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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^2).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), 157 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 9 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, 0 ms] (14) CdtProblem (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CdtProblem (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms] (18) CdtProblem (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms] (20) CdtProblem (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 63 ms] (22) CdtProblem (23) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (24) BOUNDS(1, 1) (25) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (26) TRS for Loop Detection (27) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] (28) BEST (29) proven lower bound (30) LowerBoundPropagationProof [FINISHED, 0 ms] (31) BOUNDS(n^1, INF) (32) 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: f(g(x), s(0)) -> f(g(x), g(x)) g(s(x)) -> s(g(x)) g(0) -> 0 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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 ---------------------------------------- (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: f(g(x), s(0)) -> f(g(x), g(x)) g(s(x)) -> s(g(x)) g(0) -> 0 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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: f(g(x), s(0)) -> f(g(x), g(x)) g(s(x)) -> s(g(x)) g(0) -> 0 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_g(z0) -> g(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_0 -> 0 f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(0) -> c1 ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_G(z0) -> c5(G(encArg(z0)), ENCARG(z0)) ENCODE_S(z0) -> c6(ENCARG(z0)) ENCODE_0 -> c7 F(g(z0), s(0)) -> c8(F(g(z0), g(z0)), G(z0), G(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 S tuples: F(g(z0), s(0)) -> c8(F(g(z0), g(z0)), G(z0), G(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 K tuples:none Defined Rule Symbols: f_2, g_1, encArg_1, encode_f_2, encode_g_1, encode_s_1, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_G_1, ENCODE_S_1, ENCODE_0, F_2, G_1 Compound Symbols: c_1, c1, c2_3, c3_2, c4_3, c5_2, c6_1, c7, c8_3, c9_1, c10 ---------------------------------------- (7) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: ENCODE_S(z0) -> c6(ENCARG(z0)) Removed 2 trailing nodes: ENCODE_0 -> c7 ENCARG(0) -> c1 ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_g(z0) -> g(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_0 -> 0 f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_G(z0) -> c5(G(encArg(z0)), ENCARG(z0)) F(g(z0), s(0)) -> c8(F(g(z0), g(z0)), G(z0), G(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 S tuples: F(g(z0), s(0)) -> c8(F(g(z0), g(z0)), G(z0), G(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 K tuples:none Defined Rule Symbols: f_2, g_1, encArg_1, encode_f_2, encode_g_1, encode_s_1, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_G_1, F_2, G_1 Compound Symbols: c_1, c2_3, c3_2, c4_3, c5_2, c8_3, c9_1, c10 ---------------------------------------- (9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_g(z0) -> g(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_0 -> 0 f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) ENCODE_F(z0, z1) -> c4(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCODE_G(z0) -> c5(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 F(g(z0), s(0)) -> c8(G(z0), G(z0)) S tuples: G(s(z0)) -> c9(G(z0)) G(0) -> c10 F(g(z0), s(0)) -> c8(G(z0), G(z0)) K tuples:none Defined Rule Symbols: f_2, g_1, encArg_1, encode_f_2, encode_g_1, encode_s_1, encode_0 Defined Pair Symbols: ENCARG_1, ENCODE_F_2, ENCODE_G_1, G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c4_3, c5_2, c9_1, c10, c8_2 ---------------------------------------- (11) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_g(z0) -> g(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_0 -> 0 f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_F(z0, z1) -> c1(ENCARG(z0)) ENCODE_F(z0, z1) -> c1(ENCARG(z1)) ENCODE_G(z0) -> c1(G(encArg(z0))) ENCODE_G(z0) -> c1(ENCARG(z0)) F(g(z0), s(0)) -> c1(G(z0)) S tuples: G(s(z0)) -> c9(G(z0)) G(0) -> c10 F(g(z0), s(0)) -> c1(G(z0)) K tuples:none Defined Rule Symbols: f_2, g_1, encArg_1, encode_f_2, encode_g_1, encode_s_1, encode_0 Defined Pair Symbols: ENCARG_1, G_1, ENCODE_F_2, ENCODE_G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c9_1, c10, c1_1 ---------------------------------------- (13) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: ENCODE_F(z0, z1) -> c1(ENCARG(z0)) ENCODE_F(z0, z1) -> c1(ENCARG(z1)) ENCODE_G(z0) -> c1(ENCARG(z0)) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_g(z0) -> g(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_0 -> 0 f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) S tuples: G(s(z0)) -> c9(G(z0)) G(0) -> c10 F(g(z0), s(0)) -> c1(G(z0)) K tuples:none Defined Rule Symbols: f_2, g_1, encArg_1, encode_f_2, encode_g_1, encode_s_1, encode_0 Defined Pair Symbols: ENCARG_1, G_1, ENCODE_F_2, ENCODE_G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c9_1, c10, c1_1 ---------------------------------------- (15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_f(z0, z1) -> f(encArg(z0), encArg(z1)) encode_g(z0) -> g(encArg(z0)) encode_s(z0) -> s(encArg(z0)) encode_0 -> 0 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) S tuples: G(s(z0)) -> c9(G(z0)) G(0) -> c10 F(g(z0), s(0)) -> c1(G(z0)) K tuples:none Defined Rule Symbols: encArg_1, f_2, g_1 Defined Pair Symbols: ENCARG_1, G_1, ENCODE_F_2, ENCODE_G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c9_1, c10, c1_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. F(g(z0), s(0)) -> c1(G(z0)) We considered the (Usable) Rules:none And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1 POL(ENCODE_F(x_1, x_2)) = [1] POL(ENCODE_G(x_1)) = [1] POL(F(x_1, x_2)) = [1] POL(G(x_1)) = 0 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10) = 0 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_g(x_1)) = x_1 POL(encArg(x_1)) = [3]x_1 POL(f(x_1, x_2)) = [3] + [3]x_1 POL(g(x_1)) = 0 POL(s(x_1)) = x_1 ---------------------------------------- (18) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) S tuples: G(s(z0)) -> c9(G(z0)) G(0) -> c10 K tuples: F(g(z0), s(0)) -> c1(G(z0)) Defined Rule Symbols: encArg_1, f_2, g_1 Defined Pair Symbols: ENCARG_1, G_1, ENCODE_F_2, ENCODE_G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c9_1, c10, c1_1 ---------------------------------------- (19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G(0) -> c10 We considered the (Usable) Rules:none And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = [1] POL(ENCARG(x_1)) = x_1 POL(ENCODE_F(x_1, x_2)) = [1] + x_1 POL(ENCODE_G(x_1)) = [1] POL(F(x_1, x_2)) = [1] POL(G(x_1)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10) = 0 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(cons_f(x_1, x_2)) = [1] + x_1 + x_2 POL(cons_g(x_1)) = [1] + x_1 POL(encArg(x_1)) = [1] + x_1 POL(f(x_1, x_2)) = [1] + x_2 POL(g(x_1)) = [1] POL(s(x_1)) = x_1 ---------------------------------------- (20) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) S tuples: G(s(z0)) -> c9(G(z0)) K tuples: F(g(z0), s(0)) -> c1(G(z0)) G(0) -> c10 Defined Rule Symbols: encArg_1, f_2, g_1 Defined Pair Symbols: ENCARG_1, G_1, ENCODE_F_2, ENCODE_G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c9_1, c10, c1_1 ---------------------------------------- (21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. G(s(z0)) -> c9(G(z0)) We considered the (Usable) Rules: encArg(s(z0)) -> s(encArg(z0)) g(s(z0)) -> s(g(z0)) f(g(z0), s(0)) -> f(g(z0), g(z0)) encArg(cons_g(z0)) -> g(encArg(z0)) g(0) -> 0 encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) And the Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(ENCARG(x_1)) = x_1^2 POL(ENCODE_F(x_1, x_2)) = [1] + x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2 POL(ENCODE_G(x_1)) = [2] + [2]x_1 + [2]x_1^2 POL(F(x_1, x_2)) = x_1*x_2 POL(G(x_1)) = x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10) = 0 POL(c2(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c9(x_1)) = x_1 POL(cons_f(x_1, x_2)) = [2] + x_1 + x_2 POL(cons_g(x_1)) = [1] + x_1 POL(encArg(x_1)) = x_1 POL(f(x_1, x_2)) = 0 POL(g(x_1)) = x_1 POL(s(x_1)) = [2] + x_1 ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: encArg(s(z0)) -> s(encArg(z0)) encArg(0) -> 0 encArg(cons_f(z0, z1)) -> f(encArg(z0), encArg(z1)) encArg(cons_g(z0)) -> g(encArg(z0)) f(g(z0), s(0)) -> f(g(z0), g(z0)) g(s(z0)) -> s(g(z0)) g(0) -> 0 Tuples: ENCARG(s(z0)) -> c(ENCARG(z0)) ENCARG(cons_f(z0, z1)) -> c2(F(encArg(z0), encArg(z1)), ENCARG(z0), ENCARG(z1)) ENCARG(cons_g(z0)) -> c3(G(encArg(z0)), ENCARG(z0)) G(s(z0)) -> c9(G(z0)) G(0) -> c10 ENCODE_F(z0, z1) -> c1(F(encArg(z0), encArg(z1))) ENCODE_G(z0) -> c1(G(encArg(z0))) F(g(z0), s(0)) -> c1(G(z0)) S tuples:none K tuples: F(g(z0), s(0)) -> c1(G(z0)) G(0) -> c10 G(s(z0)) -> c9(G(z0)) Defined Rule Symbols: encArg_1, f_2, g_1 Defined Pair Symbols: ENCARG_1, G_1, ENCODE_F_2, ENCODE_G_1, F_2 Compound Symbols: c_1, c2_3, c3_2, c9_1, c10, c1_1 ---------------------------------------- (23) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (24) BOUNDS(1, 1) ---------------------------------------- (25) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (26) 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: f(g(x), s(0)) -> f(g(x), g(x)) g(s(x)) -> s(g(x)) g(0) -> 0 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (27) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence g(s(x)) ->^+ s(g(x)) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [x / s(x)]. The result substitution is [ ]. ---------------------------------------- (28) Complex Obligation (BEST) ---------------------------------------- (29) 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: f(g(x), s(0)) -> f(g(x), g(x)) g(s(x)) -> s(g(x)) g(0) -> 0 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST ---------------------------------------- (30) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (31) BOUNDS(n^1, INF) ---------------------------------------- (32) 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: f(g(x), s(0)) -> f(g(x), g(x)) g(s(x)) -> s(g(x)) g(0) -> 0 The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_f(x_1, x_2)) -> f(encArg(x_1), encArg(x_2)) encArg(cons_g(x_1)) -> g(encArg(x_1)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_g(x_1) -> g(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 Rewrite Strategy: INNERMOST