/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^1)) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 174 ms] (4) CpxRelTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (8) CdtProblem (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (12) CdtProblem (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CdtProblem (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 51 ms] (16) CdtProblem (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] (18) BOUNDS(1, 1) (19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (20) CpxRelTRS (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (22) typed CpxTrs (23) OrderProof [LOWER BOUND(ID), 0 ms] (24) typed CpxTrs (25) RewriteLemmaProof [LOWER BOUND(ID), 232 ms] (26) proven lower bound (27) LowerBoundPropagationProof [FINISHED, 0 ms] (28) BOUNDS(n^1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, n^1). The TRS R consists of the following rules: f(0, 1, x) -> f(x, x, x) f(x, y, z) -> 2 0 -> 2 1 -> 2 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(2) -> 2 encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 ---------------------------------------- (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: f(0, 1, x) -> f(x, x, x) f(x, y, z) -> 2 0 -> 2 1 -> 2 The (relative) TRS S consists of the following rules: encArg(2) -> 2 encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 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^1). The TRS R consists of the following rules: f(0, 1, x) -> f(x, x, x) f(x, y, z) -> 2 0 -> 2 1 -> 2 The (relative) TRS S consists of the following rules: encArg(2) -> 2 encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 Rewrite Strategy: INNERMOST ---------------------------------------- (5) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS to CDT ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: encArg(2) -> 2 encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 f(0, 1, z0) -> f(z0, z0, z0) f(z0, z1, z2) -> 2 0 -> 2 1 -> 2 Tuples: ENCARG(2) -> c ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') ENCODE_F(z0, z1, z2) -> c4(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCODE_0 -> c5(0') ENCODE_1 -> c6(1') ENCODE_2 -> c7 F(0, 1, z0) -> c8(F(z0, z0, z0)) F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 S tuples: F(0, 1, z0) -> c8(F(z0, z0, z0)) F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 K tuples:none Defined Rule Symbols: f_3, 0, 1, encArg_1, encode_f_3, encode_0, encode_1, encode_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_3, ENCODE_0, ENCODE_1, ENCODE_2, F_3, 0', 1' Compound Symbols: c, c1_4, c2_1, c3_1, c4_4, c5_1, c6_1, c7, c8_1, c9, c10, c11 ---------------------------------------- (7) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: ENCODE_0 -> c5(0') ENCODE_1 -> c6(1') F(0, 1, z0) -> c8(F(z0, z0, z0)) Removed 2 trailing nodes: ENCODE_2 -> c7 ENCARG(2) -> c ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: encArg(2) -> 2 encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 f(0, 1, z0) -> f(z0, z0, z0) f(z0, z1, z2) -> 2 0 -> 2 1 -> 2 Tuples: ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') ENCODE_F(z0, z1, z2) -> c4(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 S tuples: F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 K tuples:none Defined Rule Symbols: f_3, 0, 1, encArg_1, encode_f_3, encode_0, encode_1, encode_2 Defined Pair Symbols: ENCARG_1, ENCODE_F_3, F_3, 0', 1' Compound Symbols: c1_4, c2_1, c3_1, c4_4, c9, c10, c11 ---------------------------------------- (9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: encArg(2) -> 2 encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 f(0, 1, z0) -> f(z0, z0, z0) f(z0, z1, z2) -> 2 0 -> 2 1 -> 2 Tuples: ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 ENCODE_F(z0, z1, z2) -> c(F(encArg(z0), encArg(z1), encArg(z2))) ENCODE_F(z0, z1, z2) -> c(ENCARG(z0)) ENCODE_F(z0, z1, z2) -> c(ENCARG(z1)) ENCODE_F(z0, z1, z2) -> c(ENCARG(z2)) S tuples: F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 K tuples:none Defined Rule