KILLED proof of input_AvQP6Lb4VN.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 333 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 248 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 148 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 231 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 105 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 507 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 950 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) CompletionProof [UPPER BOUND(ID), 0 ms] (52) CpxTypedWeightedCompleteTrs (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms] (70) CdtProblem (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 30 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms] (88) CdtProblem (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms] (94) CdtProblem (95) CdtForwardInstantiationProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (104) CdtProblem (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms] (108) CdtProblem (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms] (110) CdtProblem (111) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (138) 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: +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(*(x, y), x) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) fact(x) -> iffact(x, ge(x, s(s(0)))) iffact(x, true) -> *(x, fact(-(x, s(0)))) iffact(x, false) -> s(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(*'(x, y), x) ge(x, 0') -> true ge(0', s(y)) -> false ge(s(x), s(y)) -> ge(x, y) -(x, 0') -> x -(s(x), s(y)) -> -(x, y) fact(x) -> iffact(x, ge(x, s(s(0')))) iffact(x, true) -> *'(x, fact(-(x, s(0')))) iffact(x, false) -> s(0') S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(*(x, y), x) ge(x, 0) -> true ge(0, s(y)) -> false ge(s(x), s(y)) -> ge(x, y) -(x, 0) -> x -(s(x), s(y)) -> -(x, y) fact(x) -> iffact(x, ge(x, s(s(0)))) iffact(x, true) -> *(x, fact(-(x, s(0)))) iffact(x, false) -> s(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: +(x, 0) -> x [1] +(x, s(y)) -> s(+(x, y)) [1] *(x, 0) -> 0 [1] *(x, s(y)) -> +(*(x, y), x) [1] ge(x, 0) -> true [1] ge(0, s(y)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] fact(x) -> iffact(x, ge(x, s(s(0)))) [1] iffact(x, true) -> *(x, fact(-(x, s(0)))) [1] iffact(x, false) -> s(0) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus * => times - => minus ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] ge(x, 0) -> true [1] ge(0, s(y)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] fact(x) -> iffact(x, ge(x, s(s(0)))) [1] iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1] iffact(x, false) -> s(0) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] ge(x, 0) -> true [1] ge(0, s(y)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] fact(x) -> iffact(x, ge(x, s(s(0)))) [1] iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1] iffact(x, false) -> s(0) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false minus :: 0:s -> 0:s -> 0:s fact :: 0:s -> 0:s iffact :: 0:s -> true:false -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: times_2 ge_2 fact_1 minus_2 plus_2 iffact_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (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: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] ge(x, 0) -> true [1] ge(0, s(y)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] fact(x) -> iffact(x, ge(x, s(s(0)))) [1] iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1] iffact(x, false) -> s(0) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false minus :: 0:s -> 0:s -> 0:s fact :: 0:s -> 0:s iffact :: 0:s -> true:false -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) 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: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(0)) -> plus(0, x) [2] times(x, s(s(y'))) -> plus(plus(times(x, y'), x), x) [2] ge(x, 0) -> true [1] ge(0, s(y)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] fact(0) -> iffact(0, false) [2] fact(s(x')) -> iffact(s(x'), ge(x', s(0))) [2] iffact(s(x''), true) -> times(s(x''), fact(minus(x'', 0))) [2] iffact(x, true) -> times(x, fact(0)) [1] iffact(x, false) -> s(0) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false minus :: 0:s -> 0:s -> 0:s fact :: 0:s -> 0:s iffact :: 0:s -> true:false -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + x', ge(x', 1 + 0)) :|: z = 1 + x', x' >= 0 ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' = 1 + y, y >= 0, z = 0 iffact(z, z') -{ 1 }-> times(x, fact(0)) :|: x >= 0, z' = 1, z = x iffact(z, z') -{ 2 }-> times(1 + x'', fact(minus(x'', 0))) :|: z = 1 + x'', z' = 1, x'' >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 2 }-> plus(plus(times(x, y'), x), x) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { ge } { plus } { times } { fact, iffact } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {ge}, {plus}, {times}, {fact,iffact} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {ge}, {plus}, {times}, {fact,iffact} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {ge}, {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {ge}, {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {ge}, {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: ge after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {ge}, {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: ?, size: O(1) [1] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: ge after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0 ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (35) 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' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (37) 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' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 0 + z, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (41) 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: z + 2*z*z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 0 + z, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + z + 2*z*z' + 4*z' ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 0 + z, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (45) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 0 + z, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s6 :|: s4 >= 0, s4 <= z + 2 * ((z' - 2) * z), s5 >= 0, s5 <= s4 + z, s6 >= 0, s6 <= s5 + z, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fact after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? Computed SIZE bound using CoFloCo for: iffact after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 0 + z, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s6 :|: s4 >= 0, s4 <= z + 2 * ((z' - 2) * z), s5 >= 0, s5 <= s4 + z, s6 >= 0, s6 <= s5 + z, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] fact: runtime: ?, size: INF iffact: runtime: ?, size: INF ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fact after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0 fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0 ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0 ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0 ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0 iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1 iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0 iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 0 + z, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s6 :|: s4 >= 0, s4 <= z + 2 * ((z' - 2) * z), s5 >= 0, s5 <= s4 + z, s6 >= 0, s6 <= s5 + z, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fact,iffact} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] fact: runtime: INF, size: INF iffact: runtime: ?, size: INF ---------------------------------------- (51) 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: minus(v0, v1) -> null_minus [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] ge(v0, v1) -> null_ge [0] iffact(v0, v1) -> null_iffact [0] And the following fresh constants: null_minus, null_plus, null_times, null_ge, null_iffact ---------------------------------------- (52) 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: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] ge(x, 0) -> true [1] ge(0, s(y)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] fact(x) -> iffact(x, ge(x, s(s(0)))) [1] iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1] iffact(x, false) -> s(0) [1] minus(v0, v1) -> null_minus [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] ge(v0, v1) -> null_ge [0] iffact(v0, v1) -> null_iffact [0] The TRS has the following type information: plus :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact 0 :: 0:s:null_minus:null_plus:null_times:null_iffact s :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact times :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact ge :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> true:false:null_ge true :: true:false:null_ge false :: true:false:null_ge minus :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact fact :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact iffact :: 0:s:null_minus:null_plus:null_times:null_iffact -> true:false:null_ge -> 0:s:null_minus:null_plus:null_times:null_iffact null_minus :: 0:s:null_minus:null_plus:null_times:null_iffact null_plus :: 0:s:null_minus:null_plus:null_times:null_iffact null_times :: 0:s:null_minus:null_plus:null_times:null_iffact null_ge :: true:false:null_ge null_iffact :: 0:s:null_minus:null_plus:null_times:null_iffact Rewrite Strategy: INNERMOST ---------------------------------------- (53) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 2 false => 1 null_minus => 0 null_plus => 0 null_times => 0 null_ge => 0 null_iffact => 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: fact(z) -{ 1 }-> iffact(x, ge(x, 1 + (1 + 0))) :|: x >= 0, z = x ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 iffact(z, z') -{ 1 }-> times(x, fact(minus(x, 1 + 0))) :|: z' = 2, x >= 0, z = x iffact(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 iffact(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z' = 1, z = x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> plus(times(x, y), x) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (55) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, 0) -> c +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, 0) -> c2 *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(z0, 0) -> c4 GE(0, s(z0)) -> c5 GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(z0, 0) -> c7 -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0)))) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) IFFACT(z0, false) -> c11 S tuples: +'(z0, 0) -> c +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, 0) -> c2 *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(z0, 0) -> c4 GE(0, s(z0)) -> c5 GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(z0, 0) -> c7 -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0)))) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) IFFACT(z0, false) -> c11 K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c, c1_1, c2, c3_2, c4, c5, c6_1, c7, c8_1, c9_2, c10_3, c11 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: IFFACT(z0, false) -> c11 *'(z0, 0) -> c2 -'(z0, 0) -> c7 GE(z0, 0) -> c4 GE(0, s(z0)) -> c5 +'(z0, 0) -> c ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0)))) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0)))) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_2, c10_3 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0)))) by FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0)))) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0)))) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0)))) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0)))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) by IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_2, c10_3 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) by FACT(s(0)) -> c9(IFFACT(s(0), false), GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(0)) -> c9(IFFACT(s(0), false), GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(0)) -> c9(IFFACT(s(0), false), GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) K tuples:none Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(0)) -> c9(GE(s(0), s(s(0)))) We considered the (Usable) Rules: -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 And the Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = x_1 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = [1] POL(FACT(x_1)) = x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = x_1 POL(true) = 0 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1 ---------------------------------------- (71) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) We considered the (Usable) Rules:none And the Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = x_1 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = 0 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(FACT(x_1)) = [1] POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = 0 POL(true) = 0 ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_2, c9_1, c10_3 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) by FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2 ---------------------------------------- (79) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) We considered the (Usable) Rules: -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 And the Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(FACT(x_1)) = x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2 ---------------------------------------- (87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) We considered the (Usable) Rules: -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 And the Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = [1] POL(FACT(x_1)) = [1] + x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2 ---------------------------------------- (89) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c9_2, c10_3 ---------------------------------------- (91) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c9_2, c10_3 ---------------------------------------- (93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) We considered the (Usable) Rules: -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 And the Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(FACT(x_1)) = x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: +'(z0, s(z1)) -> c1(+'(z0, z1)) *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2 Compound Symbols: c1_1, c3_2, c6_1, c8_1, c9_1, c9_2, c10_3 ---------------------------------------- (95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(z1)) -> c1(+'(z0, z1)) by +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (97) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) by IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. -'(s(z0), s(z1)) -> c8(-'(z0, z1)) We considered the (Usable) Rules: -(z0, 0) -> z0 And the Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = x_2 POL(0) = 0 POL(FACT(x_1)) = x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) We considered the (Usable) Rules: -(z0, 0) -> z0 And the Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(FACT(x_1)) = x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) We considered the (Usable) Rules: -(z0, 0) -> z0 And the Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(FACT(x_1)) = [1] + x_1 POL(GE(x_1, x_2)) = 0 POL(IFFACT(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = 0 POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. GE(s(z0), s(z1)) -> c6(GE(z0, z1)) We considered the (Usable) Rules: -(z0, 0) -> z0 And the Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = 0 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [3] + [3]x_1 + [3]x_2 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = x_1 POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(FACT(x_1)) = [2] + [2]x_1 POL(GE(x_1, x_2)) = x_2 POL(IFFACT(x_1, x_2)) = [2]x_1 + [3]x_2 POL(c1(x_1)) = x_1 POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(c9(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [3] POL(false) = [3] POL(ge(x_1, x_2)) = 0 POL(iffact(x_1, x_2)) = [3] POL(s(x_1)) = [1] + x_1 POL(true) = 0 ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (111) CdtInstantiationProof (BOTH BOUNDS(ID, ID)) Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) by IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(z0), s(z1)) -> c6(GE(z0, z1)) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2 Compound Symbols: c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1 ---------------------------------------- (113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GE(s(z0), s(z1)) -> c6(GE(z0, z1)) by GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, -'_2, FACT_1, IFFACT_2, +'_2, GE_2 Compound Symbols: c3_2, c8_1, c9_1, c9_2, c10_3, c1_1, c6_1 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACT(s(0)) -> c9(GE(s(0), s(s(0)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, -'_2, FACT_1, IFFACT_2, +'_2, GE_2 Compound Symbols: c3_2, c8_1, c9_1, c9_2, c10_3, c1_1, c6_1 ---------------------------------------- (117) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)), -'(s(s(z0)), s(0))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)), -'(s(s(z0)), s(0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) -'(s(z0), s(z1)) -> c8(-'(z0, z1)) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, -'_2, FACT_1, +'_2, IFFACT_2, GE_2 Compound Symbols: c3_2, c8_1, c9_1, c9_2, c1_1, c10_3, c6_1 ---------------------------------------- (119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c8(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)), -'(s(s(z0)), s(0))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, +'_2, IFFACT_2, GE_2, -'_2 Compound Symbols: c3_2, c9_1, c9_2, c1_1, c10_3, c6_1, c8_1 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2 Compound Symbols: c3_2, c9_1, c9_2, c1_1, c6_1, c8_1, c10_2 ---------------------------------------- (123) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2 Compound Symbols: c3_2, c9_1, c9_2, c1_1, c6_1, c8_1, c10_2 ---------------------------------------- (125) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2 Compound Symbols: c3_2, c9_1, c9_2, c1_1, c6_1, c8_1, c10_2 ---------------------------------------- (127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) by FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) K tuples: IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2 Compound Symbols: c3_2, c9_2, c1_1, c6_1, c8_1, c10_2, c9_1 ---------------------------------------- (129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(s(y1))) -> c1(+'(z0, s(y1))) by +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) K tuples: IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, GE_2, -'_2, IFFACT_2, +'_2 Compound Symbols: c3_2, c9_2, c6_1, c8_1, c10_2, c9_1, c1_1 ---------------------------------------- (131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) by GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) K tuples: IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, -'_2, IFFACT_2, +'_2, GE_2 Compound Symbols: c3_2, c9_2, c8_1, c10_2, c9_1, c1_1, c6_1 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0)))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) K tuples: FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0)))) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, FACT_1, -'_2, IFFACT_2, +'_2, GE_2 Compound Symbols: c3_2, c9_2, c8_1, c10_2, c1_1, c6_1 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true)) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) K tuples: -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true)) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, -'_2, IFFACT_2, +'_2, GE_2, FACT_1 Compound Symbols: c3_2, c8_1, c10_2, c1_1, c6_1, c9_1 ---------------------------------------- (137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) by -'(s(s(s(y0))), s(s(s(y1)))) -> c8(-'(s(s(y0)), s(s(y1)))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) fact(z0) -> iffact(z0, ge(z0, s(s(0)))) iffact(z0, true) -> *(z0, fact(-(z0, s(0)))) iffact(z0, false) -> s(0) Tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0))) IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0))) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true)) -'(s(s(s(y0))), s(s(s(y1)))) -> c8(-'(s(s(y0)), s(s(y1)))) S tuples: *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1)))) K tuples: GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1)))) FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true)) -'(s(s(s(y0))), s(s(s(y1)))) -> c8(-'(s(s(y0)), s(s(y1)))) Defined Rule Symbols: +_2, *_2, ge_2, -_2, fact_1, iffact_2 Defined Pair Symbols: *'_2, IFFACT_2, +'_2, GE_2, FACT_1, -'_2 Compound Symbols: c3_2, c10_2, c1_1, c6_1, c9_1, c8_1