KILLED proof of input_6BenuIXuA9.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxWeightedTrs (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxTypedWeightedTrs (11) CompletionProof [UPPER BOUND(ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (16) CpxRNTS (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxRNTS (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRNTS (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 373 ms] (24) CpxRNTS (25) IntTrsBoundProof [UPPER BOUND(ID), 98 ms] (26) CpxRNTS (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 204 ms] (30) CpxRNTS (31) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] (32) CpxRNTS (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 460 ms] (36) CpxRNTS (37) IntTrsBoundProof [UPPER BOUND(ID), 155 ms] (38) CpxRNTS (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 1665 ms] (42) CpxRNTS (43) IntTrsBoundProof [UPPER BOUND(ID), 887 ms] (44) CpxRNTS (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 51 ms] (62) CdtProblem (63) CdtRewritingProof [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) 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 (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) f(s(x)) -> f(-(*(s(s(0)), s(x)), s(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: -(x, 0') -> x -(s(x), s(y)) -> -(x, y) +'(0', y) -> y +'(s(x), y) -> s(+'(x, y)) *'(x, 0') -> 0' *'(x, s(y)) -> +'(x, *'(x, y)) f(s(x)) -> f(-(*'(s(s(0')), s(x)), s(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: -(x, 0) -> x -(s(x), s(y)) -> -(x, y) +(0, y) -> y +(s(x), y) -> s(+(x, y)) *(x, 0) -> 0 *(x, s(y)) -> +(x, *(x, y)) f(s(x)) -> f(-(*(s(s(0)), s(x)), s(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: -(x, 0) -> x [1] -(s(x), s(y)) -> -(x, y) [1] +(0, y) -> y [1] +(s(x), y) -> s(+(x, y)) [1] *(x, 0) -> 0 [1] *(x, s(y)) -> +(x, *(x, y)) [1] f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x)))) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: - => minus + => plus * => times ---------------------------------------- (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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(times(s(s(0)), s(x)), s(s(x)))) [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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(times(s(s(0)), s(x)), s(s(x)))) [1] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> f 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: f_1 (c) The following functions are completely defined: times_2 minus_2 plus_2 Due to the following rules being added: minus(v0, v1) -> 0 [0] And the following fresh constants: const ---------------------------------------- (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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(times(s(s(0)), s(x)), s(s(x)))) [1] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> f const :: f 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(0)) -> plus(x, 0) [2] times(x, s(s(y'))) -> plus(x, plus(x, times(x, y'))) [2] f(s(x)) -> f(minus(plus(s(s(0)), times(s(s(0)), x)), s(s(x)))) [2] minus(v0, v1) -> 0 [0] The TRS has the following type information: minus :: 0:s -> 0:s -> 0:s 0 :: 0:s s :: 0:s -> 0:s plus :: 0:s -> 0:s -> 0:s times :: 0:s -> 0:s -> 0:s f :: 0:s -> f const :: f 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 const => 0 ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), x)), 1 + (1 + x))) :|: x >= 0, z = 1 + x minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 2 }-> plus(x, plus(x, times(x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x times(z, z') -{ 2 }-> plus(x, 0) :|: x >= 0, z' = 1 + 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (17) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 ---------------------------------------- (19) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { minus } { plus } { times } { f } ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} ---------------------------------------- (21) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {minus}, {plus}, {times}, {f} Previous analysis results are: minus: runtime: ?, size: O(n^1) [z] ---------------------------------------- (25) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: minus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (27) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (31) 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 ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (33) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] ---------------------------------------- (35) 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' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {times}, {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (37) 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' ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 2 }-> f(minus(plus(1 + (1 + 0), times(1 + (1 + 0), z - 1)), 1 + (1 + (z - 1)))) :|: z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (39) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + 9*z }-> f(s6) :|: s4 >= 0, s4 <= 1 + (1 + 0) + 2 * ((z - 1) * (1 + (1 + 0))), s5 >= 0, s5 <= 1 + (1 + 0) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + 9*z }-> f(s6) :|: s4 >= 0, s4 <= 1 + (1 + 0) + 2 * ((z - 1) * (1 + (1 + 0))), s5 >= 0, s5 <= 1 + (1 + 0) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: ?, size: O(1) [0] ---------------------------------------- (43) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: f after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 5 + 9*z }-> f(s6) :|: s4 >= 0, s4 <= 1 + (1 + 0) + 2 * ((z - 1) * (1 + (1 + 0))), s5 >= 0, s5 <= 1 + (1 + 0) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0 minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0 minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0 plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0 times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0 times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0 times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {f} Previous analysis results are: minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z] plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z'] times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z'] f: runtime: INF, size: O(1) [0] ---------------------------------------- (45) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] And the following fresh constants: null_minus, null_f, null_plus, null_times ---------------------------------------- (46) 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: minus(x, 0) -> x [1] minus(s(x), s(y)) -> minus(x, y) [1] plus(0, y) -> y [1] plus(s(x), y) -> s(plus(x, y)) [1] times(x, 0) -> 0 [1] times(x, s(y)) -> plus(x, times(x, y)) [1] f(s(x)) -> f(minus(times(s(s(0)), s(x)), s(s(x)))) [1] minus(v0, v1) -> null_minus [0] f(v0) -> null_f [0] plus(v0, v1) -> null_plus [0] times(v0, v1) -> null_times [0] The TRS has the following type information: minus :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times 0 :: 0:s:null_minus:null_plus:null_times s :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times plus :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times times :: 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times -> 0:s:null_minus:null_plus:null_times f :: 0:s:null_minus:null_plus:null_times -> null_f null_minus :: 0:s:null_minus:null_plus:null_times null_f :: null_f null_plus :: 0:s:null_minus:null_plus:null_times null_times :: 0:s:null_minus:null_plus:null_times Rewrite Strategy: INNERMOST ---------------------------------------- (47) 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_minus => 0 null_f => 0 null_plus => 0 null_times => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: f(z) -{ 1 }-> f(minus(times(1 + (1 + 0), 1 + x), 1 + (1 + x))) :|: x >= 0, z = 1 + x f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y times(z, z') -{ 1 }-> plus(x, times(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (49) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) f(s(z0)) -> f(-(*(s(s(0)), s(z0)), s(s(z0)))) Tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(0, z0) -> c2 +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, 0) -> c4 *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) S tuples: -'(z0, 0) -> c -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(0, z0) -> c2 +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, 0) -> c4 *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, f_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c, c1_1, c2, c3_1, c4, c5_2, c6_3 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: +'(0, z0) -> c2 *'(z0, 0) -> c4 -'(z0, 0) -> c ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: -(z0, 0) -> z0 -(s(z0), s(z1)) -> -(z0, z1) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) f(s(z0)) -> f(-(*(s(s(0)), s(z0)), s(s(z0)))) Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) K tuples:none Defined Rule Symbols: -_2, +_2, *_2, f_1 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3 ---------------------------------------- (53) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: f(s(z0)) -> f(-(*(s(s(0)), s(z0)), s(s(z0)))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) by F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1))) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3 ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1))) by F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c6(-'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c6(-'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c6(-'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c6_2 ---------------------------------------- (59) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) K tuples:none Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1 ---------------------------------------- (61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) We considered the (Usable) Rules:none And the Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = x_1 POL(*'(x_1, x_2)) = x_1 POL(+(x_1, x_2)) = 0 POL(+'(x_1, x_2)) = 0 POL(-(x_1, x_2)) = [1] POL(-'(x_1, x_2)) = 0 POL(0) = 0 POL(F(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(s(x_1)) = 0 ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1 ---------------------------------------- (63) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) by F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1 ---------------------------------------- (65) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) by F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c6_3, c_1 ---------------------------------------- (67) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) by F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c_1, c6_3 ---------------------------------------- (69) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) by F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0)))) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c_1, c6_3 ---------------------------------------- (71) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) by F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c_1, c6_3 ---------------------------------------- (73) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) by F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c_1, c6_3 ---------------------------------------- (75) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) by F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0)))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c_1, c6_3 ---------------------------------------- (77) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0)))) by F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) S tuples: -'(s(z0), s(z1)) -> c1(-'(z0, z1)) +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: -'_2, +'_2, *'_2, F_1 Compound Symbols: c1_1, c3_1, c5_2, c_1, c6_3 ---------------------------------------- (79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(z0), s(z1)) -> c1(-'(z0, z1)) by -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) S tuples: +'(s(z0), z1) -> c3(+'(z0, z1)) *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: +'_2, *'_2, F_1, -'_2 Compound Symbols: c3_1, c5_2, c_1, c6_3, c1_1 ---------------------------------------- (81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(z0), z1) -> c3(+'(z0, z1)) by +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c_1, c6_3, c1_1, c3_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) by -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, -'_2 Compound Symbols: c5_2, c_1, c6_3, c3_1, c1_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0))) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0))) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, +'_2, -'_2 Compound Symbols: c5_2, c_1, c6_3, c3_1, c1_1, c6_2 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) by +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: *(z0, 0) -> 0 *(z0, s(z1)) -> +(z0, *(z0, z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) -(s(z0), s(z1)) -> -(z0, z1) -(z0, 0) -> z0 Tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(x0)) -> c(*'(s(s(0)), s(x0))) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0))) +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) S tuples: *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1)) F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0))) F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1)))) F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0))) +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1)) K tuples: F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0)))) F(s(x0)) -> c(*'(s(s(0)), s(x0))) Defined Rule Symbols: *_2, +_2, -_2 Defined Pair Symbols: *'_2, F_1, -'_2, +'_2 Compound Symbols: c5_2, c_1, c6_3, c1_1, c6_2, c3_1