WORST_CASE(Omega(n^1),O(n^1)) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 392 ms] (2) CpxRelTRS (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (4) CdtProblem (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 50 ms] (12) CdtProblem (13) CdtKnowledgeProof [FINISHED, 0 ms] (14) BOUNDS(1, 1) (15) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (16) TRS for Loop Detection (17) DecreasingLoopProof [LOWER BOUND(ID), 66 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^1, INF) (22) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) The (relative) TRS S consists of the following rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) The (relative) TRS S consists of the following rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) Tuples: DEL'(.(z0, .(z1, z2))) -> c8(F'(=(z0, z1), z0, z1, z2), =''(z0, z1)) F'(true, z0, z1, z2) -> c9(DEL'(.(z1, z2))) F'(false, z0, z1, z2) -> c10(DEL'(.(z1, z2))) =''(nil, nil) -> c11 =''(.(z0, z1), nil) -> c12 =''(nil, .(z0, z1)) -> c13 =''(.(z0, z1), .(u, v)) -> c14(=''(z0, u), =''(z1, v)) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2), =''(z0, z1), ='''(z0, z1)) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) ='''(nil, nil) -> c18 ='''(.(z0, z1), nil) -> c19 ='''(nil, .(z0, z1)) -> c20 ='''(.(z0, z1), .(u, v)) -> c21(='''(z0, u)) ='''(.(z0, z1), .(u, v)) -> c22(='''(z1, v)) S tuples: DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2), =''(z0, z1), ='''(z0, z1)) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) ='''(nil, nil) -> c18 ='''(.(z0, z1), nil) -> c19 ='''(nil, .(z0, z1)) -> c20 ='''(.(z0, z1), .(u, v)) -> c21(='''(z0, u)) ='''(.(z0, z1), .(u, v)) -> c22(='''(z1, v)) K tuples:none Defined Rule Symbols: DEL_1, F_4, ='_2, del_1, f_4, =_2 Defined Pair Symbols: DEL'_1, F'_4, =''_2, DEL''_1, F''_4, ='''_2 Compound Symbols: c8_2, c9_1, c10_1, c11, c12, c13, c14_2, c15_3, c16_1, c17_1, c18, c19, c20, c21_1, c22_1 ---------------------------------------- (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing nodes: ='''(nil, nil) -> c18 =''(nil, .(z0, z1)) -> c13 ='''(.(z0, z1), .(u, v)) -> c22(='''(z1, v)) ='''(.(z0, z1), nil) -> c19 =''(.(z0, z1), .(u, v)) -> c14(=''(z0, u), =''(z1, v)) =''(.(z0, z1), nil) -> c12 =''(nil, nil) -> c11 ='''(.(z0, z1), .(u, v)) -> c21(='''(z0, u)) ='''(nil, .(z0, z1)) -> c20 ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) Tuples: DEL'(.(z0, .(z1, z2))) -> c8(F'(=(z0, z1), z0, z1, z2), =''(z0, z1)) F'(true, z0, z1, z2) -> c9(DEL'(.(z1, z2))) F'(false, z0, z1, z2) -> c10(DEL'(.(z1, z2))) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2), =''(z0, z1), ='''(z0, z1)) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) S tuples: DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2), =''(z0, z1), ='''(z0, z1)) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) K tuples:none Defined Rule Symbols: DEL_1, F_4, ='_2, del_1, f_4, =_2 Defined Pair Symbols: DEL'_1, F'_4, DEL''_1, F''_4 Compound Symbols: c8_2, c9_1, c10_1, c15_3, c16_1, c17_1 ---------------------------------------- (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) Tuples: F'(true, z0, z1, z2) -> c9(DEL'(.(z1, z2))) F'(false, z0, z1, z2) -> c10(DEL'(.(z1, z2))) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL'(.(z0, .(z1, z2))) -> c8(F'(=(z0, z1), z0, z1, z2)) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) S tuples: F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) K tuples:none Defined Rule Symbols: DEL_1, F_4, ='_2, del_1, f_4, =_2 Defined Pair Symbols: F'_4, F''_4, DEL'_1, DEL''_1 Compound Symbols: c9_1, c10_1, c16_1, c17_1, c8_1, c15_1 ---------------------------------------- (9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Tuples: F'(true, z0, z1, z2) -> c9(DEL'(.(z1, z2))) F'(false, z0, z1, z2) -> c10(DEL'(.(z1, z2))) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL'(.(z0, .