/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^3)) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). (0) CpxTRS (1) CpxTrsToCdtProof [UPPER BOUND(ID), 15 ms] (2) CdtProblem (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CdtProblem (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 123 ms] (8) CdtProblem (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 39 ms] (10) CdtProblem (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms] (12) CdtProblem (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 150 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 890 ms] (16) CdtProblem (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (18) BOUNDS(1, 1) (19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (20) TRS for Loop Detection (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (22) BEST (23) proven lower bound (24) LowerBoundPropagationProof [FINISHED, 0 ms] (25) BOUNDS(n^1, INF) (26) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (2) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) U22(dout(z0), z1, z2, z3) -> c4 U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) U32(dout(z0), z1, z2, z3) -> c6 U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) U42(dout(z0), z1, z2) -> c8 S tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) U22(dout(z0), z1, z2, z3) -> c4 U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) U32(dout(z0), z1, z2, z3) -> c6 U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) U42(dout(z0), z1, z2) -> c8 K tuples:none Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U22_4, U31_3, U32_4, U41_2, U42_3 Compound Symbols: c_2, c1_2, c2_2, c3_2, c4, c5_2, c6, c7_2, c8 ---------------------------------------- (3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: U22(dout(z0), z1, z2, z3) -> c4 U32(dout(z0), z1, z2, z3) -> c6 U42(dout(z0), z1, z2) -> c8 ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) S tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) K tuples:none Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_2, c5_2, c7_2 ---------------------------------------- (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) S tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) K tuples:none Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 ---------------------------------------- (7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) We considered the (Usable) Rules: u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) u42(dout(z0), z1, z2) -> dout(z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) din(der(der(z0))) -> u41(din(der(z0)), z0) And the Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(DIN(x_1)) = x_1 POL(U21(x_1, x_2, x_3)) = x_1 POL(U31(x_1, x_2, x_3)) = 0 POL(U41(x_1, x_2)) = 0 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(der(x_1)) = 0 POL(din(x_1)) = 0 POL(dout(x_1)) = [1] POL(plus(x_1, x_2)) = [1] POL(times(x_1, x_2)) = x_1 + x_2 POL(u21(x_1, x_2, x_3)) = 0 POL(u22(x_1, x_2, x_3, x_4)) = x_1 POL(u31(x_1, x_2, x_3)) = 0 POL(u32(x_1, x_2, x_3, x_4)) = x_1 POL(u41(x_1, x_2)) = x_1 POL(u42(x_1, x_2, x_3)) = [1] ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) S tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) K tuples: U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 ---------------------------------------- (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) We considered the (Usable) Rules: u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) u42(dout(z0), z1, z2) -> dout(z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) din(der(der(z0))) -> u41(din(der(z0)), z0) And the Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(DIN(x_1)) = x_1 POL(U21(x_1, x_2, x_3)) = 0 POL(U31(x_1, x_2, x_3)) = x_1 POL(U41(x_1, x_2)) = 0 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(der(x_1)) = 0 POL(din(x_1)) = 0 POL(dout(x_1)) = [1] POL(plus(x_1, x_2)) = [1] POL(times(x_1, x_2)) = x_2 POL(u21(x_1, x_2, x_3)) = 0 POL(u22(x_1, x_2, x_3, x_4)) = x_1 POL(u31(x_1, x_2, x_3)) = x_1 POL(u32(x_1, x_2, x_3, x_4)) = [1] POL(u41(x_1, x_2)) = x_1 POL(u42(x_1, x_2, x_3)) = [1] ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) S tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U41(dout(z0), z1) -> c7(DIN(der(z0))) K tuples: U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_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. U41(dout(z0), z1) -> c7(DIN(der(z0))) We considered the (Usable) Rules: u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) u42(dout(z0), z1, z2) -> dout(z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) din(der(der(z0))) -> u41(din(der(z0)), z0) And the Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(DIN(x_1)) = x_1 POL(U21(x_1, x_2, x_3)) = 0 POL(U31(x_1, x_2, x_3)) = x_1 POL(U41(x_1, x_2)) = x_1 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(der(x_1)) = 0 POL(din(x_1)) = 0 POL(dout(x_1)) = [1] POL(plus(x_1, x_2)) = [1] POL(times(x_1, x_2)) = x_1 + x_2 POL(u21(x_1, x_2, x_3)) = 0 POL(u22(x_1, x_2, x_3, x_4)) = x_1 POL(u31(x_1, x_2, x_3)) = x_1 POL(u32(x_1, x_2, x_3, x_4)) = [1] POL(u41(x_1, x_2)) = 0 POL(u42(x_1, x_2, x_3)) = x_1 ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) S tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) K tuples: U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 ---------------------------------------- (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) We considered the (Usable) Rules: u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) u42(dout(z0), z1, z2) -> dout(z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) din(der(der(z0))) -> u41(din(der(z0)), z0) And the Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(DIN(x_1)) = [2]x_1 + x_1^2 POL(U21(x_1, x_2, x_3)) = x_3^2 + x_2*x_3 + x_1*x_3 POL(U31(x_1, x_2, x_3)) = [1] + [2]x_2 + [2]x_3 + x_3^2 + [2]x_2*x_3 POL(U41(x_1, x_2)) = x_1^2 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(der(x_1)) = x_1 POL(din(x_1)) = 0 POL(dout(x_1)) = [2] + x_1 POL(plus(x_1, x_2)) = [2] + x_1 + x_2 POL(times(x_1, x_2)) = [1] + x_1 + x_2 POL(u21(x_1, x_2, x_3)) = [2]x_1 + x_1^2 POL(u22(x_1, x_2, x_3, x_4)) = x_1 + x_4^2 + [2]x_1*x_4 + [2]x_1^2 POL(u31(x_1, x_2, x_3)) = x_1 + x_1*x_3 + x_1^2 + [2]x_1*x_2 POL(u32(x_1, x_2, x_3, x_4)) = x_2 + x_3 + [2]x_4 + x_4^2 + [2]x_2*x_4 + [2]x_1^2 POL(u41(x_1, x_2)) = 0 POL(u42(x_1, x_2, x_3)) = [2]x_1 ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) S tuples: DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) K tuples: U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) We considered the (Usable) Rules: u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) u42(dout(z0), z1, z2) -> dout(z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) din(der(der(z0))) -> u41(din(der(z0)), z0) And the Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) The order we found is given by the following interpretation: Polynomial interpretation : POL(DIN(x_1)) = [1] + x_1 POL(U21(x_1, x_2, x_3)) = [1] + x_1*x_3 + x_1^3 POL(U31(x_1, x_2, x_3)) = x_1 + x_1*x_3 + x_1^3 POL(U41(x_1, x_2)) = x_1^2 + x_1^3 POL(c(x_1, x_2)) = x_1 + x_2 POL(c1(x_1, x_2)) = x_1 + x_2 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c3(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c7(x_1)) = x_1 POL(der(x_1)) = [1] + x_1 POL(din(x_1)) = 0 POL(dout(x_1)) = [1] + x_1 POL(plus(x_1, x_2)) = [1] + x_1 POL(times(x_1, x_2)) = x_1 POL(u21(x_1, x_2, x_3)) = 0 POL(u22(x_1, x_2, x_3, x_4)) = x_1^2 + x_1^3 + x_1^2*x_4 POL(u31(x_1, x_2, x_3)) = x_1 POL(u32(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_1^2 + x_1^2*x_2 + x_1^3 POL(u41(x_1, x_2)) = x_1^2 + x_1^3 POL(u42(x_1, x_2, x_3)) = x_1 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) din(der(der(z0))) -> u41(din(der(z0)), z0) u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) u42(dout(z0), z1, z2) -> dout(z0) Tuples: DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) S tuples:none K tuples: U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) U41(dout(z0), z1) -> c7(DIN(der(z0))) DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 ---------------------------------------- (17) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (18) BOUNDS(1, 1) ---------------------------------------- (19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (20) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (21) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence din(der(plus(X, Y))) ->^+ u21(din(der(X)), X, Y) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [X / plus(X, Y)]. The result substitution is [ ]. ---------------------------------------- (22) Complex Obligation (BEST) ---------------------------------------- (23) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (24) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (25) BOUNDS(n^1, INF) ---------------------------------------- (26) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) din(der(times(X, Y))) -> u31(din(der(X)), X, Y) u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) din(der(der(X))) -> u41(din(der(X)), X) u41(dout(DX), X) -> u42(din(der(DX)), X, DX) u42(dout(DDX), X, DX) -> dout(DDX) S is empty. Rewrite Strategy: INNERMOST