/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), ?) 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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 182 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 260 ms] (12) BEST (13) proven lower bound (14) LowerBoundPropagationProof [FINISHED, 0 ms] (15) BOUNDS(n^1, INF) (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 48 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: +(X, 0) -> X +(X, s(Y)) -> s(+(X, Y)) double(X) -> +(X, X) f(0, s(0), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> Y 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(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) 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: +(X, 0) -> X +(X, s(Y)) -> s(+(X, Y)) double(X) -> +(X, X) f(0, s(0), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> 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)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_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, INF). The TRS R consists of the following rules: +(X, 0) -> X +(X, s(Y)) -> s(+(X, Y)) double(X) -> +(X, X) f(0, s(0), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> 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)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) 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: +'(X, 0') -> X +'(X, s(Y)) -> s(+'(X, Y)) double(X) -> +'(X, X) f(0', s(0'), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> 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)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: +'(X, 0') -> X +'(X, s(Y)) -> s(+'(X, Y)) double(X) -> +'(X, X) f(0', s(0'), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> Y encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g 0' :: 0':s:cons_+:cons_double:cons_f:cons_g s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encArg :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_0 :: 0':s:cons_+:cons_double:cons_f:cons_g encode_s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g hole_0':s:cons_+:cons_double:cons_f:cons_g1_4 :: 0':s:cons_+:cons_double:cons_f:cons_g gen_0':s:cons_+:cons_double:cons_f:cons_g2_4 :: Nat -> 0':s:cons_+:cons_double:cons_f:cons_g ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: +', f, encArg They will be analysed ascendingly in the following order: +' < encArg f < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: +'(X, 0') -> X +'(X, s(Y)) -> s(+'(X, Y)) double(X) -> +'(X, X) f(0', s(0'), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> Y encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g 0' :: 0':s:cons_+:cons_double:cons_f:cons_g s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encArg :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_0 :: 0':s:cons_+:cons_double:cons_f:cons_g encode_s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g hole_0':s:cons_+:cons_double:cons_f:cons_g1_4 :: 0':s:cons_+:cons_double:cons_f:cons_g gen_0':s:cons_+:cons_double:cons_f:cons_g2_4 :: Nat -> 0':s:cons_+:cons_double:cons_f:cons_g Generator Equations: gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(0) <=> 0' gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(x)) The following defined symbols remain to be analysed: +', f, encArg They will be analysed ascendingly in the following order: +' < encArg f < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: +'(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(a), gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(n4_4)) -> gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(n4_4, a)), rt in Omega(1 + n4_4) Induction Base: +'(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(a), gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(0)) ->_R^Omega(1) gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(a) Induction Step: +'(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(a), gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(n4_4, 1))) ->_R^Omega(1) s(+'(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(a), gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(n4_4))) ->_IH s(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(a, c5_4))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (12) Complex Obligation (BEST) ---------------------------------------- (13) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: +'(X, 0') -> X +'(X, s(Y)) -> s(+'(X, Y)) double(X) -> +'(X, X) f(0', s(0'), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> Y encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g 0' :: 0':s:cons_+:cons_double:cons_f:cons_g s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encArg :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_0 :: 0':s:cons_+:cons_double:cons_f:cons_g encode_s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g hole_0':s:cons_+:cons_double:cons_f:cons_g1_4 :: 0':s:cons_+:cons_double:cons_f:cons_g gen_0':s:cons_+:cons_double:cons_f:cons_g2_4 :: Nat -> 0':s:cons_+:cons_double:cons_f:cons_g Generator Equations: gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(0) <=> 0' gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(x)) The following defined symbols remain to be analysed: +', f, encArg They will be analysed ascendingly in the following order: +' < encArg f < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: +'(X, 0') -> X +'(X, s(Y)) -> s(+'(X, Y)) double(X) -> +'(X, X) f(0', s(0'), X) -> f(X, double(X), X) g(X, Y) -> X g(X, Y) -> Y encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_+(x_1, x_2)) -> +'(encArg(x_1), encArg(x_2)) encArg(cons_double(x_1)) -> double(encArg(x_1)) encArg(cons_f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_g(x_1, x_2)) -> g(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)) encode_double(x_1) -> double(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_g(x_1, x_2) -> g(encArg(x_1), encArg(x_2)) Types: +' :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g 0' :: 0':s:cons_+:cons_double:cons_f:cons_g s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encArg :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g cons_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_+ :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_0 :: 0':s:cons_+:cons_double:cons_f:cons_g encode_s :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_double :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_f :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g encode_g :: 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g -> 0':s:cons_+:cons_double:cons_f:cons_g hole_0':s:cons_+:cons_double:cons_f:cons_g1_4 :: 0':s:cons_+:cons_double:cons_f:cons_g gen_0':s:cons_+:cons_double:cons_f:cons_g2_4 :: Nat -> 0':s:cons_+:cons_double:cons_f:cons_g Lemmas: +'(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(a), gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(n4_4)) -> gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(n4_4, a)), rt in Omega(1 + n4_4) Generator Equations: gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(0) <=> 0' gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(x, 1)) <=> s(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(x)) The following defined symbols remain to be analysed: f, encArg They will be analysed ascendingly in the following order: f < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(n878_4)) -> gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(n878_4), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(+(n878_4, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(n878_4))) ->_IH s(gen_0':s:cons_+:cons_double:cons_f:cons_g2_4(c879_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)