/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 180 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), 292 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), 83 ms] (18) BEST (19) proven lower bound (20) LowerBoundPropagationProof [FINISHED, 0 ms] (21) BOUNDS(n^2, INF) (22) typed CpxTrs (23) RewriteLemmaProof [LOWER BOUND(ID), 59 ms] (24) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^2, INF). The TRS R consists of the following rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z))) times(x, 0) -> 0 times(x, s(y)) -> plus(times(x, y), x) plus(x, 0) -> x plus(x, s(y)) -> s(plus(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(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) 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^2, INF). The TRS R consists of the following rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z))) times(x, 0) -> 0 times(x, s(y)) -> plus(times(x, y), x) plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) 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^2, INF). The TRS R consists of the following rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0))), times(x, s(z))) times(x, 0) -> 0 times(x, s(y)) -> plus(times(x, y), x) plus(x, 0) -> x plus(x, s(y)) -> s(plus(x, y)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0) -> 0 encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) 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^2, INF). The TRS R consists of the following rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) The (relative) TRS S consists of the following rules: encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) 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: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus 0' :: s:0':cons_times:cons_plus encArg :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_0 :: s:0':cons_times:cons_plus hole_s:0':cons_times:cons_plus1_3 :: s:0':cons_times:cons_plus gen_s:0':cons_times:cons_plus2_3 :: Nat -> s:0':cons_times:cons_plus ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: times, plus, encArg They will be analysed ascendingly in the following order: plus < times times < encArg plus < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus 0' :: s:0':cons_times:cons_plus encArg :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_0 :: s:0':cons_times:cons_plus hole_s:0':cons_times:cons_plus1_3 :: s:0':cons_times:cons_plus gen_s:0':cons_times:cons_plus2_3 :: Nat -> s:0':cons_times:cons_plus Generator Equations: gen_s:0':cons_times:cons_plus2_3(0) <=> 0' gen_s:0':cons_times:cons_plus2_3(+(x, 1)) <=> s(gen_s:0':cons_times:cons_plus2_3(x)) The following defined symbols remain to be analysed: plus, times, encArg They will be analysed ascendingly in the following order: plus < times times < encArg plus < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n4_3)) -> gen_s:0':cons_times:cons_plus2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Induction Base: plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(0)) ->_R^Omega(1) gen_s:0':cons_times:cons_plus2_3(a) Induction Step: plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(+(n4_3, 1))) ->_R^Omega(1) s(plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n4_3))) ->_IH s(gen_s:0':cons_times:cons_plus2_3(+(a, c5_3))) 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: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus 0' :: s:0':cons_times:cons_plus encArg :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_0 :: s:0':cons_times:cons_plus hole_s:0':cons_times:cons_plus1_3 :: s:0':cons_times:cons_plus gen_s:0':cons_times:cons_plus2_3 :: Nat -> s:0':cons_times:cons_plus Generator Equations: gen_s:0':cons_times:cons_plus2_3(0) <=> 0' gen_s:0':cons_times:cons_plus2_3(+(x, 1)) <=> s(gen_s:0':cons_times:cons_plus2_3(x)) The following defined symbols remain to be analysed: plus, times, encArg They will be analysed ascendingly in the following order: plus < times times < encArg plus < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus 0' :: s:0':cons_times:cons_plus encArg :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_0 :: s:0':cons_times:cons_plus hole_s:0':cons_times:cons_plus1_3 :: s:0':cons_times:cons_plus gen_s:0':cons_times:cons_plus2_3 :: Nat -> s:0':cons_times:cons_plus Lemmas: plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n4_3)) -> gen_s:0':cons_times:cons_plus2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Generator