KILLED proof of input_Ik5XX1y15n.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 35 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 427 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 131 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 303 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 506 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 198 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 5099 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 458 ms] (42) CpxRNTS (43) CompletionProof [UPPER BOUND(ID), 33 ms] (44) CpxTypedWeightedCompleteTrs (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (50) CdtProblem (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms] (66) CdtProblem (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (76) CdtProblem (77) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms] (80) CdtProblem (81) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtInstantiationProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms] (116) CdtProblem (117) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms] (120) CdtProblem (121) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(s(x), 0) -> false le(0, y) -> true 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)) log(x, 0) -> baseError log(x, s(0)) -> baseError log(0, s(s(b))) -> logZeroError log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) if(true, x, b, y, z) -> z if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(s(x), 0') -> false le(0', y) -> true 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)) log(x, 0') -> baseError log(x, s(0')) -> baseError log(0', s(s(b))) -> logZeroError log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0') loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) if(true, x, b, y, z) -> z if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(s(x), 0) -> false le(0, y) -> true 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)) log(x, 0) -> baseError log(x, s(0)) -> baseError log(0, s(s(b))) -> logZeroError log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) if(true, x, b, y, z) -> z if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [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] log(x, 0) -> baseError [1] log(x, s(0)) -> baseError [1] log(0, s(s(b))) -> logZeroError [1] log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1] loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1] if(true, x, b, y, z) -> z [1] if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [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] log(x, 0) -> baseError [1] log(x, s(0)) -> baseError [1] log(0, s(s(b))) -> logZeroError [1] log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1] loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1] if(true, x, b, y, z) -> z [1] if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1] The TRS has the following type information: le :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> false:true s :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError 0 :: s:0:baseError:logZeroError false :: false:true true :: false:true plus :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError times :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError log :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError baseError :: s:0:baseError:logZeroError logZeroError :: s:0:baseError:logZeroError loop :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError if :: false:true -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: log_2 loop_4 if_5 (c) The following functions are completely defined: times_2 le_2 plus_2 Due to the following rules being added: times(v0, v1) -> null_times [0] le(v0, v1) -> null_le [0] plus(v0, v1) -> null_plus [0] And the following fresh constants: null_times, null_le, null_plus ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [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] log(x, 0) -> baseError [1] log(x, s(0)) -> baseError [1] log(0, s(s(b))) -> logZeroError [1] log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1] loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1] if(true, x, b, y, z) -> z [1] if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1] times(v0, v1) -> null_times [0] le(v0, v1) -> null_le [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: le :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> false:true:null_le s :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus 0 :: s:0:baseError:logZeroError:null_times:null_plus false :: false:true:null_le true :: false:true:null_le plus :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus times :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus log :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus baseError :: s:0:baseError:logZeroError:null_times:null_plus logZeroError :: s:0:baseError:logZeroError:null_times:null_plus loop :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus if :: false:true:null_le -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus null_times :: s:0:baseError:logZeroError:null_times:null_plus null_le :: false:true:null_le null_plus :: s:0:baseError:logZeroError:null_times:null_plus Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [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] times(s(x), y) -> plus(y, null_times) [1] log(x, 0) -> baseError [1] log(x, s(0)) -> baseError [1] log(0, s(s(b))) -> logZeroError [1] log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1] loop(0, s(s(b)), s(y), z) -> if(true, 0, s(s(b)), s(y), z) [2] loop(s(x''), s(s(b)), s(y), z) -> if(le(x'', y), s(x''), s(s(b)), s(y), z) [2] loop(x, s(s(b)), s(y), z) -> if(null_le, x, s(s(b)), s(y), z) [1] if(true, x, b, y, z) -> z [1] if(false, x, 0, y, z) -> loop(x, 0, 0, s(z)) [2] if(false, x, s(x1), y, z) -> loop(x, s(x1), plus(y, times(x1, y)), s(z)) [2] if(false, x, b, y, z) -> loop(x, b, null_times, s(z)) [1] times(v0, v1) -> null_times [0] le(v0, v1) -> null_le [0] plus(v0, v1) -> null_plus [0] The TRS has the following type information: le :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> false:true:null_le s :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus 0 :: s:0:baseError:logZeroError:null_times:null_plus false :: false:true:null_le true :: false:true:null_le