/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 223 ms] (4) CpxRelTRS (5) RenamingProof [BOTH BOUNDS(ID, ID), 1 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), 245 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), 45 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 156 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 160 ms] (22) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) S is empty. Rewrite Strategy: FULL ---------------------------------------- (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(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) The (relative) TRS S consists of the following rules: encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) The (relative) TRS S consists of the following rules: encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) The (relative) TRS S consists of the following rules: encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Types: top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encArg :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check hole_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check1_0 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0 :: Nat -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: top1, top2, check, new, old, encArg They will be analysed ascendingly in the following order: top1 = top2 check < top1 new < top1 top1 < encArg check < top2 new < top2 top2 < encArg new < check old < check check < encArg new < encArg old < encArg ---------------------------------------- (10) Obligation: TRS: Rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Types: top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encArg :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check hole_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check1_0 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0 :: Nat -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check Generator Equations: gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0) <=> serve gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(x, 1)) <=> free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(x)) The following defined symbols remain to be analysed: new, top1, top2, check, old, encArg They will be analysed ascendingly in the following order: top1 = top2 check < top1 new < top1 top1 < encArg check < top2 new < top2 top2 < encArg new < check old < check check < encArg new < encArg old < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n4_0)), rt in Omega(1 + n4_0) Induction Base: new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0)) ->_R^Omega(1) free(serve) Induction Step: new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(n4_0, 1))) ->_R^Omega(1) free(new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4_0))) ->_IH free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, c5_0))) 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: TRS: Rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Types: top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encArg :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check hole_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check1_0 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0 :: Nat -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check Generator Equations: gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0) <=> serve gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(x, 1)) <=> free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(x)) The following defined symbols remain to be analysed: new, top1, top2, check, old, encArg They will be analysed ascendingly in the following order: top1 = top2 check < top1 new < top1 top1 < encArg check < top2 new < top2 top2 < encArg new < check old < check check < encArg new < encArg old < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Types: top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encArg :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check hole_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check1_0 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0 :: Nat -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check Lemmas: new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n4_0)), rt in Omega(1 + n4_0) Generator Equations: gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0) <=> serve gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(x, 1)) <=> free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(x)) The following defined symbols remain to be analysed: old, top1, top2, check, encArg They will be analysed ascendingly in the following order: top1 = top2 check < top1 top1 < encArg check < top2 top2 < encArg old < check check < encArg old < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: old(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n277_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n277_0)), rt in Omega(1 + n277_0) Induction Base: old(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0)) ->_R^Omega(1) free(serve) Induction Step: old(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(n277_0, 1))) ->_R^Omega(1) free(old(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n277_0))) ->_IH free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, c278_0))) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Types: top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encArg :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check hole_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check1_0 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0 :: Nat -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check Lemmas: new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n4_0)), rt in Omega(1 + n4_0) old(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n277_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n277_0)), rt in Omega(1 + n277_0) Generator Equations: gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0) <=> serve gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(x, 1)) <=> free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(x)) The following defined symbols remain to be analysed: check, top1, top2, encArg They will be analysed ascendingly in the following order: top1 = top2 check < top1 top1 < encArg check < top2 top2 < encArg check < encArg ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: check(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n554_0))) -> *3_0, rt in Omega(n554_0) Induction Base: check(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, 0))) Induction Step: check(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, +(n554_0, 1)))) ->_R^Omega(1) free(check(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n554_0)))) ->_IH free(*3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: top1(free(x), y) -> top2(check(new(x)), y) top1(free(x), y) -> top2(new(x), check(y)) top1(free(x), y) -> top2(check(x), new(y)) top1(free(x), y) -> top2(x, check(new(y))) top2(x, free(y)) -> top1(check(new(x)), y) top2(x, free(y)) -> top1(new(x), check(y)) top2(x, free(y)) -> top1(check(x), new(y)) top2(x, free(y)) -> top1(x, check(new(y))) new(free(x)) -> free(new(x)) old(free(x)) -> free(old(x)) new(serve) -> free(serve) old(serve) -> free(serve) check(free(x)) -> free(check(x)) check(new(x)) -> new(check(x)) check(old(x)) -> old(check(x)) check(old(x)) -> old(x) encArg(free(x_1)) -> free(encArg(x_1)) encArg(serve) -> serve encArg(cons_top1(x_1, x_2)) -> top1(encArg(x_1), encArg(x_2)) encArg(cons_top2(x_1, x_2)) -> top2(encArg(x_1), encArg(x_2)) encArg(cons_new(x_1)) -> new(encArg(x_1)) encArg(cons_old(x_1)) -> old(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top1(x_1, x_2) -> top1(encArg(x_1), encArg(x_2)) encode_free(x_1) -> free(encArg(x_1)) encode_top2(x_1, x_2) -> top2(encArg(x_1), encArg(x_2)) encode_check(x_1) -> check(encArg(x_1)) encode_new(x_1) -> new(encArg(x_1)) encode_old(x_1) -> old(encArg(x_1)) encode_serve -> serve Types: top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encArg :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check cons_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top1 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_free :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_top2 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_check :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_new :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_old :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check encode_serve :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check hole_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check1_0 :: free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0 :: Nat -> free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check Lemmas: new(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n4_0)), rt in Omega(1 + n4_0) old(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n277_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n277_0)), rt in Omega(1 + n277_0) check(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(1, n554_0))) -> *3_0, rt in Omega(n554_0) Generator Equations: gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0) <=> serve gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(x, 1)) <=> free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(x)) The following defined symbols remain to be analysed: top2, top1, encArg They will be analysed ascendingly in the following order: top1 = top2 top1 < encArg top2 < encArg ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4247_0)) -> gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4247_0), rt in Omega(0) Induction Base: encArg(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(0)) ->_R^Omega(0) serve Induction Step: encArg(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(+(n4247_0, 1))) ->_R^Omega(0) free(encArg(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(n4247_0))) ->_IH free(gen_free:serve:cons_top1:cons_top2:cons_new:cons_old:cons_check2_0(c4248_0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (22) BOUNDS(1, INF)