KILLED proof of input_FwEHoFzOiP.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 203 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 91 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 77 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 524 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 170 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 1124 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 116 ms] (42) CpxRNTS (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 725 ms] (52) CdtProblem (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 558 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) 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: plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) times(0, y) -> 0 times(x, 0) -> 0 times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 fac(s(x)) -> times(fac(p(s(x))), s(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: plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) times(0', y) -> 0' times(x, 0') -> 0' times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0')) -> 0' fac(s(x)) -> times(fac(p(s(x))), s(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: plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) times(0, y) -> 0 times(x, 0) -> 0 times(s(x), y) -> plus(times(x, y), y) p(s(s(x))) -> s(p(s(x))) p(s(0)) -> 0 fac(s(x)) -> times(fac(p(s(x))), s(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: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(x, 0) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] p(s(s(x))) -> s(p(s(x))) [1] p(s(0)) -> 0 [1] fac(s(x)) -> times(fac(p(s(x))), s(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: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(x, 0) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] p(s(s(x))) -> s(p(s(x))) [1] p(s(0)) -> 0 [1] fac(s(x)) -> times(fac(p(s(x))), s(x)) [1] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s fac :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: times_2 fac_1 p_1 plus_2 Due to the following rules being added: fac(v0) -> 0 [0] 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: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(x, 0) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] p(s(s(x))) -> s(p(s(x))) [1] p(s(0)) -> 0 [1] fac(s(x)) -> times(fac(p(s(x))), s(x)) [1] fac(v0) -> 0 [0] p(v0) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s fac :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(x, 0) -> 0 [1] times(s(0), y) -> plus(0, y) [2] times(s(x), 0) -> plus(0, 0) [2] times(s(s(x')), y) -> plus(plus(times(x', y), y), y) [2] p(s(s(x))) -> s(p(s(x))) [1] p(s(0)) -> 0 [1] fac(s(s(x''))) -> times(fac(s(p(s(x'')))), s(s(x''))) [2] fac(s(0)) -> times(fac(0), s(0)) [2] fac(s(x)) -> times(fac(0), s(x)) [1] fac(v0) -> 0 [0] p(v0) -> 0 [0] The TRS has the following type information: plus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s p :: 0:s -> 0:s fac :: 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 1 }-> times(fac(0), 1 + x) :|: x >= 0, z = 1 + x fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + x'')), 1 + (1 + x'')) :|: x'' >= 0, z = 1 + (1 + x'') fac(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 2 }-> plus(plus(times(x', y), y), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y times(z, z') -{ 2 }-> plus(0, y) :|: z = 1 + 0, y >= 0, z' = y times(z, z') -{ 2 }-> plus(0, 0) :|: x >= 0, z = 1 + x, z' = 0 times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, 0) :|: z - 1 >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { p } { times } { fac } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, 0) :|: z - 1 >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {fac} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, 0) :|: z - 1 >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {fac} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, 0) :|: z - 1 >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {p}, {times}, {fac} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0 times(z, z') -{ 2 }-> plus(0, 0) :|: z - 1 >= 0, z' = 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {p}, {times}, {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {p}, {times}, {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {p}, {times}, {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: p after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 }-> times(fac(1 + p(1 + (z - 2))), 1 + (1 + (z - 2))) :|: z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 1 + p(1 + (z - 2)) :|: z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2*z*z' + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] times: runtime: ?, size: O(n^2) [2*z*z' + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 8 + 4*z + 2*z*z' + z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 4 + 4*z + 2*z*z' + -1*z' }-> s5 :|: s3 >= 0, s3 <= z' + 2 * (z' * (z - 2)), s4 >= 0, s4 <= s3 + z', s5 >= 0, s5 <= s4 + z', z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: fac after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 4 + 4*z + 2*z*z' + -1*z' }-> s5 :|: s3 >= 0, s3 <= z' + 2 * (z' * (z - 2)), s4 >= 0, s4 <= s3 + z', s5 >= 0, s5 <= s4 + z', z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] fac: runtime: ?, size: INF ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: fac after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0 fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0 fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0 fac(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: z >= 0 p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0 times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0 times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0 times(z, z') -{ 4 + 4*z + 2*z*z' + -1*z' }-> s5 :|: s3 >= 0, s3 <= z' + 2 * (z' * (z - 2)), s4 >= 0, s4 <= s3 + z', s5 >= 0, s5 <= s4 + z', z - 2 >= 0, z' >= 0 times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] p: runtime: O(n^1) [1 + z], size: O(n^1) [z] times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z'] fac: runtime: INF, size: INF ---------------------------------------- (43) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: p(v0) -> null_p [0] fac(v0) -> null_fac [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] And the following fresh constants: null_p, null_fac, null_plus, null_times ---------------------------------------- (44) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] times(0, y) -> 0 [1] times(x, 0) -> 0 [1] times(s(x), y) -> plus(times(x, y), y) [1] p(s(s(x))) -> s(p(s(x))) [1] p(s(0)) -> 0 [1] fac(s(x)) -> times(fac(p(s(x))), s(x)) [1] p(v0) -> null_p [0] fac(v0) -> null_fac [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] The TRS has the following type information: plus :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times 0 :: 0:s:null_p:null_fac:null_plus:null_times s :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times times :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times p :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times fac :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times null_p :: 0:s:null_p:null_fac:null_plus:null_times null_fac :: 0:s:null_p:null_fac:null_plus:null_times null_plus :: 0:s:null_p:null_fac:null_plus:null_times null_times :: 0:s:null_p:null_fac:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (45) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 null_p => 0 null_fac => 0 null_plus => 0 null_times => 0 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: fac(z) -{ 1 }-> times(fac(p(1 + x)), 1 + x) :|: x >= 0, z = 1 + x fac(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 0 :|: z = 1 + 0 p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 p(z) -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z = 1 + (1 + x) plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> plus(times(x, y), y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (47) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, 0) -> c PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(0, z0) -> c2 TIMES(z0, 0) -> c3 TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) P(s(0)) -> c6 FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) S tuples: PLUS(z0, 0) -> c PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(0, z0) -> c2 TIMES(z0, 0) -> c3 TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) P(s(0)) -> c6 FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c, c1_1, c2, c3, c4_2, c5_1, c6, c7_3 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: TIMES(0, z0) -> c2 P(s(0)) -> c6 PLUS(z0, 0) -> c TIMES(z0, 0) -> c3 ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) S tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) K tuples:none Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3 ---------------------------------------- (51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) We considered the (Usable) Rules: plus(z0, s(z1)) -> s(plus(z0, z1)) times(s(z0), z1) -> plus(times(z0, z1), z1) plus(z0, 0) -> z0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) times(0, z0) -> 0 times(z0, 0) -> 0 And the Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1)) = 0 POL(P(x_1)) = 0 POL(PLUS(x_1, x_2)) = x_2 POL(TIMES(x_1, x_2)) = x_1*x_2 POL(c1(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(fac(x_1)) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = x_1 + [2]x_2 POL(s(x_1)) = [1] + x_1 POL(times(x_1, x_2)) = [2]x_1*x_2 ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) S tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3 ---------------------------------------- (53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) We considered the (Usable) Rules: plus(z0, s(z1)) -> s(plus(z0, z1)) times(s(z0), z1) -> plus(times(z0, z1), z1) plus(z0, 0) -> z0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) times(0, z0) -> 0 times(z0, 0) -> 0 And the Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1)) = 0 POL(P(x_1)) = 0 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_1 POL(c1(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(fac(x_1)) = 0 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = x_1 + x_2 POL(s(x_1)) = [2] + x_1 POL(times(x_1, x_2)) = x_1*x_2 ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) by FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) FAC(s(0)) -> c7(TIMES(fac(0), s(0)), FAC(p(s(0))), P(s(0))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) FAC(s(0)) -> c7(TIMES(fac(0), s(0)), FAC(p(s(0))), P(s(0))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) FAC(s(0)) -> c7(TIMES(fac(0), s(0)), FAC(p(s(0))), P(s(0))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) FAC(s(0)) -> c7(FAC(p(s(0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) FAC(s(0)) -> c7(FAC(p(s(0)))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3, c7_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) by FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(0)) -> c7(FAC(p(s(0)))) FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(0)) -> c7(FAC(p(s(0)))) FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_1, c7_3, c7_2 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(0)) -> c7(FAC(p(s(0)))) by FAC(s(0)) -> c7(FAC(0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(0)) -> c7(FAC(0)) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(0)) -> c7(FAC(0)) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3, c7_2, c7_1 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: FAC(s(0)) -> c7(FAC(0)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3, c7_2 ---------------------------------------- (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) by FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3, c7_2 ---------------------------------------- (67) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) by FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3, c7_2 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) by FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_2, c7_3 ---------------------------------------- (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) by FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) K tuples: PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: PLUS_2, TIMES_2, P_1, FAC_1 Compound Symbols: c1_1, c4_2, c5_1, c7_3, c7_2 ---------------------------------------- (73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(z1)) -> c1(PLUS(z0, z1)) by PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) S tuples: P(s(s(z0))) -> c5(P(s(z0))) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, P_1, FAC_1, PLUS_2 Compound Symbols: c4_2, c5_1, c7_3, c7_2, c1_1 ---------------------------------------- (75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(z0))) -> c5(P(s(z0))) by P(s(s(s(y0)))) -> c5(P(s(s(y0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, PLUS_2, P_1 Compound Symbols: c4_2, c7_3, c7_2, c1_1, c5_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, PLUS_2, P_1 Compound Symbols: c4_2, c7_3, c7_2, c1_1, c5_1 ---------------------------------------- (79) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) by FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, PLUS_2, P_1 Compound Symbols: c4_2, c7_3, c7_2, c1_1, c5_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) by PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, P_1, PLUS_2 Compound Symbols: c4_2, c7_3, c7_2, c5_1, c1_1 ---------------------------------------- (83) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) by FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(0))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(0))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, P_1, PLUS_2 Compound Symbols: c4_2, c7_3, c7_2, c5_1, c1_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, P_1, PLUS_2 Compound Symbols: c4_2, c7_3, c7_2, c5_1, c1_1, c7_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. FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) We considered the (Usable) Rules:none And the Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(FAC(x_1)) = [1] + x_1 POL(P(x_1)) = x_1 POL(PLUS(x_1, x_2)) = 0 POL(TIMES(x_1, x_2)) = x_2 POL(c1(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(fac(x_1)) = x_1 POL(p(x_1)) = 0 POL(plus(x_1, x_2)) = x_1 POL(s(x_1)) = 0 POL(times(x_1, x_2)) = x_2 ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) P(s(s(s(y0)))) -> c5(P(s(s(y0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, P_1, PLUS_2 Compound Symbols: c4_2, c7_3, c7_2, c5_1, c1_1, c7_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace P(s(s(s(y0)))) -> c5(P(s(s(y0)))) by P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: plus(z0, 0) -> z0 plus(z0, s(z1)) -> s(plus(z0, z1)) times(0, z0) -> 0 times(z0, 0) -> 0 times(s(z0), z1) -> plus(times(z0, z1), z1) p(s(s(z0))) -> s(p(s(z0))) p(s(0)) -> 0 fac(s(z0)) -> times(fac(p(s(z0))), s(z0)) Tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) S tuples: FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0)))) FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0))))) P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0))))) K tuples: TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1)) PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1)))) FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0)))) Defined Rule Symbols: plus_2, times_2, p_1, fac_1 Defined Pair Symbols: TIMES_2, FAC_1, PLUS_2, P_1 Compound Symbols: c4_2, c7_3, c7_2, c1_1, c7_1, c5_1