KILLED proof of input_JOMpMVynbq.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) InliningProof [UPPER BOUND(ID), 50 ms] (18) CpxRNTS (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 262 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 29 ms] (28) CpxRNTS (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 88 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] (34) CpxRNTS (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 436 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 155 ms] (40) CpxRNTS (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 428 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (46) CpxRNTS (47) CompletionProof [UPPER BOUND(ID), 0 ms] (48) CpxTypedWeightedCompleteTrs (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (52) CdtProblem (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 115 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtRewritingProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (80) 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: fac(0) -> 1 fac(s(x)) -> *(s(x), fac(x)) floop(0, y) -> y floop(s(x), y) -> floop(x, *(s(x), y)) *(x, 0) -> 0 *(x, s(y)) -> +(*(x, y), x) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) 1 -> s(0) fac(0) -> s(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fac(0') -> 1' fac(s(x)) -> *'(s(x), fac(x)) floop(0', y) -> y floop(s(x), y) -> floop(x, *'(s(x), y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(*'(x, y), x) +'(x, 0') -> x +'(x, s(y)) -> s(+'(x, y)) 1' -> s(0') fac(0') -> s(0') S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fac(0) -> 1 fac(s(x)) -> *(s(x), fac(x)) floop(0, y) -> y floop(s(x), y) -> floop(x, *(s(x), y)) *(x, 0) -> 0 *(x, s(y)) -> +(*(x, y), x) +(x, 0) -> x +(x, s(y)) -> s(+(x, y)) 1 -> s(0) fac(0) -> s(0) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fac(0) -> 1 [1] fac(s(x)) -> *(s(x), fac(x)) [1] floop(0, y) -> y [1] floop(s(x), y) -> floop(x, *(s(x), y)) [1] *(x, 0) -> 0 [1] *(x, s(y)) -> +(*(x, y), x) [1] +(x, 0) -> x [1] +(x, s(y)) -> s(+(x, y)) [1] 1 -> s(0) [1] fac(0) -> s(0) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: 1 => 1' * => times + => plus ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: fac(0) -> 1' [1] fac(s(x)) -> times(s(x), fac(x)) [1] floop(0, y) -> y [1] floop(s(x), y) -> floop(x, times(s(x), y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] 1' -> s(0) [1] fac(0) -> s(0) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (9) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fac(0) -> 1' [1] fac(s(x)) -> times(s(x), fac(x)) [1] floop(0, y) -> y [1] floop(s(x), y) -> floop(x, times(s(x), y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] 1' -> s(0) [1] fac(0) -> s(0) [1] The TRS has the following type information: fac :: 0:s -> 0:s 0 :: 0:s 1' :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s floop :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (11) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: floop_2 (c) The following functions are completely defined: times_2 fac_1 plus_2 1' Due to the following rules being added: none And the following fresh constants: none ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fac(0) -> 1' [1] fac(s(x)) -> times(s(x), fac(x)) [1] floop(0, y) -> y [1] floop(s(x), y) -> floop(x, times(s(x), y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] 1' -> s(0) [1] fac(0) -> s(0) [1] The TRS has the following type information: fac :: 0:s -> 0:s 0 :: 0:s 1' :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s floop :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (13) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (14) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: fac(0) -> 1' [1] fac(s(0)) -> times(s(0), 1') [2] fac(s(s(x'))) -> times(s(s(x')), times(s(x'), fac(x'))) [2] fac(s(0)) -> times(s(0), s(0)) [2] floop(0, y) -> y [1] floop(s(x), 0) -> floop(x, 0) [2] floop(s(x), s(y')) -> floop(x, plus(times(s(x), y'), s(x))) [2] times(x, 0) -> 0 [1] times(x, s(0)) -> plus(0, x) [2] times(x, s(s(y''))) -> plus(plus(times(x, y''), x), x) [2] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] 1' -> s(0) [1] fac(0) -> s(0) [1] The TRS has the following type information: fac :: 0:s -> 0:s 0 :: 0:s 1' :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s floop :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (15) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1') :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + x'), times(1 + x', fac(x'))) :|: x' >= 0, z = 1 + (1 + x') fac(z) -{ 1 }-> 1' :|: z = 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y floop(z, z') -{ 2 }-> floop(x, plus(times(1 + x, y'), 1 + x)) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y' floop(z, z') -{ 2 }-> floop(x, 0) :|: x >= 0, z = 1 + x, z' = 0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 2 }-> plus(plus(times(x, y''), x), x) :|: z' = 1 + (1 + y''), x >= 0, y'' >= 0, z = x times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (17) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: 1' -{ 1 }-> 1 + 0 :|: ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + x'), times(1 + x', fac(x'))) :|: x' >= 0, z = 1 + (1 + x') fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y floop(z, z') -{ 2 }-> floop(x, plus(times(1 + x, y'), 1 + x)) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y' floop(z, z') -{ 2 }-> floop(x, 0) :|: x >= 0, z = 1 + x, z' = 0 plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 2 }-> plus(plus(times(x, y''), x), x) :|: z' = 1 + (1 + y''), x >= 0, y'' >= 0, z = x times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (19) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (21) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { 1' } { times } { fac } { floop } ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {1'}, {times}, {fac}, {floop} ---------------------------------------- (23) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {1'}, {times}, {fac}, {floop} ---------------------------------------- (25) 