/export/starexec/sandbox/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 165 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), 283 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), 172 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 28 ms] (20) 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: quot(0, s(y), s(z)) -> 0 quot(s(x), s(y), z) -> quot(x, y, z) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) 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_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(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: quot(0, s(y), s(z)) -> 0 quot(s(x), s(y), z) -> quot(x, y, z) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(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: quot(0, s(y), s(z)) -> 0 quot(s(x), s(y), z) -> quot(x, y, z) plus(0, y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0, s(z)) -> s(quot(x, plus(z, s(0)), s(z))) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0 encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(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: quot(0', s(y), s(z)) -> 0' quot(s(x), s(y), z) -> quot(x, y, z) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0', s(z)) -> s(quot(x, plus(z, s(0')), s(z))) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: quot(0', s(y), s(z)) -> 0' quot(s(x), s(y), z) -> quot(x, y, z) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0', s(z)) -> s(quot(x, plus(z, s(0')), s(z))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus 0' :: 0':s:cons_quot:cons_plus s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encArg :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_0 :: 0':s:cons_quot:cons_plus encode_s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus hole_0':s:cons_quot:cons_plus1_4 :: 0':s:cons_quot:cons_plus gen_0':s:cons_quot:cons_plus2_4 :: Nat -> 0':s:cons_quot:cons_plus ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: quot, plus, encArg They will be analysed ascendingly in the following order: plus < quot quot < encArg plus < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: quot(0', s(y), s(z)) -> 0' quot(s(x), s(y), z) -> quot(x, y, z) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0', s(z)) -> s(quot(x, plus(z, s(0')), s(z))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus 0' :: 0':s:cons_quot:cons_plus s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encArg :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_0 :: 0':s:cons_quot:cons_plus encode_s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus hole_0':s:cons_quot:cons_plus1_4 :: 0':s:cons_quot:cons_plus gen_0':s:cons_quot:cons_plus2_4 :: Nat -> 0':s:cons_quot:cons_plus Generator Equations: gen_0':s:cons_quot:cons_plus2_4(0) <=> 0' gen_0':s:cons_quot:cons_plus2_4(+(x, 1)) <=> s(gen_0':s:cons_quot:cons_plus2_4(x)) The following defined symbols remain to be analysed: plus, quot, encArg They will be analysed ascendingly in the following order: plus < quot quot < encArg plus < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_0':s:cons_quot:cons_plus2_4(n4_4), gen_0':s:cons_quot:cons_plus2_4(b)) -> gen_0':s:cons_quot:cons_plus2_4(+(n4_4, b)), rt in Omega(1 + n4_4) Induction Base: plus(gen_0':s:cons_quot:cons_plus2_4(0), gen_0':s:cons_quot:cons_plus2_4(b)) ->_R^Omega(1) gen_0':s:cons_quot:cons_plus2_4(b) Induction Step: plus(gen_0':s:cons_quot:cons_plus2_4(+(n4_4, 1)), gen_0':s:cons_quot:cons_plus2_4(b)) ->_R^Omega(1) s(plus(gen_0':s:cons_quot:cons_plus2_4(n4_4), gen_0':s:cons_quot:cons_plus2_4(b))) ->_IH s(gen_0':s:cons_quot:cons_plus2_4(+(b, 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: quot(0', s(y), s(z)) -> 0' quot(s(x), s(y), z) -> quot(x, y, z) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0', s(z)) -> s(quot(x, plus(z, s(0')), s(z))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus 0' :: 0':s:cons_quot:cons_plus s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encArg :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_0 :: 0':s:cons_quot:cons_plus encode_s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus hole_0':s:cons_quot:cons_plus1_4 :: 0':s:cons_quot:cons_plus gen_0':s:cons_quot:cons_plus2_4 :: Nat -> 0':s:cons_quot:cons_plus Generator Equations: