WORST_CASE(Omega(n^1), O(n^2)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).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), 154 ms] (4) CpxRelTRS (5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (8) CdtProblem (9) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (10) CdtProblem (11) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (14) CdtProblem (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 61 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 51 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 (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (5) RcToIrcProof (BOTH BOUNDS(ID, ID)) Converted rc-obligation to irc-obligation. As the TRS is a non-duplicating overlay system, we have rc = irc. ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_0 -> 0 encode_s(z0) -> s(encArg(z0)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(0) -> c ENCARG(s(z0)) -> c1(ENCARG(z0)) 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_0 -> c4 ENCODE_S(z0) -> c5(ENCARG(z0)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) S tuples: +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_0, encode_s_1 Defined Pair Symbols: ENCARG_1, ENCODE_+_2, ENCODE_0, ENCODE_S_1, +'_2 Compound Symbols: c, c1_1, c2_3, c3_3, c4, c5_1, c6, c7_1, c8_1 ---------------------------------------- (9) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_S(z0) -> c5(ENCARG(z0)) Removed 2 trailing nodes: ENCODE_0 -> c4 ENCARG(0) -> c ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_0 -> 0 encode_s(z0) -> s(encArg(z0)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) 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)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) S tuples: +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_0, encode_s_1 Defined Pair Symbols: ENCARG_1, ENCODE_+_2, +'_2 Compound Symbols: c1_1, c2_3, c3_3, c6, c7_1, 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(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_0 -> 0 encode_s(z0) -> s(encArg(z0)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1))) ENCODE_+(z0, z1) -> c(ENCARG(z0)) ENCODE_+(z0, z1) -> c(ENCARG(z1)) S tuples: +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_0, encode_s_1 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c1_1, c2_3, c6, c7_1, c8_1, c_1 ---------------------------------------- (13) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 2 leading nodes: ENCODE_+(z0, z1) -> c(ENCARG(z0)) ENCODE_+(z0, z1) -> c(ENCARG(z1)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_0 -> 0 encode_s(z0) -> s(encArg(z0)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1))) S tuples: +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) K tuples:none Defined Rule Symbols: +_2, encArg_1, encode_+_2, encode_0, encode_s_1 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c1_1, c2_3, c6, c7_1, c8_1, c_1 ---------------------------------------- (15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_+(z0, z1) -> +(encArg(z0), encArg(z1)) encode_0 -> 0 encode_s(z0) -> s(encArg(z0)) ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1))) S tuples: +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) K tuples:none Defined Rule Symbols: encArg_1, +_2 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c1_1, c2_3, c6, c7_1, c8_1, c_1 ---------------------------------------- (17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. +'(0, z0) -> c6 We considered the (Usable) Rules:none And the Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(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] POL(0) = [1] POL(ENCARG(x_1)) = x_1 POL(ENCODE_+(x_1, x_2)) = [1] + x_2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c6) = 0 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons_+(x_1, x_2)) = [1] + x_1 + x_2 POL(encArg(x_1)) = [1] + x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1))) S tuples: +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) K tuples: +'(0, z0) -> c6 Defined Rule Symbols: encArg_1, +_2 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c1_1, c2_3, c6, c7_1, c8_1, c_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. +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) We considered the (Usable) Rules: encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) encArg(s(z0)) -> s(encArg(z0)) +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) encArg(0) -> 0 +(0, z0) -> z0 And the Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1))) The order we found is given by the following interpretation: Polynomial interpretation : POL(+(x_1, x_2)) = [1] + x_1 + x_2 POL(+'(x_1, x_2)) = x_1 POL(0) = 0 POL(ENCARG(x_1)) = [2]x_1^2 POL(ENCODE_+(x_1, x_2)) = [2] + x_1 + [2]x_2 + [2]x_2^2 + x_1*x_2 + [2]x_1^2 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c6) = 0 POL(c7(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(cons_+(x_1, x_2)) = [2] + x_1 + x_2 POL(encArg(x_1)) = x_1 POL(s(x_1)) = [1] + x_1 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(0) -> 0 encArg(s(z0)) -> s(encArg(z0)) encArg(cons_+(z0, z1)) -> +(encArg(z0), encArg(z1)) +(0, z0) -> z0 +(s(z0), z1) -> s(+(z0, z1)) +(s(z0), z1) -> +(z0, s(z1)) Tuples: ENCARG(s(z0)) -> c1(ENCARG(z0)) ENCARG(cons_+(z0, z1)) -> c2(+'(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) ENCODE_+(z0, z1) -> c(+'(encArg(z0), encArg(z1))) S tuples:none K tuples: +'(0, z0) -> c6 +'(s(z0), z1) -> c7(+'(z0, z1)) +'(s(z0), z1) -> c8(+'(z0, s(z1))) Defined Rule Symbols: encArg_1, +_2 Defined Pair Symbols: ENCARG_1, +'_2, ENCODE_+_2 Compound Symbols: c1_1, c2_3, c6, c7_1, c8_1, c_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 (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (25) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence +(s(x), y) ->^+ +(x, s(y)) gives rise to a decreasing loop by considering the right hand sides subterm at position []. The pumping substitution is [x / s(x)]. The result substitution is [y / s(y)]. ---------------------------------------- (26) Complex Obligation (BEST) ---------------------------------------- (27) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL ---------------------------------------- (28) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (29) BOUNDS(n^1, INF) ---------------------------------------- (30) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: +(0, y) -> y +(s(x), y) -> s(+(x, y)) +(s(x), y) -> +(x, s(y)) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encode_+(x_1, x_2) -> +(encArg(x_1), encArg(x_2)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) Rewrite Strategy: FULL