plus :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus times :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus log :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus baseError :: s:0:baseError:logZeroError:null_times:null_plus logZeroError :: s:0:baseError:logZeroError:null_times:null_plus loop :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus if :: false:true:null_le -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus null_times :: s:0:baseError:logZeroError:null_times:null_plus null_le :: false:true:null_le null_plus :: s:0:baseError:logZeroError:null_times:null_plus 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 => 1 true => 2 baseError => 1 logZeroError => 2 null_times => 0 null_le => 0 null_plus => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z :|: z1 = b, b >= 0, z2 = y, z >= 0, z' = 2, z3 = z, x >= 0, y >= 0, z'' = x if(z', z'', z1, z2, z3) -{ 1 }-> loop(x, b, 0, 1 + z) :|: z1 = b, b >= 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(x, 0, 0, 1 + z) :|: z1 = 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(x, 1 + x1, plus(y, times(x1, y)), 1 + z) :|: z2 = y, x1 >= 0, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1, z1 = 1 + x1 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 log(z', z'') -{ 1 }-> loop(1 + x, 1 + (1 + b), 1 + 0, 0) :|: b >= 0, z' = 1 + x, x >= 0, z'' = 1 + (1 + b) log(z', z'') -{ 1 }-> 2 :|: b >= 0, z'' = 1 + (1 + b), z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0 log(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 2 }-> if(le(x'', y), 1 + x'', 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z' = 1 + x'', z2 = z, y >= 0, z1 = 1 + y, x'' >= 0, z'' = 1 + (1 + b) loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z2 = z, y >= 0, z1 = 1 + y, z'' = 1 + (1 + b), z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, x, 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z' = x, z2 = z, x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + (1 + b) plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0 times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(y, 0) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { le } { plus } { times } { if, loop } { log } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {le}, {plus}, {times}, {if,loop}, {log} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {le}, {plus}, {times}, {if,loop}, {log} ---------------------------------------- (21) 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: 2 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {le}, {plus}, {times}, {if,loop}, {log} Previous analysis results are: le: runtime: ?, size: O(1) [2] ---------------------------------------- (23) 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'' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (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: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {plus}, {times}, {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: 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'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0 times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {times}, {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 + 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 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {times}, {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] ---------------------------------------- (33) 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'' + 2*z'' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 + 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 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {times}, {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] times: runtime: ?, size: O(n^2) [2*z'*z'' + 2*z''] ---------------------------------------- (35) 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: 6 + 4*z' + 2*z'*z'' + 2*z'' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 + 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 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z''] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 5 + 4*z1 + 2*z1*z2 + z2 }-> loop(z'', 1 + (z1 - 1), s7, 1 + z3) :|: s6 >= 0, s6 <= 2 * z2 + 2 * (z2 * (z1 - 1)), s7 >= 0, s7 <= z2 + s6, z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 + 4*z' + 2*z'*z'' }-> s5 :|: s3 >= 0, s3 <= 2 * z'' + 2 * (z'' * (z' - 2)), s4 >= 0, s4 <= z'' + s3, s5 >= 0, s5 <= z'' + s4, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z''] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: loop after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 5 + 4*z1 + 2*z1*z2 + z2 }-> loop(z'', 1 + (z1 - 1), s7, 1 + z3) :|: s6 >= 0, s6 <= 2 * z2 + 2 * (z2 * (z1 - 1)), s7 >= 0, s7 <= z2 + s6, z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 + 4*z' + 2*z'*z'' }-> s5 :|: s3 >= 0, s3 <= 2 * z'' + 2 * (z'' * (z' - 2)), s4 >= 0, s4 <= z'' + s3, s5 >= 0, s5 <= z'' + s4, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z''] if: runtime: ?, size: INF loop: runtime: ?, size: INF ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0 if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 if(z', z'', z1, z2, z3) -{ 5 + 4*z1 + 2*z1*z2 + z2 }-> loop(z'', 1 + (z1 - 1), s7, 1 + z3) :|: s6 >= 0, s6 <= 2 * z2 + 2 * (z2 * (z1 - 1)), s7 >= 0, s7 <= z2 + s6, z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1 le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0 le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0 le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0 log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0 log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0 loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0 loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0 loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0 plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0 times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0 times(z', z'') -{ 2 + 4*z' + 2*z'*z'' }-> s5 :|: s3 >= 0, s3 <= 2 * z'' + 2 * (z'' * (z' - 2)), s4 >= 0, s4 <= z'' + s3, s5 >= 0, s5 <= z'' + s4, z' - 2 >= 0, z'' >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0 Function symbols to be analyzed: {if,loop}, {log} Previous analysis results are: le: runtime: O(n^1) [2 + z''], size: O(1) [2] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z''] times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z''] if: runtime: INF, size: INF loop: runtime: ?, size: INF ---------------------------------------- (43) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: le(v0, v1) -> null_le [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] log(v0, v1) -> null_log [0] loop(v0, v1, v2, v3) -> null_loop [0] if(v0, v1, v2, v3, v4) -> null_if [0] And the following fresh constants: null_le, null_plus, null_times, null_log, null_loop, null_if ---------------------------------------- (44) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: le(s(x), 0) -> false [1] le(0, y) -> true [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] log(x, 0) -> baseError [1] log(x, s(0)) -> baseError [1] log(0, s(s(b))) -> logZeroError [1] log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1] loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1] if(true, x, b, y, z) -> z [1] if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1] le(v0, v1) -> null_le [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] log(v0, v1) -> null_log [0] loop(v0, v1, v2, v3) -> null_loop [0] if(v0, v1, v2, v3, v4) -> null_if [0] The TRS has the following type information: le :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> false:true:null_le s :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if 0 :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if false :: false:true:null_le true :: false:true:null_le plus :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if times :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if log :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if baseError :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if logZeroError :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if loop :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if if :: false:true:null_le -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if null_le :: false:true:null_le null_plus :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if null_times :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if null_log :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if null_loop :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if null_if :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 false => 1 true => 2 baseError => 1 logZeroError => 2 null_le => 0 null_plus => 0 null_times => 0 null_log => 0 null_loop => 0 null_if => 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: if(z', z'', z1, z2, z3) -{ 1 }-> z :|: z1 = b, b >= 0, z2 = y, z >= 0, z' = 2, z3 = z, x >= 0, y >= 0, z'' = x if(z', z'', z1, z2, z3) -{ 1 }-> loop(x, b, times(b, y), 1 + z) :|: z1 = b, b >= 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1 if(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0 le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0 le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0 le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 log(z', z'') -{ 1 }-> loop(1 + x, 1 + (1 + b), 1 + 0, 0) :|: b >= 0, z' = 1 + x, x >= 0, z'' = 1 + (1 + b) log(z', z'') -{ 1 }-> 2 :|: b >= 0, z'' = 1 + (1 + b), z' = 0 log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0 log(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + 0 log(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 loop(z', z'', z1, z2) -{ 1 }-> if(le(x, 1 + y), x, 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z' = x, z2 = z, x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + (1 + b) loop(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0 plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0 plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> plus(y, times(x, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0 times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0 times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true 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)) log(z0, 0) -> baseError log(z0, s(0)) -> baseError log(0, s(s(z0))) -> logZeroError log(s(z0), s(s(z1))) -> loop(s(z0), s(s(z1)), s(0), 0) loop(z0, s(s(z1)), s(z2), z3) -> if(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3) if(true, z0, z1, z2, z3) -> z3 if(false, z0, z1, z2, z3) -> loop(z0, z1, times(z1, z2), s(z3)) Tuples: LE(s(z0), 0) -> c LE(0, z0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(0, z0) -> c3 PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(0, z0) -> c5 TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOG(z0, 0) -> c7 LOG(z0, s(0)) -> c8 LOG(0, s(s(z0))) -> c9 LOG(s(z0), s(s(z1))) -> c10(LOOP(s(z0), s(s(z1)), s(0), 0)) LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) IF(true, z0, z1, z2, z3) -> c12 IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) S tuples: LE(s(z0), 0) -> c LE(0, z0) -> c1 LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(0, z0) -> c3 PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(0, z0) -> c5 TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOG(z0, 0) -> c7 LOG(z0, s(0)) -> c8 LOG(0, s(s(z0))) -> c9 LOG(s(z0), s(s(z1))) -> c10(LOOP(s(z0), s(s(z1)), s(0), 0)) LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) IF(true, z0, z1, z2, z3) -> c12 IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) K tuples:none Defined Rule Symbols: le_2, plus_2, times_2, log_2, loop_4, if_5 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOG_2, LOOP_4, IF_5 Compound Symbols: c, c1, c2_1, c3, c4_1, c5, c6_2, c7, c8, c9, c10_1, c11_2, c12, c13_2 ---------------------------------------- (49) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: LOG(s(z0), s(s(z1))) -> c10(LOOP(s(z0), s(s(z1)), s(0), 0)) Removed 8 trailing nodes: LE(0, z0) -> c1 LOG(z0, 0) -> c7 IF(true, z0, z1, z2, z3) -> c12 LOG(z0, s(0)) -> c8 LOG(0, s(s(z0))) -> c9 TIMES(0, z0) -> c5 LE(s(z0), 0) -> c PLUS(0, z0) -> c3 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: le(s(z0), 0) -> false le(0, z0) -> true 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)) log(z0, 0) -> baseError log(z0, s(0)) -> baseError log(0, s(s(z0))) -> logZeroError log(s(z0), s(s(z1))) -> loop(s(z0), s(s(z1)), s(0), 0) loop(z0, s(s(z1)), s(z2), z3) -> if(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3) if(true, z0, z1, z2, z3) -> z3 