(z1, z2))) -> c8(F'(=(z0, z1), z0, z1, z2)) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) S tuples: F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) K tuples:none Defined Rule Symbols: =_2 Defined Pair Symbols: F'_4, F''_4, DEL'_1, DEL''_1 Compound Symbols: c9_1, c10_1, c16_1, c17_1, c8_1, c15_1 ---------------------------------------- (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) We considered the (Usable) Rules:none And the Tuples: F'(true, z0, z1, z2) -> c9(DEL'(.(z1, z2))) F'(false, z0, z1, z2) -> c10(DEL'(.(z1, z2))) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL'(.(z0, .(z1, z2))) -> c8(F'(=(z0, z1), z0, z1, z2)) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) The order we found is given by the following interpretation: Polynomial interpretation : POL(.(x_1, x_2)) = [1] + x_2 POL(=(x_1, x_2)) = [1] POL(DEL'(x_1)) = 0 POL(DEL''(x_1)) = x_1 POL(F'(x_1, x_2, x_3, x_4)) = 0 POL(F''(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(and(x_1, x_2)) = [1] + x_1 + x_2 POL(c10(x_1)) = x_1 POL(c15(x_1)) = x_1 POL(c16(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c8(x_1)) = x_1 POL(c9(x_1)) = x_1 POL(false) = [1] POL(nil) = [1] POL(true) = [1] POL(u) = [1] POL(v) = [1] ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Tuples: F'(true, z0, z1, z2) -> c9(DEL'(.(z1, z2))) F'(false, z0, z1, z2) -> c10(DEL'(.(z1, z2))) F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL'(.(z0, .(z1, z2))) -> c8(F'(=(z0, z1), z0, z1, z2)) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) S tuples: F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) K tuples: DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) Defined Rule Symbols: =_2 Defined Pair Symbols: F'_4, F''_4, DEL'_1, DEL''_1 Compound Symbols: c9_1, c10_1, c16_1, c17_1, c8_1, c15_1 ---------------------------------------- (13) CdtKnowledgeProof (FINISHED) The following tuples could be moved from S to K by knowledge propagation: F''(true, z0, z1, z2) -> c16(DEL''(.(z1, z2))) F''(false, z0, z1, z2) -> c17(DEL''(.(z1, z2))) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) DEL''(.(z0, .(z1, z2))) -> c15(F''(=(z0, z1), z0, z1, z2)) Now S is empty ---------------------------------------- (14) BOUNDS(1, 1) ---------------------------------------- (15) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (16) 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^1). The TRS R consists of the following rules: DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) The (relative) TRS S consists of the following rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Rewrite Strategy: INNERMOST ---------------------------------------- (17) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence F(true, z0, nil, .(nil, z23_0)) ->^+ c1(c(F(true, nil, nil, z23_0), ='(nil, nil))) gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0]. The pumping substitution is [z23_0 / .(nil, z23_0)]. The result substitution is [z0 / nil]. ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) 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^1). The TRS R consists of the following rules: DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) The (relative) TRS S consists of the following rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Rewrite Strategy: INNERMOST ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^1, INF) ---------------------------------------- (22) 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^1). The TRS R consists of the following rules: DEL(.(z0, .(z1, z2))) -> c(F(=(z0, z1), z0, z1, z2), ='(z0, z1)) F(true, z0, z1, z2) -> c1(DEL(.(z1, z2))) F(false, z0, z1, z2) -> c2(DEL(.(z1, z2))) ='(nil, nil) -> c3 ='(.(z0, z1), nil) -> c4 ='(nil, .(z0, z1)) -> c5 ='(.(z0, z1), .(u, v)) -> c6(='(z0, u)) ='(.(z0, z1), .(u, v)) -> c7(='(z1, v)) The (relative) TRS S consists of the following rules: del(.(z0, .(z1, z2))) -> f(=(z0, z1), z0, z1, z2) f(true, z0, z1, z2) -> del(.(z1, z2)) f(false, z0, z1, z2) -> .(z0, del(.(z1, z2))) =(nil, nil) -> true =(.(z0, z1), nil) -> false =(nil, .(z0, z1)) -> false =(.(z0, z1), .(u, v)) -> and(=(z0, u), =(z1, v)) Rewrite Strategy: INNERMOST