KILLED proof of input_BbcN10rF1P.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 121 ms] (2) CpxRelTRS (3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxRelTRS (5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxTRS (7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (8) CpxWeightedTrs (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CpxWeightedTrs (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedTrs (13) CompletionProof [UPPER BOUND(ID), 0 ms] (14) CpxTypedWeightedCompleteTrs (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxTypedWeightedCompleteTrs (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (18) CpxRNTS (19) InliningProof [UPPER BOUND(ID), 121 ms] (20) CpxRNTS (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (22) CpxRNTS (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 190 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 9 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 453 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 110 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 418 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 35 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 613 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) CompletionProof [UPPER BOUND(ID), 0 ms] (50) CpxTypedWeightedCompleteTrs (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (52) CpxRNTS (53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (54) CdtProblem (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 70 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms] (82) CdtProblem (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', S(x)) -> mult(x', power(x', x)) mult(x', S(x)) -> add0(x', mult(x', x)) add0(x', S(x)) -> +(S(0), add0(x', x)) power(x, 0) -> S(0) mult(x, 0) -> 0 add0(x, 0) -> x The (relative) TRS S consists of the following rules: +(x, S(0)) -> S(x) +(S(0), y) -> S(y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', S(x)) -> mult(x', power(x', x)) mult(x', S(x)) -> add0(x', mult(x', x)) add0(x', S(x)) -> +(S(0), add0(x', x)) power(x, 0) -> S(0) mult(x, 0) -> 0 add0(x, 0) -> x The (relative) TRS S consists of the following rules: +(x, S(0)) -> S(x) +(S(0), y) -> S(y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: power(x', S(x)) -> mult(x', power(x', x)) mult(x', S(x)) -> add0(x', mult(x', x)) add0(x', S(x)) -> +'(S(0'), add0(x', x)) power(x, 0') -> S(0') mult(x, 0') -> 0' add0(x, 0') -> x The (relative) TRS S consists of the following rules: +'(x, S(0')) -> S(x) +'(S(0'), y) -> S(y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (6) 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: power(x', S(x)) -> mult(x', power(x', x)) mult(x', S(x)) -> add0(x', mult(x', x)) add0(x', S(x)) -> +(S(0), add0(x', x)) power(x, 0) -> S(0) mult(x, 0) -> 0 add0(x, 0) -> x +(x, S(0)) -> S(x) +(S(0), y) -> S(y) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (7) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (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: power(x', S(x)) -> mult(x', power(x', x)) [1] mult(x', S(x)) -> add0(x', mult(x', x)) [1] add0(x', S(x)) -> +(S(0), add0(x', x)) [1] power(x, 0) -> S(0) [1] mult(x, 0) -> 0 [1] add0(x, 0) -> x [1] +(x, S(0)) -> S(x) [0] +(S(0), y) -> S(y) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID)) Renamed defined symbols to avoid conflicts with arithmetic symbols: + => plus ---------------------------------------- (10) 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: power(x', S(x)) -> mult(x', power(x', x)) [1] mult(x', S(x)) -> add0(x', mult(x', x)) [1] add0(x', S(x)) -> plus(S(0), add0(x', x)) [1] power(x, 0) -> S(0) [1] mult(x, 0) -> 0 [1] add0(x, 0) -> x [1] plus(x, S(0)) -> S(x) [0] plus(S(0), y) -> S(y) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (11) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: power(x', S(x)) -> mult(x', power(x', x)) [1] mult(x', S(x)) -> add0(x', mult(x', x)) [1] add0(x', S(x)) -> plus(S(0), add0(x', x)) [1] power(x, 0) -> S(0) [1] mult(x, 0) -> 0 [1] add0(x, 0) -> x [1] plus(x, S(0)) -> S(x) [0] plus(S(0), y) -> S(y) [0] The TRS has the following type information: power :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 mult :: S:0 -> S:0 -> S:0 add0 :: S:0 -> S:0 -> S:0 plus :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (13) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: power_2 mult_2 add0_2 plus_2 Due to the following rules being added: plus(v0, v1) -> 0 [0] And the following fresh constants: none ---------------------------------------- (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: power(x', S(x)) -> mult(x', power(x', x)) [1] mult(x', S(x)) -> add0(x', mult(x', x)) [1] add0(x', S(x)) -> plus(S(0), add0(x', x)) [1] power(x, 0) -> S(0) [1] mult(x, 0) -> 0 [1] add0(x, 0) -> x [1] plus(x, S(0)) -> S(x) [0] plus(S(0), y) -> S(y) [0] plus(v0, v1) -> 0 [0] The TRS has the following type information: power :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 mult :: S:0 -> S:0 -> S:0 add0 :: S:0 -> S:0 -> S:0 plus :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (15) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (16) 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: power(x', S(S(x1))) -> mult(x', mult(x', power(x', x1))) [2] power(x', S(0)) -> mult(x', S(0)) [2] mult(x', S(S(x2))) -> add0(x', add0(x', mult(x', x2))) [2] mult(x', S(0)) -> add0(x', 0) [2] add0(x', S(S(x3))) -> plus(S(0), plus(S(0), add0(x', x3))) [2] add0(x', S(0)) -> plus(S(0), x') [2] power(x, 