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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 184 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), 1229 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), 5034 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__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(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(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(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__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(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__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(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__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) The (relative) TRS S consists of the following rules: encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) Types: a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark 2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encArg :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0 :: Nat -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a__2nd, mark, a__from, encArg They will be analysed ascendingly in the following order: a__2nd = mark a__2nd = a__from a__2nd < encArg mark = a__from mark < encArg a__from < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: a__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) Types: a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark 2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encArg :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0 :: Nat -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark Generator Equations: gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(0) <=> hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(x, 1)) <=> cons1(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0, gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(x)) The following defined symbols remain to be analysed: mark, a__2nd, a__from, encArg They will be analysed ascendingly in the following order: a__2nd = mark a__2nd = a__from a__2nd < encArg mark = a__from mark < encArg a__from < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: mark(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, 0))) Induction Step: mark(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) cons1(mark(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0), mark(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, n4_0)))) ->_IH cons1(mark(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0), *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__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) Types: a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark 2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encArg :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0 :: Nat -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark Generator Equations: gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(0) <=> hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(x, 1)) <=> cons1(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0, gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(x)) The following defined symbols remain to be analysed: mark, a__2nd, a__from, encArg They will be analysed ascendingly in the following order: a__2nd = mark a__2nd = a__from a__2nd < encArg mark = a__from mark < encArg a__from < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: a__2nd(cons1(X, cons(Y, Z))) -> mark(Y) a__2nd(cons(X, X1)) -> a__2nd(cons1(mark(X), mark(X1))) a__from(X) -> cons(mark(X), from(s(X))) mark(2nd(X)) -> a__2nd(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(cons1(X1, X2)) -> cons1(mark(X1), mark(X2)) a__2nd(X) -> 2nd(X) a__from(X) -> from(X) encArg(cons1(x_1, x_2)) -> cons1(encArg(x_1), encArg(x_2)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(from(x_1)) -> from(encArg(x_1)) encArg(s(x_1)) -> s(encArg(x_1)) encArg(2nd(x_1)) -> 2nd(encArg(x_1)) encArg(cons_a__2nd(x_1)) -> a__2nd(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__2nd(x_1) -> a__2nd(encArg(x_1)) encode_cons1(x_1, x_2) -> cons1(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_2nd(x_1) -> 2nd(encArg(x_1)) Types: a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark 2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encArg :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark cons_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons1 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_cons :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_mark :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_a__from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_from :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_s :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark encode_2nd :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 :: cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0 :: Nat -> cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark Lemmas: mark(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(0) <=> hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0 gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(x, 1)) <=> cons1(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0, gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(x)) The following defined symbols remain to be analysed: a__2nd, a__from, encArg They will be analysed ascendingly in the following order: a__2nd = mark a__2nd = a__from a__2nd < encArg mark = a__from mark < encArg a__from < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, n225115_0))) -> *3_0, rt in Omega(0) Induction Base: encArg(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, 0))) Induction Step: encArg(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, +(n225115_0, 1)))) ->_R^Omega(0) cons1(encArg(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0), encArg(gen_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark2_0(+(1, n225115_0)))) ->_IH cons1(encArg(hole_cons:cons1:s:from:2nd:cons_a__2nd:cons_a__from:cons_mark1_0), *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)