/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 224 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxRelTRS (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 7 ms] (8) typed CpxTrs (9) OrderProof [LOWER BOUND(ID), 0 ms] (10) typed CpxTrs (11) RewriteLemmaProof [LOWER BOUND(ID), 516 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), 155 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: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0, y) -> tt lt(s(x), s(y)) -> lt(x, 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(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt 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, INF). The TRS R consists of the following rules: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0, y) -> tt lt(s(x), s(y)) -> lt(x, y) The (relative) TRS S consists of the following rules: encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt 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, INF). The TRS R consists of the following rules: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0, y) -> tt lt(s(x), s(y)) -> lt(x, y) The (relative) TRS S consists of the following rules: encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0 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: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0', y) -> tt lt(s(x), s(y)) -> lt(x, y) The (relative) TRS S consists of the following rules: encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0', y) -> tt lt(s(x), s(y)) -> lt(x, y) encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt tt :: tt:s:0':cons_f:cons_cond:cons_lt s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt 0' :: tt:s:0':cons_f:cons_cond:cons_lt encArg :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_tt :: tt:s:0':cons_f:cons_cond:cons_lt encode_s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_0 :: tt:s:0':cons_f:cons_cond:cons_lt hole_tt:s:0':cons_f:cons_cond:cons_lt1_4 :: tt:s:0':cons_f:cons_cond:cons_lt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4 :: Nat -> tt:s:0':cons_f:cons_cond:cons_lt ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: f, cond, lt, encArg They will be analysed ascendingly in the following order: f = cond lt < f f < encArg cond < encArg lt < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0', y) -> tt lt(s(x), s(y)) -> lt(x, y) encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt tt :: tt:s:0':cons_f:cons_cond:cons_lt s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt 0' :: tt:s:0':cons_f:cons_cond:cons_lt encArg :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_tt :: tt:s:0':cons_f:cons_cond:cons_lt encode_s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_0 :: tt:s:0':cons_f:cons_cond:cons_lt hole_tt:s:0':cons_f:cons_cond:cons_lt1_4 :: tt:s:0':cons_f:cons_cond:cons_lt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4 :: Nat -> tt:s:0':cons_f:cons_cond:cons_lt Generator Equations: gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(0) <=> tt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(x)) The following defined symbols remain to be analysed: lt, f, cond, encArg They will be analysed ascendingly in the following order: f = cond lt < f f < encArg cond < encArg lt < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: lt(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, n4_4)), gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) Induction Base: lt(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, 0)), gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, 0))) Induction Step: lt(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, +(n4_4, 1))), gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, +(n4_4, 1)))) ->_R^Omega(1) lt(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, n4_4)), gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, n4_4))) ->_IH *3_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: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0', y) -> tt lt(s(x), s(y)) -> lt(x, y) encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt tt :: tt:s:0':cons_f:cons_cond:cons_lt s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt 0' :: tt:s:0':cons_f:cons_cond:cons_lt encArg :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_tt :: tt:s:0':cons_f:cons_cond:cons_lt encode_s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_0 :: tt:s:0':cons_f:cons_cond:cons_lt hole_tt:s:0':cons_f:cons_cond:cons_lt1_4 :: tt:s:0':cons_f:cons_cond:cons_lt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4 :: Nat -> tt:s:0':cons_f:cons_cond:cons_lt Generator Equations: gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(0) <=> tt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(x)) The following defined symbols remain to be analysed: lt, f, cond, encArg They will be analysed ascendingly in the following order: f = cond lt < f f < encArg cond < encArg lt < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: f(x, y) -> cond(lt(x, y), x, y) cond(tt, x, y) -> f(s(x), s(y)) lt(0', y) -> tt lt(s(x), s(y)) -> lt(x, y) encArg(tt) -> tt 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_cond(x_1, x_2, x_3)) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_lt(x_1, x_2)) -> lt(encArg(x_1), encArg(x_2)) encode_f(x_1, x_2) -> f(encArg(x_1), encArg(x_2)) encode_cond(x_1, x_2, x_3) -> cond(encArg(x_1), encArg(x_2), encArg(x_3)) encode_lt(x_1, x_2) -> lt(encArg(x_1), encArg(x_2)) encode_tt -> tt encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt tt :: tt:s:0':cons_f:cons_cond:cons_lt s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt 0' :: tt:s:0':cons_f:cons_cond:cons_lt encArg :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt cons_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_f :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_cond :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_lt :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_tt :: tt:s:0':cons_f:cons_cond:cons_lt encode_s :: tt:s:0':cons_f:cons_cond:cons_lt -> tt:s:0':cons_f:cons_cond:cons_lt encode_0 :: tt:s:0':cons_f:cons_cond:cons_lt hole_tt:s:0':cons_f:cons_cond:cons_lt1_4 :: tt:s:0':cons_f:cons_cond:cons_lt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4 :: Nat -> tt:s:0':cons_f:cons_cond:cons_lt Lemmas: lt(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, n4_4)), gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(1, n4_4))) -> *3_4, rt in Omega(n4_4) Generator Equations: gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(0) <=> tt gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(x, 1)) <=> s(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(x)) The following defined symbols remain to be analysed: cond, f, encArg They will be analysed ascendingly in the following order: f = cond f < encArg cond < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(n1862_4)) -> gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(n1862_4), rt in Omega(0) Induction Base: encArg(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(0)) ->_R^Omega(0) tt Induction Step: encArg(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(+(n1862_4, 1))) ->_R^Omega(0) s(encArg(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(n1862_4))) ->_IH s(gen_tt:s:0':cons_f:cons_cond:cons_lt2_4(c1863_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)