WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 164 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (8) CdtProblem (9) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 1 ms] (14) CdtProblem (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 219 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 102 ms] (20) CdtProblem (21) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (22) BOUNDS(1, 1) (23) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (24) TRS for Loop Detection (25) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (26) BEST (27) proven lower bound (28) LowerBoundPropagationProof [FINISHED, 0 ms] (29) BOUNDS(n^1, INF) (30) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(u) -> u encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_u -> u ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) The (relative) TRS S consists of the following rules: encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(u) -> u encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_u -> u Rewrite Strategy: INNERMOST ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) The (relative) TRS S consists of the following rules: encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(u) -> u encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_u -> u Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_*(z0, z1) -> *(encArg(z0), encArg(z1)) encode_u -> u +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(u) -> c1 ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_*(z0, z1) -> c4(ENCARG(z0), ENCARG(z1)) ENCODE_U -> c5 +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(*(z0, +(z1, z2)), u), +'(z1, z2)) S tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(*(z0, +(z1, z2)), u), +'(z1, z2)) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_*_2, encode_u Defined Pair Symbols: ENCARG_1, ENCODE_+_2, ENCODE_*_2, ENCODE_U, +'_2 Compound Symbols: c_2, c1, c2_3, c3_3, c4_2, c5, c6_1, c7_2, c8_2 ---------------------------------------- (7) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_*(z0, z1) -> c4(ENCARG(z0), ENCARG(z1)) Removed 2 trailing nodes: ENCARG(u) -> c1 ENCODE_U -> c5 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_*(z0, z1) -> *(encArg(z0), encArg(z1)) encode_u -> u +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(*(z0, +(z1, z2)), u), +'(z1, z2)) S tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(*(z0, +(z1, z2)), u), +'(z1, z2)) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_*_2, encode_u Defined Pair Symbols: ENCARG_1, ENCODE_+_2, +'_2 Compound Symbols: c_2, c2_3, c3_3, c6_1, c7_2, c8_2 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_*(z0, z1) -> *(encArg(z0), encArg(z1)) encode_u -> u +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_+(z0, z1) -> c3(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) S tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_*_2, encode_u Defined Pair Symbols: ENCARG_1, ENCODE_+_2, +'_2 Compound Symbols: c_2, c2_3, c3_3, c6_1, c7_2, c8_1 ---------------------------------------- (11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_*(z0, z1) -> *(encArg(z0), encArg(z1)) encode_u -> u +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) ENCODE_+(z0, z1) -> c1(ENCARG(z0)) ENCODE_+(z0, z1) -> c1(ENCARG(z1)) S tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_*_2, encode_u Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c_2, c2_3, c6_1, c7_2, c8_1, c1_1 ---------------------------------------- (13) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_+(z0, z1) -> c1(ENCARG(z0)) ENCODE_+(z0, z1) -> c1(ENCARG(z1)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_*(z0, z1) -> *(encArg(z0), encArg(z1)) encode_u -> u +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) S tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_*_2, encode_u Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c_2, c2_3, c6_1, c7_2, c8_1, c1_1 ---------------------------------------- (15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_*(z0, z1) -> *(encArg(z0), encArg(z1)) encode_u -> u ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) S tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) K tuples:none Defined Rule Symbols: encArg_1, +_2 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c_2, c2_3, c6_1, c7_2, c8_1, c1_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) We considered the (Usable) Rules: encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) And the Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = [2] + x_1 + x_2 POL(+(x_1, x_2)) = x_1 + x_2 POL(+'(x_1, x_2)) = [2]x_1 POL(ENCARG(x_1)) = [2]x_1^2 POL(ENCODE_+(x_1, x_2)) = [2] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c6(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons_+(x_1, x_2)) = [2] + x_1 + x_2 POL(encArg(x_1)) = x_1 POL(u) = 0 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) S tuples: +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) K tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) Defined Rule Symbols: encArg_1, +_2 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c_2, c2_3, c6_1, c7_2, c8_1, c1_1 ---------------------------------------- (19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) We considered the (Usable) Rules: encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) And the Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(*(x_1, x_2)) = x_1 + x_2 POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = x_1 POL(ENCARG(x_1)) = x_1 + x_1^2 POL(ENCODE_+(x_1, x_2)) = [2] + x_1 + x_2 + [2]x_2^2 + [2]x_1*x_2 + x_1^2 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c6(x_1)) = x_1 POL(c7(x_1, x_2)) = x_1 + x_2 POL(c8(x_1)) = x_1 POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2 POL(encArg(x_1)) = x_1 POL(u) = 0 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(*(z0, z1)) -> *(encArg(z0), encArg(z1)) encArg(u) -> u encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(*(z0, z1), *(z0, z2)) -> *(z0, +(z1, z2)) +(+(z0, z1), z2) -> +(z0, +(z1, z2)) +(*(z0, z1), +(*(z0, z2), u)) -> +(*(z0, +(z1, z2)), u) Tuples: ENCARG(*(z0, z1)) -> c(ENCARG(z0), ENCARG(z1)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) ENCODE_+(z0, z1) -> c1(+'(encArg(z0), encArg(z1))) S tuples:none K tuples: +'(*(z0, z1), *(z0, z2)) -> c6(+'(z1, z2)) +'(*(z0, z1), +(*(z0, z2), u)) -> c8(+'(z1, z2)) +'(+(z0, z1), z2) -> c7(+'(z0, +(z1, z2)), +'(z1, z2)) Defined Rule Symbols: encArg_1, +_2 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c_2, c2_3, c6_1, c7_2, c8_1, c1_1 ---------------------------------------- (21) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (22) BOUNDS(1, 1) ---------------------------------------- (23) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (24) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) The (relative) TRS S consists of the following rules: encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(u) -> u encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_u -> u Rewrite Strategy: INNERMOST ---------------------------------------- (25) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence +(*(x1_0, y), *(x1_0, z)) ->^+ *(x1_0, +(y, z)) gives rise to a decreasing loop by considering the right hand sides subterm at position [1]. The pumping substitution is [y / *(x1_0, y), z / *(x1_0, z)]. The result substitution is [ ]. ---------------------------------------- (26) Complex Obligation (BEST) ---------------------------------------- (27) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) The (relative) TRS S consists of the following rules: encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(u) -> u encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_u -> u Rewrite Strategy: INNERMOST ---------------------------------------- (28) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (29) BOUNDS(n^1, INF) ---------------------------------------- (30) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(*(x, y), *(x, z)) -> *(x, +(y, z)) +(+(x, y), z) -> +(x, +(y, z)) +(*(x, y), +(*(x, z), u)) -> +(*(x, +(y, z)), u) The (relative) TRS S consists of the following rules: encArg(*(x_1, x_2)) -> *(encArg(x_1), encArg(x_2)) encArg(u) -> u encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_*(x_1, x_2) -> *(encArg(x_1), encArg(x_2)) encode_u -> u Rewrite Strategy: INNERMOST