/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 181 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), 226 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), 95 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: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c 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(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) ---------------------------------------- (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: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) 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: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) 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: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c The (relative) TRS S consists of the following rules: encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) Types: a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark b :: b:c:f:cons_a__f:cons_a__c:cons_mark c :: b:c:f:cons_a__f:cons_a__c:cons_mark a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encArg :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark cons_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_b :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark hole_b:c:f:cons_a__f:cons_a__c:cons_mark1_0 :: b:c:f:cons_a__f:cons_a__c:cons_mark gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0 :: Nat -> b:c:f:cons_a__f:cons_a__c:cons_mark ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a__f, mark, encArg They will be analysed ascendingly in the following order: a__f < mark a__f < encArg mark < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) Types: a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark b :: b:c:f:cons_a__f:cons_a__c:cons_mark c :: b:c:f:cons_a__f:cons_a__c:cons_mark a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encArg :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark cons_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_b :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark hole_b:c:f:cons_a__f:cons_a__c:cons_mark1_0 :: b:c:f:cons_a__f:cons_a__c:cons_mark gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0 :: Nat -> b:c:f:cons_a__f:cons_a__c:cons_mark Generator Equations: gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(0) <=> b gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(+(x, 1)) <=> f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(x), b) The following defined symbols remain to be analysed: a__f, mark, encArg They will be analysed ascendingly in the following order: a__f < mark a__f < encArg mark < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n15_0)) -> gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n15_0), rt in Omega(1 + n15_0) Induction Base: mark(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(0)) ->_R^Omega(1) b Induction Step: mark(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(+(n15_0, 1))) ->_R^Omega(1) a__f(b, mark(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n15_0)), b) ->_IH a__f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(c16_0), b) ->_R^Omega(1) f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n15_0), b) 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: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) Types: a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark b :: b:c:f:cons_a__f:cons_a__c:cons_mark c :: b:c:f:cons_a__f:cons_a__c:cons_mark a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encArg :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark cons_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_b :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark hole_b:c:f:cons_a__f:cons_a__c:cons_mark1_0 :: b:c:f:cons_a__f:cons_a__c:cons_mark gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0 :: Nat -> b:c:f:cons_a__f:cons_a__c:cons_mark Generator Equations: gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(0) <=> b gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(+(x, 1)) <=> f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(x), b) The following defined symbols remain to be analysed: mark, encArg They will be analysed ascendingly in the following order: mark < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: a__f(b, X, c) -> a__f(X, a__c, X) a__c -> b mark(f(X1, X2, X3)) -> a__f(X1, mark(X2), X3) mark(c) -> a__c mark(b) -> b a__f(X1, X2, X3) -> f(X1, X2, X3) a__c -> c encArg(b) -> b encArg(c) -> c encArg(f(x_1, x_2, x_3)) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__f(x_1, x_2, x_3)) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encArg(cons_a__c) -> a__c encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1, x_2, x_3) -> a__f(encArg(x_1), encArg(x_2), encArg(x_3)) encode_b -> b encode_c -> c encode_a__c -> a__c encode_mark(x_1) -> mark(encArg(x_1)) encode_f(x_1, x_2, x_3) -> f(encArg(x_1), encArg(x_2), encArg(x_3)) Types: a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark b :: b:c:f:cons_a__f:cons_a__c:cons_mark c :: b:c:f:cons_a__f:cons_a__c:cons_mark a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encArg :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark cons_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark cons_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_b :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_a__c :: b:c:f:cons_a__f:cons_a__c:cons_mark encode_mark :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark encode_f :: b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark -> b:c:f:cons_a__f:cons_a__c:cons_mark hole_b:c:f:cons_a__f:cons_a__c:cons_mark1_0 :: b:c:f:cons_a__f:cons_a__c:cons_mark gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0 :: Nat -> b:c:f:cons_a__f:cons_a__c:cons_mark Lemmas: mark(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n15_0)) -> gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n15_0), rt in Omega(1 + n15_0) Generator Equations: gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(0) <=> b gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(+(x, 1)) <=> f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(x), b) The following defined symbols remain to be analysed: encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n425_0)) -> gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n425_0), rt in Omega(0) Induction Base: encArg(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(0)) ->_R^Omega(0) b Induction Step: encArg(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(+(n425_0, 1))) ->_R^Omega(0) f(encArg(b), encArg(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n425_0)), encArg(b)) ->_R^Omega(0) f(b, encArg(gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n425_0)), encArg(b)) ->_IH f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(c426_0), encArg(b)) ->_R^Omega(0) f(b, gen_b:c:f:cons_a__f:cons_a__c:cons_mark2_0(n425_0), b) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)