(x0, s(s(0)))) by IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_1, c1_2, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (177) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_1, c1_2, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (179) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_1, c1_2, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GEN(s(x0)) -> c1(LE(s(x0), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by GEN(s(s(y0))) -> c1(LE(s(s(y0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (183) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (185) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (187) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (189) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (191) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (193) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1 ---------------------------------------- (195) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF2(x0, s(s(s(s(x1))))) -> c4(LE(s(s(s(s(x1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) by IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: times(0, z0) -> 0 times(s(z0), z1) -> plus(z1, times(z0, z1)) plus(0, z0) -> z0 plus(s(z0), z1) -> s(plus(z0, z1)) le(0, z0) -> true le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) S tuples: TIMES(s(z0), z1) -> c14(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) GEN(s(s(0))) -> c1(IF1(true, s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) GEN(s(s(s(z0)))) -> c1(IF1(le(z0, s(s(s(s(s(s(s(0)))))))), s(s(s(z0)))), LE(s(s(s(z0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF1(true, s(0)) -> c3(IF2(s(0), s(0))) IF1(true, s(s(0))) -> c3(IF2(s(s(0)), s(s(0)))) IF1(true, s(s(s(x0)))) -> c3(IF2(s(s(s(x0))), s(s(s(x0))))) LE(s(s(y0)), s(s(y1))) -> c10(LE(s(y0), s(y1))) GEN(s(0)) -> c1(IF1(true, s(0))) PLUS(s(s(y0)), z1) -> c12(PLUS(s(y0), z1)) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(IF3(le(x0, s(s(s(s(s(s(s(0)))))))), s(s(s(x0))), s(s(s(x0)))), LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(0)))) -> c6(IF2(x0, s(s(s(s(0)))))) IF3(true, x0, s(s(s(s(x1))))) -> c6(IF2(x0, s(s(s(s(s(x1))))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c6(IF2(s(s(s(x0))), s(s(s(s(x0)))))) IF2(z0, s(s(s(0)))) -> c4(IF3(true, z0, s(s(s(0)))), LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(false, x0, s(s(s(s(x1))))) -> c7(GEN(s(x0))) IF3(false, s(s(s(x0))), s(s(s(x0)))) -> c7(GEN(s(s(s(s(x0)))))) IF2(z0, s(s(s(s(z1))))) -> c4(IF3(le(z1, s(s(s(s(s(s(0))))))), z0, s(s(s(s(z1))))), LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(0)) -> c4(IF3(true, s(0), s(0))) IF3(true, s(0), s(0)) -> c6(IF2(s(0), s(s(0)))) IF2(s(s(0)), s(s(0))) -> c4(IF3(true, s(s(0)), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(0))) -> c4(IF3(true, s(0), s(s(0))), LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, s(s(0)), s(s(0))) -> c6(IF2(s(s(0)), s(s(s(0))))) IF3(true, s(0), s(s(0))) -> c6(IF2(s(0), s(s(s(0))))) K tuples: GEN(s(s(x0))) -> c1(LE(s(s(x0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(x0)))) -> c4(LE(s(s(s(x0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF3(true, x0, s(s(s(s(x1))))) -> c5(TIMES(x0, s(s(s(s(x1)))))) IF3(true, s(s(s(x0))), s(s(s(x0)))) -> c5(TIMES(s(s(s(x0))), s(s(s(x0))))) IF3(true, x0, s(s(s(0)))) -> c5(TIMES(x0, s(s(s(0))))) IF3(true, s(0), s(0)) -> c5(TIMES(s(0), s(0))) IF3(true, s(s(0)), s(s(0))) -> c5(TIMES(s(s(0)), s(s(0)))) IF3(true, s(0), s(s(0))) -> c5(TIMES(s(0), s(s(0)))) IF2(s(0), s(s(0))) -> c4(LE(s(s(0)), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(0))), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(0)), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(0), s(s(s(0)))) -> c4(LE(s(s(s(0))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(s(z1)))), s(s(s(s(z1))))) -> c4(LE(s(s(s(s(z1)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(0))))) -> c4(LE(s(s(s(s(0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(x0, s(s(s(s(s(x1)))))) -> c4(LE(s(s(s(s(s(x1))))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) IF2(s(s(s(x0))), s(s(s(s(x0))))) -> c4(LE(s(s(s(s(x0)))), s(s(s(s(s(s(s(s(s(s(0)))))))))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, GEN_1, IF1_2, LE_2, IF2_2, PLUS_2, IF3_3 Compound Symbols: c14_2, c1_2, c1_1, c3_1, c10_1, c4_1, c12_1, c4_2, c6_1, c7_1, c5_1