Equations: gen_s:0':cons_times:cons_plus2_3(0) <=> 0' gen_s:0':cons_times:cons_plus2_3(+(x, 1)) <=> s(gen_s:0':cons_times:cons_plus2_3(x)) The following defined symbols remain to be analysed: times, encArg They will be analysed ascendingly in the following order: times < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: times(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n669_3)) -> gen_s:0':cons_times:cons_plus2_3(*(n669_3, a)), rt in Omega(1 + a*n669_3 + n669_3) Induction Base: times(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(0)) ->_R^Omega(1) 0' Induction Step: times(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(+(n669_3, 1))) ->_R^Omega(1) plus(times(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n669_3)), gen_s:0':cons_times:cons_plus2_3(a)) ->_IH plus(gen_s:0':cons_times:cons_plus2_3(*(c670_3, a)), gen_s:0':cons_times:cons_plus2_3(a)) ->_L^Omega(1 + a) gen_s:0':cons_times:cons_plus2_3(+(a, *(n669_3, a))) We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2). ---------------------------------------- (18) Complex Obligation (BEST) ---------------------------------------- (19) Obligation: Proved the lower bound n^2 for the following obligation: Innermost TRS: Rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus 0' :: s:0':cons_times:cons_plus encArg :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_0 :: s:0':cons_times:cons_plus hole_s:0':cons_times:cons_plus1_3 :: s:0':cons_times:cons_plus gen_s:0':cons_times:cons_plus2_3 :: Nat -> s:0':cons_times:cons_plus Lemmas: plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n4_3)) -> gen_s:0':cons_times:cons_plus2_3(+(n4_3, a)), rt in Omega(1 + n4_3) Generator Equations: gen_s:0':cons_times:cons_plus2_3(0) <=> 0' gen_s:0':cons_times:cons_plus2_3(+(x, 1)) <=> s(gen_s:0':cons_times:cons_plus2_3(x)) The following defined symbols remain to be analysed: times, encArg They will be analysed ascendingly in the following order: times < encArg ---------------------------------------- (20) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (21) BOUNDS(n^2, INF) ---------------------------------------- (22) Obligation: Innermost TRS: Rules: times(x, plus(y, s(z))) -> plus(times(x, plus(y, times(s(z), 0'))), times(x, s(z))) times(x, 0') -> 0' times(x, s(y)) -> plus(times(x, y), x) plus(x, 0') -> x plus(x, s(y)) -> s(plus(x, y)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(0') -> 0' encArg(cons_times(x_1, x_2)) -> times(encArg(x_1), encArg(x_2)) encArg(cons_plus(x_1, x_2)) -> plus(encArg(x_1), encArg(x_2)) encode_times(x_1, x_2) -> times(encArg(x_1), encArg(x_2)) encode_plus(x_1, x_2) -> plus(encArg(x_1), encArg(x_2)) encode_s(x_1) -> s(encArg(x_1)) encode_0 -> 0' Types: times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus 0' :: s:0':cons_times:cons_plus encArg :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus cons_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_times :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_plus :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_s :: s:0':cons_times:cons_plus -> s:0':cons_times:cons_plus encode_0 :: s:0':cons_times:cons_plus hole_s:0':cons_times:cons_plus1_3 :: s:0':cons_times:cons_plus gen_s:0':cons_times:cons_plus2_3 :: Nat -> s:0':cons_times:cons_plus Lemmas: plus(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n4_3)) -> gen_s:0':cons_times:cons_plus2_3(+(n4_3, a)), rt in Omega(1 + n4_3) times(gen_s:0':cons_times:cons_plus2_3(a), gen_s:0':cons_times:cons_plus2_3(n669_3)) -> gen_s:0':cons_times:cons_plus2_3(*(n669_3, a)), rt in Omega(1 + a*n669_3 + n669_3) Generator Equations: gen_s:0':cons_times:cons_plus2_3(0) <=> 0' gen_s:0':cons_times:cons_plus2_3(+(x, 1)) <=> s(gen_s:0':cons_times:cons_plus2_3(x)) The following defined symbols remain to be analysed: encArg ---------------------------------------- (23) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:0':cons_times:cons_plus2_3(n1625_3)) -> gen_s:0':cons_times:cons_plus2_3(n1625_3), rt in Omega(0) Induction Base: encArg(gen_s:0':cons_times:cons_plus2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_s:0':cons_times:cons_plus2_3(+(n1625_3, 1))) ->_R^Omega(0) s(encArg(gen_s:0':cons_times:cons_plus2_3(n1625_3))) ->_IH s(gen_s:0':cons_times:cons_plus2_3(c1626_3)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (24) BOUNDS(1, INF)