0) -> S(0) [1] mult(x, 0) -> 0 [1] add0(x, 0) -> x [1] plus(x, S(0)) -> S(x) [0] plus(S(0), y) -> S(y) [0] plus(v0, v1) -> 0 [0] The TRS has the following type information: power :: S:0 -> S:0 -> S:0 S :: S:0 -> S:0 mult :: S:0 -> S:0 -> S:0 add0 :: S:0 -> S:0 -> S:0 plus :: S:0 -> S:0 -> S:0 0 :: S:0 Rewrite Strategy: INNERMOST ---------------------------------------- (17) 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 ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, x') :|: x' >= 0, z' = 1 + 0, z = x' add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(x', x3))) :|: z' = 1 + (1 + x3), x' >= 0, x3 >= 0, z = x' mult(z, z') -{ 2 }-> add0(x', add0(x', mult(x', x2))) :|: z' = 1 + (1 + x2), x' >= 0, x2 >= 0, z = x' mult(z, z') -{ 2 }-> add0(x', 0) :|: x' >= 0, z' = 1 + 0, z = x' mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y power(z, z') -{ 2 }-> mult(x', mult(x', power(x', x1))) :|: x1 >= 0, z' = 1 + (1 + x1), x' >= 0, z = x' power(z, z') -{ 2 }-> mult(x', 1 + 0) :|: x' >= 0, z' = 1 + 0, z = x' power(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (19) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(x', x3))) :|: z' = 1 + (1 + x3), x' >= 0, x3 >= 0, z = x' add0(z, z') -{ 2 }-> 0 :|: x' >= 0, z' = 1 + 0, z = x', v0 >= 0, v1 >= 0, 1 + 0 = v0, x' = v1 add0(z, z') -{ 2 }-> 1 + x :|: x' >= 0, z' = 1 + 0, z = x', x >= 0, x' = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + y :|: x' >= 0, z' = 1 + 0, z = x', 1 + 0 = 1 + 0, y >= 0, x' = y mult(z, z') -{ 2 }-> add0(x', add0(x', mult(x', x2))) :|: z' = 1 + (1 + x2), x' >= 0, x2 >= 0, z = x' mult(z, z') -{ 2 }-> add0(x', 0) :|: x' >= 0, z' = 1 + 0, z = x' mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y power(z, z') -{ 2 }-> mult(x', mult(x', power(x', x1))) :|: x1 >= 0, z' = 1 + (1 + x1), x' >= 0, z = x' power(z, z') -{ 2 }-> mult(x', 1 + 0) :|: x' >= 0, z' = 1 + 0, z = x' power(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0 ---------------------------------------- (21) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 ---------------------------------------- (23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { plus } { add0 } { mult } { power } ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {add0}, {mult}, {power} ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {add0}, {mult}, {power} ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: plus after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {plus}, {add0}, {mult}, {power} Previous analysis results are: plus: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: plus after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {add0}, {mult}, {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {add0}, {mult}, {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: add0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {add0}, {mult}, {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: add0 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z' ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0 add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: mult after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: z + 2*z*z' ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0 add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {mult}, {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] mult: runtime: ?, size: O(n^2) [z + 2*z*z'] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: mult after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 7 + 8*z*z'^2 + 10*z' ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0 add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0 mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0 add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0 mult(z, z') -{ -3 + 2*s3 + 2*s4 + 32*z + -32*z*z' + 8*z*z'^2 + 10*z' }-> s5 :|: s3 >= 0, s3 <= 2 * ((z' - 2) * z) + z, s4 >= 0, s4 <= z + s3, s5 >= 0, s5 <= z + s4, z >= 0, z' - 2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 19 + 8*z }-> s2 :|: s2 >= 0, s2 <= 2 * ((1 + 0) * z) + z, z >= 0, z' = 1 + 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z'] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: power after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0 add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0 mult(z, z') -{ -3 + 2*s3 + 2*s4 + 32*z + -32*z*z' + 8*z*z'^2 + 10*z' }-> s5 :|: s3 >= 0, s3 <= 2 * ((z' - 2) * z) + z, s4 >= 0, s4 <= z + s3, s5 >= 0, s5 <= z + s4, z >= 0, z' - 2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 19 + 8*z }-> s2 :|: s2 >= 0, s2 <= 2 * ((1 + 0) * z) + z, z >= 0, z' = 1 + 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z'] power: runtime: ?, size: INF ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: power after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0 add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0 add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0 add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0 mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0 mult(z, z') -{ -3 + 2*s3 + 2*s4 + 32*z + -32*z*z' + 8*z*z'^2 + 10*z' }-> s5 :|: s3 >= 0, s3 <= 2 * ((z' - 2) * z) + z, s4 >= 0, s4 <= z + s3, s5 >= 0, s5 <= z + s4, z >= 0, z' - 2 >= 0 mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0 plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0 power(z, z') -{ 19 + 8*z }-> s2 :|: s2 >= 0, s2 <= 2 * ((1 + 0) * z) + z, z >= 0, z' = 1 + 0 power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0 