gen_0':s:cons_quot:cons_plus2_4(0) <=> 0' gen_0':s:cons_quot:cons_plus2_4(+(x, 1)) <=> s(gen_0':s:cons_quot:cons_plus2_4(x)) The following defined symbols remain to be analysed: plus, quot, encArg They will be analysed ascendingly in the following order: plus < quot quot < encArg plus < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: quot(0', s(y), s(z)) -> 0' quot(s(x), s(y), z) -> quot(x, y, z) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0', s(z)) -> s(quot(x, plus(z, s(0')), s(z))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus 0' :: 0':s:cons_quot:cons_plus s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encArg :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_0 :: 0':s:cons_quot:cons_plus encode_s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus hole_0':s:cons_quot:cons_plus1_4 :: 0':s:cons_quot:cons_plus gen_0':s:cons_quot:cons_plus2_4 :: Nat -> 0':s:cons_quot:cons_plus Lemmas: plus(gen_0':s:cons_quot:cons_plus2_4(n4_4), gen_0':s:cons_quot:cons_plus2_4(b)) -> gen_0':s:cons_quot:cons_plus2_4(+(n4_4, b)), rt in Omega(1 + n4_4) Generator Equations: gen_0':s:cons_quot:cons_plus2_4(0) <=> 0' gen_0':s:cons_quot:cons_plus2_4(+(x, 1)) <=> s(gen_0':s:cons_quot:cons_plus2_4(x)) The following defined symbols remain to be analysed: quot, encArg They will be analysed ascendingly in the following order: quot < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: quot(gen_0':s:cons_quot:cons_plus2_4(n669_4), gen_0':s:cons_quot:cons_plus2_4(+(1, n669_4)), gen_0':s:cons_quot:cons_plus2_4(1)) -> gen_0':s:cons_quot:cons_plus2_4(0), rt in Omega(1 + n669_4) Induction Base: quot(gen_0':s:cons_quot:cons_plus2_4(0), gen_0':s:cons_quot:cons_plus2_4(+(1, 0)), gen_0':s:cons_quot:cons_plus2_4(1)) ->_R^Omega(1) 0' Induction Step: quot(gen_0':s:cons_quot:cons_plus2_4(+(n669_4, 1)), gen_0':s:cons_quot:cons_plus2_4(+(1, +(n669_4, 1))), gen_0':s:cons_quot:cons_plus2_4(1)) ->_R^Omega(1) quot(gen_0':s:cons_quot:cons_plus2_4(n669_4), gen_0':s:cons_quot:cons_plus2_4(+(1, n669_4)), gen_0':s:cons_quot:cons_plus2_4(1)) ->_IH gen_0':s:cons_quot:cons_plus2_4(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: Innermost TRS: Rules: quot(0', s(y), s(z)) -> 0' quot(s(x), s(y), z) -> quot(x, y, z) plus(0', y) -> y plus(s(x), y) -> s(plus(x, y)) quot(x, 0', s(z)) -> s(quot(x, plus(z, s(0')), s(z))) encArg(0') -> 0' encArg(s(x_1)) -> s(encArg(x_1)) encArg(cons_quot(x_1, x_2, x_3)) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_quot(x_1, x_2, x_3) -> quot(encArg(x_1), encArg(x_2), encArg(x_3)) encode_0 -> 0' encode_s(x_1) -> s(encArg(x_1)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) Types: quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus 0' :: 0':s:cons_quot:cons_plus s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encArg :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus cons_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_quot :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_0 :: 0':s:cons_quot:cons_plus encode_s :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus encode_plus :: 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus -> 0':s:cons_quot:cons_plus hole_0':s:cons_quot:cons_plus1_4 :: 0':s:cons_quot:cons_plus gen_0':s:cons_quot:cons_plus2_4 :: Nat -> 0':s:cons_quot:cons_plus Lemmas: plus(gen_0':s:cons_quot:cons_plus2_4(n4_4), gen_0':s:cons_quot:cons_plus2_4(b)) -> gen_0':s:cons_quot:cons_plus2_4(+(n4_4, b)), rt in Omega(1 + n4_4) quot(gen_0':s:cons_quot:cons_plus2_4(n669_4), gen_0':s:cons_quot:cons_plus2_4(+(1, n669_4)), gen_0':s:cons_quot:cons_plus2_4(1)) -> gen_0':s:cons_quot:cons_plus2_4(0), rt in Omega(1 + n669_4) Generator Equations: gen_0':s:cons_quot:cons_plus2_4(0) <=> 0' gen_0':s:cons_quot:cons_plus2_4(+(x, 1)) <=> s(gen_0':s:cons_quot:cons_plus2_4(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:cons_quot:cons_plus2_4(n2490_4)) -> gen_0':s:cons_quot:cons_plus2_4(n2490_4), rt in Omega(0) Induction Base: encArg(gen_0':s:cons_quot:cons_plus2_4(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:cons_quot:cons_plus2_4(+(n2490_4, 1))) ->_R^Omega(0) s(encArg(gen_0':s:cons_quot:cons_plus2_4(n2490_4))) ->_IH s(gen_0':s:cons_quot:cons_plus2_4(c2491_4)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (20) BOUNDS(1, INF)