Symbols: f_3, 0, 1, encArg_1, encode_f_3, encode_0, encode_1, encode_2 Defined Pair Symbols: ENCARG_1, F_3, 0', 1', ENCODE_F_3 Compound Symbols: c1_4, c2_1, c3_1, c9, c10, c11, c_1 ---------------------------------------- (11) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 3 leading nodes: ENCODE_F(z0, z1, z2) -> c(ENCARG(z0)) ENCODE_F(z0, z1, z2) -> c(ENCARG(z1)) ENCODE_F(z0, z1, z2) -> c(ENCARG(z2)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: encArg(2) -> 2 encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 f(0, 1, z0) -> f(z0, z0, z0) f(z0, z1, z2) -> 2 0 -> 2 1 -> 2 Tuples: ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 ENCODE_F(z0, z1, z2) -> c(F(encArg(z0), encArg(z1), encArg(z2))) S tuples: F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 K tuples:none Defined Rule Symbols: f_3, 0, 1, encArg_1, encode_f_3, encode_0, encode_1, encode_2 Defined Pair Symbols: ENCARG_1, F_3, 0', 1', ENCODE_F_3 Compound Symbols: c1_4, c2_1, c3_1, c9, c10, c11, c_1 ---------------------------------------- (13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: encode_f(z0, z1, z2) -> f(encArg(z0), encArg(z1), encArg(z2)) encode_0 -> 0 encode_1 -> 1 encode_2 -> 2 f(0, 1, z0) -> f(z0, z0, z0) ---------------------------------------- (14) Obligation: Complexity Dependency Tuples Problem Rules: encArg(2) -> 2 encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 f(z0, z1, z2) -> 2 0 -> 2 1 -> 2 Tuples: ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 ENCODE_F(z0, z1, z2) -> c(F(encArg(z0), encArg(z1), encArg(z2))) S tuples: F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 K tuples:none Defined Rule Symbols: encArg_1, f_3, 0, 1 Defined Pair Symbols: ENCARG_1, F_3, 0', 1', ENCODE_F_3 Compound Symbols: c1_4, c2_1, c3_1, c9, c10, c11, c_1 ---------------------------------------- (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 We considered the (Usable) Rules:none And the Tuples: ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 ENCODE_F(z0, z1, z2) -> c(F(encArg(z0), encArg(z1), encArg(z2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(0') = [1] POL(1) = 0 POL(1') = [1] POL(2) = 0 POL(ENCARG(x_1)) = x_1 POL(ENCODE_F(x_1, x_2, x_3)) = [1] + x_1 POL(F(x_1, x_2, x_3)) = [1] POL(c(x_1)) = x_1 POL(c1(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(c10) = 0 POL(c11) = 0 POL(c2(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c9) = 0 POL(cons_0) = [1] POL(cons_1) = [1] POL(cons_f(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3 POL(encArg(x_1)) = [1] + x_1 POL(f(x_1, x_2, x_3)) = x_1 + x_2 ---------------------------------------- (16) Obligation: Complexity Dependency Tuples Problem Rules: encArg(2) -> 2 encArg(cons_f(z0, z1, z2)) -> f(encArg(z0), encArg(z1), encArg(z2)) encArg(cons_0) -> 0 encArg(cons_1) -> 1 f(z0, z1, z2) -> 2 0 -> 2 1 -> 2 Tuples: ENCARG(cons_f(z0, z1, z2)) -> c1(F(encArg(z0), encArg(z1), encArg(z2)), ENCARG(z0), ENCARG(z1), ENCARG(z2)) ENCARG(cons_0) -> c2(0') ENCARG(cons_1) -> c3(1') F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 ENCODE_F(z0, z1, z2) -> c(F(encArg(z0), encArg(z1), encArg(z2))) S tuples:none K tuples: F(z0, z1, z2) -> c9 0' -> c10 1' -> c11 Defined Rule Symbols: encArg_1, f_3, 0, 1 Defined Pair Symbols: ENCARG_1, F_3, 0', 1', ENCODE_F_3 Compound Symbols: c1_4, c2_1, c3_1, c9, c10, c11, c_1 ---------------------------------------- (17) SIsEmptyProof (BOTH BOUNDS(ID, ID)) The set S is empty ---------------------------------------- (18) BOUNDS(1, 1) ---------------------------------------- (19) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (20) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: f(0', 1', x) -> f(x, x, x) f(x, y, z) -> 2' 0' -> 2' 1' -> 2' The (relative) TRS S consists of the following rules: encArg(2') -> 2' encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0' encArg(cons_1) -> 1' encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_1 -> 1' encode_2 -> 2' Rewrite Strategy: INNERMOST ---------------------------------------- (21) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (22) Obligation: Innermost TRS: Rules: f(0', 1', x) -> f(x, x, x) f(x, y, z) -> 2' 0' -> 2' 1' -> 2' encArg(2') -> 2' encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0' encArg(cons_1) -> 1' encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_1 -> 1' encode_2 -> 2' Types: f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 0' :: 2':cons_f:cons_0:cons_1 1' :: 2':cons_f:cons_0:cons_1 2' :: 2':cons_f:cons_0:cons_1 encArg :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 cons_f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 cons_0 :: 2':cons_f:cons_0:cons_1 cons_1 :: 2':cons_f:cons_0:cons_1 encode_f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 encode_0 :: 2':cons_f:cons_0:cons_1 encode_1 :: 2':cons_f:cons_0:cons_1 encode_2 :: 2':cons_f:cons_0:cons_1 hole_2':cons_f:cons_0:cons_11_4 :: 2':cons_f:cons_0:cons_1 gen_2':cons_f:cons_0:cons_12_4 :: Nat -> 2':cons_f:cons_0:cons_1 ---------------------------------------- (23) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (24) Obligation: Innermost TRS: Rules: f(0', 1', x) -> f(x, x, x) f(x, y, z) -> 2' 0' -> 2' 1' -> 2' encArg(2') -> 2' encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0' encArg(cons_1) -> 1' encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_1 -> 1' encode_2 -> 2' Types: f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 0' :: 2':cons_f:cons_0:cons_1 1' :: 2':cons_f:cons_0:cons_1 2' :: 2':cons_f:cons_0:cons_1 encArg :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 cons_f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 cons_0 :: 2':cons_f:cons_0:cons_1 cons_1 :: 2':cons_f:cons_0:cons_1 encode_f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 encode_0 :: 2':cons_f:cons_0:cons_1 encode_1 :: 2':cons_f:cons_0:cons_1 encode_2 :: 2':cons_f:cons_0:cons_1 hole_2':cons_f:cons_0:cons_11_4 :: 2':cons_f:cons_0:cons_1 gen_2':cons_f:cons_0:cons_12_4 :: Nat -> 2':cons_f:cons_0:cons_1 Generator Equations: gen_2':cons_f:cons_0:cons_12_4(0) <=> 2' gen_2':cons_f:cons_0:cons_12_4(+(x, 1)) <=> cons_f(2', 2', gen_2':cons_f:cons_0:cons_12_4(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (25) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_2':cons_f:cons_0:cons_12_4(n14_4)) -> gen_2':cons_f:cons_0:cons_12_4(0), rt in Omega(n14_4) Induction Base: encArg(gen_2':cons_f:cons_0:cons_12_4(0)) ->_R^Omega(0) 2' Induction Step: encArg(gen_2':cons_f:cons_0:cons_12_4(+(n14_4, 1))) ->_R^Omega(0) f(encArg(2'), encArg(2'), encArg(gen_2':cons_f:cons_0:cons_12_4(n14_4))) ->_R^Omega(0) f(2', encArg(2'), encArg(gen_2':cons_f:cons_0:cons_12_4(n14_4))) ->_R^Omega(0) f(2', 2', encArg(gen_2':cons_f:cons_0:cons_12_4(n14_4))) ->_IH f(2', 2', gen_2':cons_f:cons_0:cons_12_4(0)) ->_R^Omega(1) 2' We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (26) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: f(0', 1', x) -> f(x, x, x) f(x, y, z) -> 2' 0' -> 2' 1' -> 2' encArg(2') -> 2' encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_0) -> 0' encArg(cons_1) -> 1' encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_1 -> 1' encode_2 -> 2' Types: f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 0' :: 2':cons_f:cons_0:cons_1 1' :: 2':cons_f:cons_0:cons_1 2' :: 2':cons_f:cons_0:cons_1 encArg :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 cons_f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 cons_0 :: 2':cons_f:cons_0:cons_1 cons_1 :: 2':cons_f:cons_0:cons_1 encode_f :: 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 -> 2':cons_f:cons_0:cons_1 encode_0 :: 2':cons_f:cons_0:cons_1 encode_1 :: 2':cons_f:cons_0:cons_1 encode_2 :: 2':cons_f:cons_0:cons_1 hole_2':cons_f:cons_0:cons_11_4 :: 2':cons_f:cons_0:cons_1 gen_2':cons_f:cons_0:cons_12_4 :: Nat -> 2':cons_f:cons_0:cons_1 Generator Equations: gen_2':cons_f:cons_0:cons_12_4(0) <=> 2' gen_2':cons_f:cons_0:cons_12_4(+(x, 1)) <=> cons_f(2', 2', gen_2':cons_f:cons_0:cons_12_4(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (27) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (28) BOUNDS(n^1, INF)