if(false, z0, z1, z2, z3) -> loop(z0, z1, times(z1, z2), s(z3)) Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) K tuples:none Defined Rule Symbols: le_2, plus_2, times_2, log_2, loop_4, if_5 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c13_2 ---------------------------------------- (51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: log(z0, 0) -> baseError log(z0, s(0)) -> baseError log(0, s(s(z0))) -> logZeroError log(s(z0), s(s(z1))) -> loop(s(z0), s(s(z1)), s(0), 0) loop(z0, s(s(z1)), s(z2), z3) -> if(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3) if(true, z0, z1, z2, z3) -> z3 if(false, z0, z1, z2, z3) -> loop(z0, z1, times(z1, z2), s(z3)) ---------------------------------------- (52) 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)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c13_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) by LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2))) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) ---------------------------------------- (54) 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)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2))) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2))) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4 Compound Symbols: c2_1, c4_1, c6_2, c13_2, c11_2 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2))) ---------------------------------------- (56) 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)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4 Compound Symbols: c2_1, c4_1, c6_2, c13_2, c11_2 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) by IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) ---------------------------------------- (58) 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)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c13_2 ---------------------------------------- (59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0)) ---------------------------------------- (60) 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)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c13_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) by LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(IF(true, s(0), s(s(x1)), s(z0), x3), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(IF(true, s(0), s(s(x1)), s(z0), x3), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(IF(true, s(0), s(s(x1)), s(z0), x3), LE(s(0), s(z0))) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4 Compound Symbols: c2_1, c4_1, c6_2, c13_2, c11_2, c11_1 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) K tuples:none Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4 Compound Symbols: c2_1, c4_1, c6_2, c13_2, c11_2, c11_1 ---------------------------------------- (65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) We considered the (Usable) Rules: le(s(z0), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false le(0, z0) -> true And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_5 POL(LE(x_1, x_2)) = 0 POL(LOOP(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(le(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = x_1 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = 0 ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4 Compound Symbols: c2_1, c4_1, c6_2, c13_2, c11_2, c11_1 ---------------------------------------- (67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_2 POL(LE(x_1, x_2)) = 0 POL(LOOP(x_1, x_2, x_3, x_4)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(plus(x_1, x_2)) = [1] + x_2 POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4 Compound Symbols: c2_1, c4_1, c6_2, c13_2, c11_2, c11_1 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) by IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(0), z0, x3) -> c13(LOOP(x0, s(0), plus(z0, 0), s(x3)), TIMES(s(0), z0)) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(0), z0, x3) -> c13(LOOP(x0, s(0), plus(z0, 0), s(x3)), TIMES(s(0), z0)) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(0), z0, x3) -> c13(LOOP(x0, s(0), plus(z0, 0), s(x3)), TIMES(s(0), z0)) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0)) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1 ---------------------------------------- (73) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), 0, x3) -> c(TIMES(s(x1), 0)) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), 0, x3) -> c(TIMES(s(x1), 0)) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1 ---------------------------------------- (75) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0)) IF(false, x0, s(x1), 0, x3) -> c(TIMES(s(x1), 0)) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1 ---------------------------------------- (77) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) We considered the (Usable) Rules:none And the Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_3 POL(LE(x_1, x_2)) = 0 POL(LOOP(x_1, x_2, x_3, x_4)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = [1] POL(le(x_1, x_2)) = [1] + x_1 + x_2 POL(plus(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = [1] ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1 ---------------------------------------- (81) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) by LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) S tuples: LE(s(z0), s(z1)) -> c2(LE(z0, z1)) PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5 Compound Symbols: c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, LOOP_4, IF_5, LE_2 Compound Symbols: c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1, c2_1 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) K tuples: LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, LOOP_4, IF_5, LE_2 Compound Symbols: c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1, c2_1 ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) K tuples: LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, LOOP_4, IF_5, LE_2 Compound Symbols: c4_1, c6_2, c11_1, c13_2, c13_1, c_1, c11_2, c2_1 ---------------------------------------- (89) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) by LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c13_2, c13_1, c_1, c11_2, c2_1, c11_1 ---------------------------------------- (91) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) by IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c13_2, c13_1, c_1, c11_2, c2_1, c11_1 ---------------------------------------- (93) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) by IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), 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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c13_1, c_1, c11_2, c2_1, c11_1, c13_2 ---------------------------------------- (95) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c13_1, c_1, c11_2, c2_1, c11_1, c13_2 ---------------------------------------- (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) by IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), 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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2 Compound Symbols: c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1 ---------------------------------------- (105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) by PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) ---------------------------------------- (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), s(z1)) -> le(z0, z1) le(s(z0), 0) -> false Tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1, c4_1 ---------------------------------------- (107) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c13_2, c4_1 ---------------------------------------- (109) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), 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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c13_2, c4_1 ---------------------------------------- (111) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c13_2, c4_1 ---------------------------------------- (113) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), 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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2 ---------------------------------------- (115) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2 ---------------------------------------- (117) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1 ---------------------------------------- (119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), 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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3 POL(LE(x_1, x_2)) = 0 POL(LOOP(x_1, x_2, x_3, x_4)) = x_2 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c11(x_1)) = x_1 POL(c11(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c13(x_1, x_2)) = x_1 + x_2 POL(c2(x_1)) = x_1 POL(c4(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(false) = 0 POL(le(x_1, x_2)) = 0 POL(plus(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(true) = 0 ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1 ---------------------------------------- (121) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) by LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1 ---------------------------------------- (123) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) by IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1 ---------------------------------------- (127) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1 ---------------------------------------- (129) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) by LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2 Compound Symbols: c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1, c2_1 ---------------------------------------- (133) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) by IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2 Compound Symbols: c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1, c2_1 ---------------------------------------- (135) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) by IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2 Compound Symbols: c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c3_1, c2_1, c1_1 ---------------------------------------- (137) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) by IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2 Compound Symbols: c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c3_1, c2_1, c1_1 ---------------------------------------- (139) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) by IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), 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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2 Compound Symbols: c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c2_1, c1_1, c3_1 ---------------------------------------- (141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) by LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1))))) ---------------------------------------- (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: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0))) LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1))))) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0))) LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2 Compound Symbols: c6_2, c_1, c11_2, c13_1, c4_1, c13_2, c11_1, c2_1, c1_1, c3_1 ---------------------------------------- (143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) by PLUS(s(s(s(y0))), z1) -> c4(PLUS(s(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) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0))) LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1))))) PLUS(s(s(s(y0))), z1) -> c4(PLUS(s(s(y0)), z1)) S tuples: TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1)) LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2)))) IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1)))) PLUS(s(s(s(y0))), z1) -> c4(PLUS(s(s(y0)), z1)) K tuples: IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3))) IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0))) LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2)))) IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0))) LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1))))) Defined Rule Symbols: times_2, plus_2, le_2 Defined Pair Symbols: TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2 Compound Symbols: c6_2, c_1, c11_2, c13_1, c13_2, c11_1, c2_1, c1_1, c3_1, c4_1