KILLED proof of input_fIqo96VtF1.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [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), 299 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 246 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 6 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 89 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 2 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 119 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 52 ms] (44) CpxRNTS (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 454 ms] (48) CpxRNTS (49) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (50) CpxRNTS (51) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 884 ms] (54) CpxRNTS (55) IntTrsBoundProof [UPPER BOUND(ID), 177 ms] (56) CpxRNTS (57) CompletionProof [UPPER BOUND(ID), 0 ms] (58) CpxTypedWeightedCompleteTrs (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (60) CpxRNTS (61) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms] (80) CdtProblem (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 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)), 29 ms] (88) CdtProblem (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 27 ms] (92) CdtProblem (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (96) CdtProblem (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (108) 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: fact(X) -> if(zero(X), s(0), prod(X, fact(p(X)))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) prod(0, X) -> 0 prod(s(X), Y) -> add(Y, prod(X, Y)) if(true, X, Y) -> X if(false, X, Y) -> Y zero(0) -> true zero(s(X)) -> false p(s(X)) -> X 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: fact(X) -> if(zero(X), s(0'), prod(X, fact(p(X)))) add(0', X) -> X add(s(X), Y) -> s(add(X, Y)) prod(0', X) -> 0' prod(s(X), Y) -> add(Y, prod(X, Y)) if(true, X, Y) -> X if(false, X, Y) -> Y zero(0') -> true zero(s(X)) -> false p(s(X)) -> X 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: fact(X) -> if(zero(X), s(0), prod(X, fact(p(X)))) add(0, X) -> X add(s(X), Y) -> s(add(X, Y)) prod(0, X) -> 0 prod(s(X), Y) -> add(Y, prod(X, Y)) if(true, X, Y) -> X if(false, X, Y) -> Y zero(0) -> true zero(s(X)) -> false p(s(X)) -> X 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: fact(X) -> if(zero(X), s(0), prod(X, fact(p(X)))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] prod(0, X) -> 0 [1] prod(s(X), Y) -> add(Y, prod(X, Y)) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] zero(0) -> true [1] zero(s(X)) -> false [1] p(s(X)) -> X [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fact(X) -> if(zero(X), s(0), prod(X, fact(p(X)))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] prod(0, X) -> 0 [1] prod(s(X), Y) -> add(Y, prod(X, Y)) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] zero(0) -> true [1] zero(s(X)) -> false [1] p(s(X)) -> X [1] The TRS has the following type information: fact :: 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s zero :: 0:s -> true:false s :: 0:s -> 0:s 0 :: 0:s prod :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s add :: 0:s -> 0:s -> 0:s true :: true:false false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: prod_2 zero_1 fact_1 p_1 add_2 if_3 Due to the following rules being added: p(v0) -> 0 [0] And the following fresh constants: none ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fact(X) -> if(zero(X), s(0), prod(X, fact(p(X)))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] prod(0, X) -> 0 [1] prod(s(X), Y) -> add(Y, prod(X, Y)) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] zero(0) -> true [1] zero(s(X)) -> false [1] p(s(X)) -> X [1] p(v0) -> 0 [0] The TRS has the following type information: fact :: 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s zero :: 0:s -> true:false s :: 0:s -> 0:s 0 :: 0:s prod :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s add :: 0:s -> 0:s -> 0:s true :: true:false false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fact(0) -> if(true, s(0), prod(0, fact(0))) [2] fact(s(X')) -> if(false, s(0), prod(s(X'), fact(X'))) [3] fact(s(X')) -> if(false, s(0), prod(s(X'), fact(0))) [2] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] prod(0, X) -> 0 [1] prod(s(0), Y) -> add(Y, 0) [2] prod(s(s(X'')), Y) -> add(Y, add(Y, prod(X'', Y))) [2] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] zero(0) -> true [1] zero(s(X)) -> false [1] p(s(X)) -> X [1] p(v0) -> 0 [0] The TRS has the following type information: fact :: 0:s -> 0:s if :: true:false -> 0:s -> 0:s -> 0:s zero :: 0:s -> true:false s :: 0:s -> 0:s 0 :: 0:s prod :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s add :: 0:s -> 0:s -> 0:s true :: true:false false :: true:false Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 1 false => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + X', fact(X'))) :|: X' >= 0, z = 1 + X' fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + X', fact(0))) :|: X' >= 0, z = 1 + X' if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 prod(z, z') -{ 2 }-> add(Y, add(Y, prod(X'', Y))) :|: z' = Y, Y >= 0, z = 1 + (1 + X''), X'' >= 0 prod(z, z') -{ 2 }-> add(Y, 0) :|: z' = Y, Y >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z = 1 + X, X >= 0 ---------------------------------------- (15) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + X', fact(X'))) :|: X' >= 0, z = 1 + X' fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + X', fact(0))) :|: X' >= 0, z = 1 + X' if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 prod(z, z') -{ 2 }-> add(Y, add(Y, prod(X'', Y))) :|: z' = Y, Y >= 0, z = 1 + (1 + X''), X'' >= 0 prod(z, z') -{ 2 }-> add(Y, 0) :|: z' = Y, Y >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z = 1 + X, X >= 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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { if } { add } { zero } { p } { prod } { fact } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if}, {add}, {zero}, {p}, {prod}, {fact} ---------------------------------------- (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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if}, {add}, {zero}, {p}, {prod}, {fact} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' + z'' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {if}, {add}, {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: ?, size: O(n^1) [z' + z''] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: if after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {add}, {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + 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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {add}, {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: add after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {add}, {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: add after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 2 }-> add(z', 0) :|: z' >= 0, z = 1 + 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: zero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {zero}, {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: ?, size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: zero after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {p}, {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: p after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [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: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: prod after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + z' ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {prod}, {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] prod: runtime: ?, size: O(n^2) [2*z*z' + z'] ---------------------------------------- (49) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: prod after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + 4*z + 2*z*z' + z' ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 2 }-> add(z', add(z', prod(z - 2, z'))) :|: z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] prod: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (51) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= z' + s'', s2 >= 0, s2 <= z' + s1, z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] prod: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fact after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= z' + s'', s2 >= 0, s2 <= z' + s1, z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] prod: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] fact: runtime: ?, size: O(1) [0] ---------------------------------------- (55) 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: ? ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 add(z, z') -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1 + z', z' >= 0, z - 1 >= 0 fact(z) -{ 2 }-> if(1, 1 + 0, prod(0, fact(0))) :|: z = 0 fact(z) -{ 2 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(0))) :|: z - 1 >= 0 fact(z) -{ 3 }-> if(0, 1 + 0, prod(1 + (z - 1), fact(z - 1))) :|: z - 1 >= 0 if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0 if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0 prod(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= z' + 0, z' >= 0, z = 1 + 0 prod(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= z' + s'', s2 >= 0, s2 <= z' + s1, z' >= 0, z - 2 >= 0 prod(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 zero(z) -{ 1 }-> 1 :|: z = 0 zero(z) -{ 1 }-> 0 :|: z - 1 >= 0 Function symbols to be analyzed: {fact} Previous analysis results are: if: runtime: O(1) [1], size: O(n^1) [z' + z''] add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] zero: runtime: O(1) [1], size: O(1) [1] p: runtime: O(1) [1], size: O(n^1) [z] prod: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] fact: runtime: INF, size: O(1) [0] ---------------------------------------- (57) 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: p(v0) -> null_p [0] add(v0, v1) -> null_add [0] prod(v0, v1) -> null_prod [0] zero(v0) -> null_zero [0] if(v0, v1, v2) -> null_if [0] And the following fresh constants: null_p, null_add, null_prod, null_zero, null_if ---------------------------------------- (58) 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: fact(X) -> if(zero(X), s(0), prod(X, fact(p(X)))) [1] add(0, X) -> X [1] add(s(X), Y) -> s(add(X, Y)) [1] prod(0, X) -> 0 [1] prod(s(X), Y) -> add(Y, prod(X, Y)) [1] if(true, X, Y) -> X [1] if(false, X, Y) -> Y [1] zero(0) -> true [1] zero(s(X)) -> false [1] p(s(X)) -> X [1] p(v0) -> null_p [0] add(v0, v1) -> null_add [0] prod(v0, v1) -> null_prod [0] zero(v0) -> null_zero [0] if(v0, v1, v2) -> null_if [0] The TRS has the following type information: fact :: 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if if :: true:false:null_zero -> 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if zero :: 0:s:null_p:null_add:null_prod:null_if -> true:false:null_zero s :: 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if 0 :: 0:s:null_p:null_add:null_prod:null_if prod :: 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if p :: 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if