power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0 Function symbols to be analyzed: {power} Previous analysis results are: plus: runtime: O(1) [0], size: O(n^1) [z + z'] add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z'] mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z'] power: runtime: INF, size: INF ---------------------------------------- (49) 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: plus(v0, v1) -> null_plus [0] power(v0, v1) -> null_power [0] mult(v0, v1) -> null_mult [0] add0(v0, v1) -> null_add0 [0] And the following fresh constants: null_plus, null_power, null_mult, null_add0 ---------------------------------------- (50) 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: power(x', S(x)) -> mult(x', power(x', x)) [1] mult(x', S(x)) -> add0(x', mult(x', x)) [1] add0(x', S(x)) -> plus(S(0), add0(x', x)) [1] power(x, 0) -> S(0) [1] mult(x, 0) -> 0 [1] add0(x, 0) -> x [1] plus(x, S(0)) -> S(x) [0] plus(S(0), y) -> S(y) [0] plus(v0, v1) -> null_plus [0] power(v0, v1) -> null_power [0] mult(v0, v1) -> null_mult [0] add0(v0, v1) -> null_add0 [0] The TRS has the following type information: power :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 S :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 mult :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 add0 :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 plus :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 0 :: S:0:null_plus:null_power:null_mult:null_add0 null_plus :: S:0:null_plus:null_power:null_mult:null_add0 null_power :: S:0:null_plus:null_power:null_mult:null_add0 null_mult :: S:0:null_plus:null_power:null_mult:null_add0 null_add0 :: S:0:null_plus:null_power:null_mult:null_add0 Rewrite Strategy: INNERMOST ---------------------------------------- (51) 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_plus => 0 null_power => 0 null_mult => 0 null_add0 => 0 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0 add0(z, z') -{ 1 }-> plus(1 + 0, add0(x', x)) :|: z' = 1 + x, x' >= 0, x >= 0, z = x' add0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 mult(z, z') -{ 1 }-> add0(x', mult(x', x)) :|: z' = 1 + x, x' >= 0, x >= 0, z = x' mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0 mult(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y power(z, z') -{ 1 }-> mult(x', power(x', x)) :|: z' = 1 + x, x' >= 0, x >= 0, z = x' power(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 power(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (53) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: +'(z0, S(0)) -> c +'(S(0), z0) -> c1 POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) POWER(z0, 0) -> c3 MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) MULT(z0, 0) -> c5 ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1)) ADD0(z0, 0) -> c7 S tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) POWER(z0, 0) -> c3 MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) MULT(z0, 0) -> c5 ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1)) ADD0(z0, 0) -> c7 K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: +'_2, POWER_2, MULT_2, ADD0_2 Compound Symbols: c, c1, c2_2, c3, c4_2, c5, c6_2, c7 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing nodes: +'(S(0), z0) -> c1 +'(z0, S(0)) -> c POWER(z0, 0) -> c3 ADD0(z0, 0) -> c7 MULT(z0, 0) -> c5 ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1)) S tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1)) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c4_2, c6_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) S tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c4_2, c6_1 ---------------------------------------- (59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) We considered the (Usable) Rules:none And the Tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(0) = [1] POL(ADD0(x_1, x_2)) = 0 POL(MULT(x_1, x_2)) = 0 POL(POWER(x_1, x_2)) = x_2 POL(S(x_1)) = [1] + x_1 POL(add0(x_1, x_2)) = [1] + x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(mult(x_1, x_2)) = [1] + x_2 POL(power(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) S tuples: MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c4_2, c6_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) by POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0)), POWER(z0, 0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0)), POWER(z0, 0)) S tuples: MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: MULT_2, ADD0_2, POWER_2 Compound Symbols: c4_2, c6_1, c2_2 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0))) S tuples: MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: MULT_2, ADD0_2, POWER_2 Compound Symbols: c4_2, c6_1, c2_2, c2_1 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) by MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4(ADD0(z0, 0), MULT(z0, 0)) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0))) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4(ADD0(z0, 0), MULT(z0, 0)) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4(ADD0(z0, 0), MULT(z0, 0)) K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_2, c2_1, c4_2 ---------------------------------------- (67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0))) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4 S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4 K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_2, c2_1, c4_2, c4 ---------------------------------------- (69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MULT(z0, S(0)) -> c4 We considered the (Usable) Rules:none And the Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0))) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4 The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = x_1 + x_2 POL(0) = 0 POL(ADD0(x_1, x_2)) = 0 POL(MULT(x_1, x_2)) = [1] POL(POWER(x_1, x_2)) = [1] + x_2 POL(S(x_1)) = [1] + x_1 POL(add0(x_1, x_2)) = [1] + x_1 + x_2 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4) = 0 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(mult(x_1, x_2)) = [1] + x_2 POL(power(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) POWER(z0, S(0)) -> c2(MULT(z0, S(0))) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4 S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(0)) -> c4 Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_2, c2_1, c4_2, c4 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) by POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))), POWER(z0, S(0))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(0)) -> c2(MULT(z0, S(0))) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) MULT(z0, S(0)) -> c4 POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))), POWER(z0, S(0))) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) K tuples: POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) MULT(z0, S(0)) -> c4 Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_1, c4_2, c4, c2_2 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: POWER(z0, S(0)) -> c2(MULT(z0, S(0))) MULT(z0, S(0)) -> c4 ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))), POWER(z0, S(0))) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, MULT_2, POWER_2 Compound Symbols: c6_1, c4_2, c2_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, MULT_2, POWER_2 Compound Symbols: c6_1, c4_2, c2_2, c2_1 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) by MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)), MULT(z0, S(0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)), MULT(z0, S(0))) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)), MULT(z0, S(0))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_2, c2_1, c4_2 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) K tuples:none Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_2, c2_1, c4_2, c4_1 ---------------------------------------- (81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) We considered the (Usable) Rules:none And the Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = x_1 + x_2 POL(0) = 0 POL(ADD0(x_1, x_2)) = 0 POL(MULT(x_1, x_2)) = [1] POL(POWER(x_1, x_2)) = [1] + x_2 POL(S(x_1)) = [1] + x_1 POL(add0(x_1, x_2)) = [1] + x_2 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c4(x_1)) = x_1 POL(c4(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(mult(x_1, x_2)) = [1] + x_2 POL(power(x_1, x_2)) = [1] + x_1 + x_2 ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) S tuples: ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) K tuples: MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: ADD0_2, POWER_2, MULT_2 Compound Symbols: c6_1, c2_2, c2_1, c4_2, c4_1 ---------------------------------------- (83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) by ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) S tuples: MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) K tuples: MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c2_1, c4_2, c4_1, c6_1 ---------------------------------------- (85) CdtRewritingProof (BOTH BOUNDS(ID, ID)) Used rewriting to replace MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) by MULT(z0, S(S(0))) -> c4(ADD0(z0, z0)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) MULT(z0, S(S(0))) -> c4(ADD0(z0, z0)) S tuples: MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) K tuples: MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c2_1, c4_2, c6_1, c4_1 ---------------------------------------- (87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) by ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1)))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) MULT(z0, S(S(0))) -> c4(ADD0(z0, z0)) ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1)))) S tuples: MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1)))) K tuples: MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c2_1, c4_2, c4_1, c6_1 ---------------------------------------- (89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MULT(z0, S(S(0))) -> c4(ADD0(z0, z0)) by MULT(S(S(S(y1))), S(S(0))) -> c4(ADD0(S(S(S(y1))), S(S(S(y1))))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: +(z0, S(0)) -> S(z0) +(S(0), z0) -> S(z0) power(z0, S(z1)) -> mult(z0, power(z0, z1)) power(z0, 0) -> S(0) mult(z0, S(z1)) -> add0(z0, mult(z0, z1)) mult(z0, 0) -> 0 add0(z0, S(z1)) -> +(S(0), add0(z0, z1)) add0(z0, 0) -> z0 Tuples: POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1)))) POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0)))) MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1)))) MULT(S(S(S(y1))), S(S(0))) -> c4(ADD0(S(S(S(y1))), S(S(S(y1))))) S tuples: MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1)))) ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1)))) K tuples: MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) Defined Rule Symbols: power_2, mult_2, add0_2, +_2 Defined Pair Symbols: POWER_2, MULT_2, ADD0_2 Compound Symbols: c2_2, c2_1, c4_2, c6_1, c4_1