/export/starexec/sandbox2/solver/bin/starexec_run_rcdcRelativeAlsoLower /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 183 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), 317 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), 269 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(0) -> cons(0, f(s(0))) a__f(s(0)) -> a__f(a__p(s(0))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) 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(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) ---------------------------------------- (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(0) -> cons(0, f(s(0))) a__f(s(0)) -> a__f(a__p(s(0))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) 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(0) -> cons(0, f(s(0))) a__f(s(0)) -> a__f(a__p(s(0))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) The (relative) TRS S consists of the following rules: encArg(0) -> 0 encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0 encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) 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(0') -> cons(0', f(s(0'))) a__f(s(0')) -> a__f(a__p(s(0'))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0') -> 0' mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) The (relative) TRS S consists of the following rules: encArg(0') -> 0' encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0' encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a__f(0') -> cons(0', f(s(0'))) a__f(s(0')) -> a__f(a__p(s(0'))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0') -> 0' mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) encArg(0') -> 0' encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0' encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) Types: a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark 0' :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encArg :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_0 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark hole_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark1_3 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3 :: Nat -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a__f, a__p, mark, encArg They will be analysed ascendingly in the following order: a__f = a__p a__f = mark a__f < encArg a__p = mark a__p < encArg mark < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: a__f(0') -> cons(0', f(s(0'))) a__f(s(0')) -> a__f(a__p(s(0'))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0') -> 0' mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) encArg(0') -> 0' encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0' encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) Types: a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark 0' :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encArg :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_0 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark hole_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark1_3 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3 :: Nat -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark Generator Equations: gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(0) <=> 0' gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(+(x, 1)) <=> cons(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(x), 0') The following defined symbols remain to be analysed: a__p, a__f, mark, encArg They will be analysed ascendingly in the following order: a__f = a__p a__f = mark a__f < encArg a__p = mark a__p < encArg mark < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n10_3)) -> gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n10_3), rt in Omega(1 + n10_3) Induction Base: mark(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(0)) ->_R^Omega(1) 0' Induction Step: mark(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(+(n10_3, 1))) ->_R^Omega(1) cons(mark(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n10_3)), 0') ->_IH cons(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(c11_3), 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: Innermost TRS: Rules: a__f(0') -> cons(0', f(s(0'))) a__f(s(0')) -> a__f(a__p(s(0'))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0') -> 0' mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) encArg(0') -> 0' encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0' encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) Types: a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark 0' :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encArg :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_0 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark hole_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark1_3 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3 :: Nat -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark Generator Equations: gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(0) <=> 0' gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(+(x, 1)) <=> cons(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(x), 0') The following defined symbols remain to be analysed: mark, a__f, encArg They will be analysed ascendingly in the following order: a__f = a__p a__f = mark a__f < encArg a__p = mark a__p < encArg mark < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: a__f(0') -> cons(0', f(s(0'))) a__f(s(0')) -> a__f(a__p(s(0'))) a__p(s(X)) -> mark(X) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(0') -> 0' mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) a__f(X) -> f(X) a__p(X) -> p(X) encArg(0') -> 0' encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(f(x_1)) -> f(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(p(x_1)) -> p(encArg(x_1)) encArg(cons_a__f(x_1)) -> a__f(encArg(x_1)) encArg(cons_a__p(x_1)) -> a__p(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__f(x_1) -> a__f(encArg(x_1)) encode_0 -> 0' encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_f(x_1) -> f(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__p(x_1) -> a__p(encArg(x_1)) encode_mark(x_1) -> mark(encArg(x_1)) encode_p(x_1) -> p(encArg(x_1)) Types: a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark 0' :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encArg :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark cons_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_0 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_cons :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_f :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_s :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_a__p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_mark :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark encode_p :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark hole_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark1_3 :: 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3 :: Nat -> 0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark Lemmas: mark(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n10_3)) -> gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n10_3), rt in Omega(1 + n10_3) Generator Equations: gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(0) <=> 0' gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(+(x, 1)) <=> cons(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(x), 0') The following defined symbols remain to be analysed: a__f, a__p, encArg They will be analysed ascendingly in the following order: a__f = a__p a__f = mark a__f < encArg a__p = mark a__p < encArg mark < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n953_3)) -> gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n953_3), rt in Omega(0) Induction Base: encArg(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(0)) ->_R^Omega(0) 0' Induction Step: encArg(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(+(n953_3, 1))) ->_R^Omega(0) cons(encArg(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n953_3)), encArg(0')) ->_IH cons(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(c954_3), encArg(0')) ->_R^Omega(0) cons(gen_0':s:f:cons:p:cons_a__f:cons_a__p:cons_mark2_3(n953_3), 0') We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)