add :: 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if -> 0:s:null_p:null_add:null_prod:null_if true :: true:false:null_zero false :: true:false:null_zero null_p :: 0:s:null_p:null_add:null_prod:null_if null_add :: 0:s:null_p:null_add:null_prod:null_if null_prod :: 0:s:null_p:null_add:null_prod:null_if null_zero :: true:false:null_zero null_if :: 0:s:null_p:null_add:null_prod:null_if Rewrite Strategy: INNERMOST ---------------------------------------- (59) 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_p => 0 null_add => 0 null_prod => 0 null_zero => 0 null_if => 0 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0 add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 fact(z) -{ 1 }-> if(zero(X), 1 + 0, prod(X, fact(p(X)))) :|: X >= 0, z = X if(z, z', z'') -{ 1 }-> X :|: z = 2, z' = X, Y >= 0, z'' = Y, X >= 0 if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 p(z) -{ 1 }-> X :|: z = 1 + X, X >= 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 prod(z, z') -{ 1 }-> add(Y, prod(X, Y)) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0 prod(z, z') -{ 1 }-> 0 :|: z' = X, X >= 0, z = 0 prod(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 zero(z) -{ 1 }-> 2 :|: z = 0 zero(z) -{ 1 }-> 1 :|: z = 1 + X, X >= 0 zero(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (61) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: FACT(z0) -> c(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), ZERO(z0)) FACT(z0) -> c1(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), PROD(z0, fact(p(z0))), FACT(p(z0)), P(z0)) ADD(0, z0) -> c2 ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(0, z0) -> c4 PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) IF(true, z0, z1) -> c6 IF(false, z0, z1) -> c7 ZERO(0) -> c8 ZERO(s(z0)) -> c9 P(s(z0)) -> c10 S tuples: FACT(z0) -> c(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), ZERO(z0)) FACT(z0) -> c1(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), PROD(z0, fact(p(z0))), FACT(p(z0)), P(z0)) ADD(0, z0) -> c2 ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(0, z0) -> c4 PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) IF(true, z0, z1) -> c6 IF(false, z0, z1) -> c7 ZERO(0) -> c8 ZERO(s(z0)) -> c9 P(s(z0)) -> c10 K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: FACT_1, ADD_2, PROD_2, IF_3, ZERO_1, P_1 Compound Symbols: c_2, c1_4, c2, c3_1, c4, c5_2, c6, c7, c8, c9, c10 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 8 trailing nodes: ZERO(0) -> c8 IF(true, z0, z1) -> c6 IF(false, z0, z1) -> c7 ADD(0, z0) -> c2 FACT(z0) -> c(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), ZERO(z0)) PROD(0, z0) -> c4 P(s(z0)) -> c10 ZERO(s(z0)) -> c9 ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: FACT(z0) -> c1(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), PROD(z0, fact(p(z0))), FACT(p(z0)), P(z0)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) S tuples: FACT(z0) -> c1(IF(zero(z0), s(0), prod(z0, fact(p(z0)))), PROD(z0, fact(p(z0))), FACT(p(z0)), P(z0)) ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: FACT_1, ADD_2, PROD_2 Compound Symbols: c1_4, c3_1, c5_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(z0) -> c1(PROD(z0, fact(p(z0))), FACT(p(z0))) S tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(z0) -> c1(PROD(z0, fact(p(z0))), FACT(p(z0))) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: ADD_2, PROD_2, FACT_1 Compound Symbols: c3_1, c5_2, c1_2 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(z0) -> c1(PROD(z0, fact(p(z0))), FACT(p(z0))) by FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), fact(p(p(x0)))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), fact(z0)), FACT(p(s(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), fact(p(p(x0)))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), fact(z0)), FACT(p(s(z0)))) S tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), fact(p(p(x0)))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), fact(z0)), FACT(p(s(z0)))) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: ADD_2, PROD_2, FACT_1 Compound Symbols: c3_1, c5_2, c1_2 ---------------------------------------- (69) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), fact(p(p(x0)))))), FACT(p(x0))) by FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), fact(z0)), FACT(p(s(z0)))) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) S tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), fact(z0)), FACT(p(s(z0)))) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: ADD_2, PROD_2, FACT_1 Compound Symbols: c3_1, c5_2, c1_2, c1_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FACT(s(z0)) -> c1(PROD(s(z0), fact(z0)), FACT(p(s(z0)))) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) S tuples: ADD(s(z0), z1) -> c3(ADD(z0, z1)) PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: ADD_2, PROD_2, FACT_1 Compound Symbols: c3_1, c5_2, c1_2, c1_1 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(z0), z1) -> c3(ADD(z0, z1)) by ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_2, c1_1, c3_1 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(x0) -> c1(PROD(x0, if(zero(p(x0)), s(0), prod(p(x0), if(zero(p(p(x0))), s(0), prod(p(p(x0)), fact(p(p(p(x0))))))))), FACT(p(x0))) by FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_2, c1_1, c3_1 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(p(s(z0)))) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) K tuples:none Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_2, c1_1, c3_1 ---------------------------------------- (79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) We considered the (Usable) Rules: p(s(z0)) -> z0 And the Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(FACT(x_1)) = x_1 POL(PROD(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] POL(false) = [1] POL(if(x_1, x_2, x_3)) = [1] + x_2 POL(p(x_1)) = x_1 POL(prod(x_1, x_2)) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = [1] POL(zero(x_1)) = [1] ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_2, c1_1, c3_1 ---------------------------------------- (81) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(p(s(z0)))) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_2, c1_1, c3_1 ---------------------------------------- (83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) We considered the (Usable) Rules: p(s(z0)) -> z0 And the Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(FACT(x_1)) = x_1 POL(PROD(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] + x_1 POL(false) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_3 POL(p(x_1)) = x_1 POL(prod(x_1, x_2)) = x_2 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_2, c1_1, c3_1 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(p(s(z0)))) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c1_2, c3_1 ---------------------------------------- (87) 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(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) We considered the (Usable) Rules: p(s(z0)) -> z0 And the Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(FACT(x_1)) = x_1 POL(PROD(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] + x_1 POL(false) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_3 POL(p(x_1)) = x_1 POL(prod(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c1_2, c3_1 ---------------------------------------- (89) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(p(s(z0)))) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (91) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) We considered the (Usable) Rules: p(s(z0)) -> z0 And the Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ADD(x_1, x_2)) = 0 POL(FACT(x_1)) = x_1 POL(PROD(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] + x_1 POL(false) = [1] POL(if(x_1, x_2, x_3)) = x_1 + x_2 POL(p(x_1)) = x_1 POL(prod(x_1, x_2)) = [1] + x_1 + x_2 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = x_1 ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(x0)) -> c1(FACT(p(s(x0)))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (93) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(x0)) -> c1(FACT(p(s(x0)))) by FACT(s(z0)) -> c1(FACT(z0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(FACT(z0)) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (95) 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(z0)) -> c1(FACT(z0)) We considered the (Usable) Rules: p(s(z0)) -> z0 And the Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ADD(x_1, x_2)) = 0 POL(FACT(x_1)) = x_1 POL(PROD(x_1, x_2)) = 0 POL(add(x_1, x_2)) = [1] + x_1 + x_2 POL(c1(x_1)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(fact(x_1)) = [1] + x_1 POL(false) = 0 POL(if(x_1, x_2, x_3)) = x_1 + x_3 POL(p(x_1)) = x_1 POL(prod(x_1, x_2)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(zero(x_1)) = 0 ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (97) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), fact(p(p(p(p(z0)))))))))))), FACT(p(z0))) by FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (99) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (101) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (103) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (105) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) by FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(s(z0)) -> c1(FACT(z0)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c3_1, c1_2 ---------------------------------------- (107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) by ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: fact(z0) -> if(zero(z0), s(0), prod(z0, fact(p(z0)))) add(0, z0) -> z0 add(s(z0), z1) -> s(add(z0, z1)) prod(0, z0) -> 0 prod(s(z0), z1) -> add(z1, prod(z0, z1)) if(true, z0, z1) -> z0 if(false, z0, z1) -> z1 zero(0) -> true zero(s(z0)) -> false p(s(z0)) -> z0 Tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(s(z0)) -> c1(FACT(z0)) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), if(zero(p(p(s(z0)))), s(0), prod(p(p(s(z0))), fact(p(p(p(s(z0)))))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, if(zero(p(z0)), s(0), prod(p(z0), fact(p(p(z0)))))))), FACT(z0)) ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) S tuples: PROD(s(z0), z1) -> c5(ADD(z1, prod(z0, z1)), PROD(z0, z1)) FACT(x0) -> c1(FACT(p(x0))) FACT(z0) -> c1(PROD(z0, if(zero(p(z0)), s(0), prod(p(z0), if(zero(p(p(z0))), s(0), prod(p(p(z0)), if(zero(p(p(p(z0)))), s(0), prod(p(p(p(z0))), if(zero(p(p(p(p(z0))))), s(0), prod(p(p(p(p(z0)))), fact(p(p(p(p(p(z0))))))))))))))), FACT(p(z0))) ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1)) K tuples: FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(p(s(z0)), fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(p(s(z0))), s(0), prod(z0, fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(p(s(z0)), fact(p(p(s(z0))))))), FACT(z0)) FACT(s(z0)) -> c1(PROD(s(z0), if(zero(z0), s(0), prod(z0, fact(p(z0))))), FACT(z0)) FACT(s(z0)) -> c1(FACT(z0)) Defined Rule Symbols: fact_1, add_2, prod_2, if_3, zero_1, p_1 Defined Pair Symbols: PROD_2, FACT_1, ADD_2 Compound Symbols: c5_2, c1_1, c1_2, c3_1