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' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {1'}, {times}, {fac}, {floop} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (27) 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' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 0 plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(0, z) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {1'}, {times}, {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (29) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {1'}, {times}, {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] ---------------------------------------- (31) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: 1' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {1'}, {times}, {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: ?, size: O(1) [1] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: 1' after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (35) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] ---------------------------------------- (37) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + 2*z*z' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: times after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 4 + z + 2*z*z' + 4*z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 2 }-> floop(z - 1, plus(times(1 + (z - 1), z' - 1), 1 + (z - 1))) :|: z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(plus(times(z, z' - 2), z), z) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (41) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 13 }-> s'' :|: s'' >= 0, s'' <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0 fac(z) -{ 14 }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 3 + 2*z*z' + 4*z' }-> floop(z - 1, s2) :|: s1 >= 0, s1 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s2 >= 0, s2 <= s1 + (1 + (z - 1)), z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s5 :|: s3 >= 0, s3 <= z + 2 * ((z' - 2) * z), s4 >= 0, s4 <= s3 + z, s5 >= 0, s5 <= s4 + z, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (43) 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: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 13 }-> s'' :|: s'' >= 0, s'' <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0 fac(z) -{ 14 }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 3 + 2*z*z' + 4*z' }-> floop(z - 1, s2) :|: s1 >= 0, s1 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s2 >= 0, s2 <= s1 + (1 + (z - 1)), z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s5 :|: s3 >= 0, s3 <= z + 2 * ((z' - 2) * z), s4 >= 0, s4 <= s3 + z, s5 >= 0, s5 <= s4 + z, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] fac: runtime: ?, size: INF ---------------------------------------- (45) 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: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 13 }-> s'' :|: s'' >= 0, s'' <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0 fac(z) -{ 14 }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0 fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 fac(z) -{ 2 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 floop(z, z') -{ 3 + 2*z*z' + 4*z' }-> floop(z - 1, s2) :|: s1 >= 0, s1 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s2 >= 0, s2 <= s1 + (1 + (z - 1)), z - 1 >= 0, z' - 1 >= 0 floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s5 :|: s3 >= 0, s3 <= z + 2 * ((z' - 2) * z), s4 >= 0, s4 <= s3 + z, s5 >= 0, s5 <= s4 + z, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {fac}, {floop} Previous analysis results are: plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z'] 1': runtime: O(1) [1], size: O(1) [1] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] fac: runtime: INF, size: INF ---------------------------------------- (47) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (48) 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: fac(0) -> 1' [1] fac(s(x)) -> times(s(x), fac(x)) [1] floop(0, y) -> y [1] floop(s(x), y) -> floop(x, times(s(x), y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(times(x, y), x) [1] plus(x, 0) -> x [1] plus(x, s(y)) -> s(plus(x, y)) [1] 1' -> s(0) [1] fac(0) -> s(0) [1] The TRS has the following type information: fac :: 0:s -> 0:s 0 :: 0:s 1' :: 0:s s :: 0:s -> 0:s times :: 0:s -> 0:s -> 0:s floop :: 0:s -> 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s Rewrite Strategy: INNERMOST ---------------------------------------- (49) 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 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: 1' -{ 1 }-> 1 + 0 :|: fac(z) -{ 1 }-> times(1 + x, fac(x)) :|: x >= 0, z = 1 + x fac(z) -{ 1 }-> 1' :|: z = 0 fac(z) -{ 1 }-> 1 + 0 :|: z = 0 floop(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y floop(z, z') -{ 1 }-> floop(x, times(1 + x, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y 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') -{ 1 }-> plus(times(x, y), x) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (51) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) floop(0, z0) -> z0 floop(s(z0), z1) -> floop(z0, *(s(z0), z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) 1 -> s(0) Tuples: FAC(0) -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0) -> c2 FLOOP(0, z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0) -> c5 *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, 0) -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 S tuples: FAC(0) -> c(1') FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FAC(0) -> c2 FLOOP(0, z0) -> c3 FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, 0) -> c5 *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, 0) -> c7 +'(z0, s(z1)) -> c8(+'(z0, z1)) 1' -> c9 K tuples:none Defined Rule Symbols: fac_1, floop_2, *_2, +_2, 1 Defined Pair Symbols: FAC_1, FLOOP_2, *'_2, +'_2, 1' Compound Symbols: c_1, c1_2, c2, c3, c4_2, c5, c6_2, c7, c8_1, c9 ---------------------------------------- (53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: FAC(0) -> c2 +'(z0, 0) -> c7 1' -> c9 *'(z0, 0) -> c5 FAC(0) -> c(1') FLOOP(0, z0) -> c3 ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) floop(0, z0) -> z0 floop(s(z0), z1) -> floop(z0, *(s(z0), z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) 1 -> s(0) Tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) S tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples:none Defined Rule Symbols: fac_1, floop_2, *_2, +_2, 1 Defined Pair Symbols: FAC_1, FLOOP_2, *'_2, +'_2 Compound Symbols: c1_2, c4_2, c6_2, c8_1 ---------------------------------------- (55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: floop(0, z0) -> z0 floop(s(z0), z1) -> floop(z0, *(s(z0), z1)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) S tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples:none Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FAC_1, FLOOP_2, *'_2, +'_2 Compound Symbols: c1_2, c4_2, c6_2, c8_1 ---------------------------------------- (57) 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(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) We considered the (Usable) Rules:none And the Tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [1] + x_1 POL(*'(x_1, x_2)) = 0 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = 0 POL(0) = 0 POL(1) = [1] POL(FAC(x_1)) = x_1 POL(FLOOP(x_1, x_2)) = x_1 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(fac(x_1)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FAC_1, FLOOP_2, *'_2, +'_2 Compound Symbols: c1_2, c4_2, c6_2, c8_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) by FAC(s(0)) -> c1(*'(s(0), 1), FAC(0)) FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0))) FAC(s(0)) -> c1(*'(s(0), s(0)), FAC(0)) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), 1), FAC(0)) FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0))) FAC(s(0)) -> c1(*'(s(0), s(0)), FAC(0)) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, +'_2, FAC_1 Compound Symbols: c4_2, c6_2, c8_1, c1_2 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0))) FAC(s(0)) -> c1(*'(s(0), 1)) FAC(s(0)) -> c1(*'(s(0), s(0))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, +'_2, FAC_1 Compound Symbols: c4_2, c6_2, c8_1, c1_2, c1_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0))) by FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), 1)), FAC(s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), s(0))), FAC(s(0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), 1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), 1)), FAC(s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), s(0))), FAC(s(0))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, +'_2, FAC_1 Compound Symbols: c4_2, c6_2, c8_1, c1_1, c1_2 ---------------------------------------- (65) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), 1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), 1))) FAC(s(s(0))) -> c(FAC(s(0))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, +'_2, FAC_1 Compound Symbols: c4_2, c6_2, c8_1, c1_1, c1_2, c_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FAC(s(0)) -> c1(*'(s(0), 1)) by FAC(s(0)) -> c1(*'(s(0), s(0))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), 1))) FAC(s(s(0))) -> c(FAC(s(0))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, +'_2, FAC_1 Compound Symbols: c4_2, c6_2, c8_1, c1_1, c1_2, c_1 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), 1))) by FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(FAC(s(0))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(z1)) -> c8(+'(z0, z1)) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, +'_2, FAC_1 Compound Symbols: c4_2, c6_2, c8_1, c1_1, c1_2, c_1 ---------------------------------------- (71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(z1)) -> c8(+'(z0, z1)) by +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(FAC(s(0))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, FAC_1, +'_2 Compound Symbols: c4_2, c6_2, c1_1, c1_2, c_1, c8_1 ---------------------------------------- (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) by FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(FAC(s(0))) FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, FAC_1, +'_2 Compound Symbols: c4_2, c6_2, c1_1, c1_2, c_1, c8_1 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0)))) by FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(FAC(s(0))) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, FAC_1, +'_2 Compound Symbols: c4_2, c6_2, c1_1, c1_2, c_1, c8_1 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) by FAC(s(s(0))) -> c(*'(s(s(0)), s(+(*(s(0), 0), 0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(FAC(s(0))) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) FAC(s(s(0))) -> c(*'(s(s(0)), s(+(*(s(0), 0), 0)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, FAC_1, +'_2 Compound Symbols: c4_2, c6_2, c1_1, c1_2, c_1, c8_1 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) by +'(z0, s(s(s(y1)))) -> c8(+'(z0, s(s(y1)))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: fac(0) -> 1 fac(s(z0)) -> *(s(z0), fac(z0)) fac(0) -> s(0) 1 -> s(0) *(z0, 0) -> 0 *(z0, s(z1)) -> +(*(z0, z1), z0) +(z0, 0) -> z0 +(z0, s(z1)) -> s(+(z0, z1)) Tuples: FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) FAC(s(0)) -> c1(*'(s(0), s(0))) FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0)))) FAC(s(s(0))) -> c(FAC(s(0))) FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) FAC(s(s(0))) -> c(*'(s(s(0)), s(+(*(s(0), 0), 0)))) +'(z0, s(s(s(y1)))) -> c8(+'(z0, s(s(y1)))) S tuples: *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1)) +'(z0, s(s(s(y1)))) -> c8(+'(z0, s(s(y1)))) K tuples: FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1)) Defined Rule Symbols: fac_1, 1, *_2, +_2 Defined Pair Symbols: FLOOP_2, *'_2, FAC_1, +'_2 Compound Symbols: c4_2, c6_2, c1_1, c1_2, c_1, c8_1