/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 250 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), 316 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), 1618 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__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0 a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0) -> 0 a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(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(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(nil) -> nil encArg(0) -> 0 encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0 encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(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__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0 a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0) -> 0 a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) The (relative) TRS S consists of the following rules: 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(nil) -> nil encArg(0) -> 0 encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0 encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(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__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0 a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0) -> 0 a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) The (relative) TRS S consists of the following rules: 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(nil) -> nil encArg(0) -> 0 encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0 encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(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__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0' a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0') -> 0' a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) The (relative) TRS S consists of the following rules: 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(nil) -> nil encArg(0') -> 0' encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0' encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(encArg(x_1)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0' a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0') -> 0' a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) 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(nil) -> nil encArg(0') -> 0' encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0' encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(encArg(x_1)) Types: a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark 0' :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encArg :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_0 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark hole_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark1_3 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3 :: Nat -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a__from, mark, a__length, a__length1, encArg They will be analysed ascendingly in the following order: a__from = mark a__from < encArg a__length < mark a__length1 < mark mark < encArg a__length = a__length1 a__length < encArg a__length1 < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: a__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0' a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0') -> 0' a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) 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(nil) -> nil encArg(0') -> 0' encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0' encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(encArg(x_1)) Types: a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark 0' :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encArg :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_0 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark hole_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark1_3 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3 :: Nat -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark Generator Equations: gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(0) <=> nil gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(+(x, 1)) <=> cons(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(x), nil) The following defined symbols remain to be analysed: a__length1, a__from, mark, a__length, encArg They will be analysed ascendingly in the following order: a__from = mark a__from < encArg a__length < mark a__length1 < mark mark < encArg a__length = a__length1 a__length < encArg a__length1 < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n58_3)) -> gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n58_3), rt in Omega(1 + n58_3) Induction Base: mark(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(0)) ->_R^Omega(1) nil Induction Step: mark(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(+(n58_3, 1))) ->_R^Omega(1) cons(mark(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n58_3)), nil) ->_IH cons(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(c59_3), nil) 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__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0' a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0') -> 0' a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) 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(nil) -> nil encArg(0') -> 0' encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0' encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(encArg(x_1)) Types: a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark 0' :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encArg :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_0 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark hole_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark1_3 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3 :: Nat -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark Generator Equations: gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(0) <=> nil gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(+(x, 1)) <=> cons(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(x), nil) The following defined symbols remain to be analysed: mark, a__from, encArg They will be analysed ascendingly in the following order: a__from = mark a__from < encArg mark < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: a__from(X) -> cons(mark(X), from(s(X))) a__length(nil) -> 0' a__length(cons(X, Y)) -> s(a__length1(Y)) a__length1(X) -> a__length(X) mark(from(X)) -> a__from(mark(X)) mark(length(X)) -> a__length(X) mark(length1(X)) -> a__length1(X) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(s(X)) -> s(mark(X)) mark(nil) -> nil mark(0') -> 0' a__from(X) -> from(X) a__length(X) -> length(X) a__length1(X) -> length1(X) 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(nil) -> nil encArg(0') -> 0' encArg(length(x_1)) -> length(encArg(x_1)) encArg(length1(x_1)) -> length1(encArg(x_1)) encArg(cons_a__from(x_1)) -> a__from(encArg(x_1)) encArg(cons_a__length(x_1)) -> a__length(encArg(x_1)) encArg(cons_a__length1(x_1)) -> a__length1(encArg(x_1)) encArg(cons_mark(x_1)) -> mark(encArg(x_1)) encode_a__from(x_1) -> a__from(encArg(x_1)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_mark(x_1) -> mark(encArg(x_1)) encode_from(x_1) -> from(encArg(x_1)) encode_s(x_1) -> s(encArg(x_1)) encode_a__length(x_1) -> a__length(encArg(x_1)) encode_nil -> nil encode_0 -> 0' encode_a__length1(x_1) -> a__length1(encArg(x_1)) encode_length(x_1) -> length(encArg(x_1)) encode_length1(x_1) -> length1(encArg(x_1)) Types: a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark 0' :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encArg :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark cons_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_cons :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_mark :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_from :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_s :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_nil :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_0 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_a__length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark encode_length1 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark hole_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark1_3 :: s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3 :: Nat -> s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark Lemmas: mark(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n58_3)) -> gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n58_3), rt in Omega(1 + n58_3) Generator Equations: gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(0) <=> nil gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(+(x, 1)) <=> cons(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(x), nil) The following defined symbols remain to be analysed: a__from, encArg They will be analysed ascendingly in the following order: a__from = mark a__from < encArg mark < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n1198_3)) -> gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n1198_3), rt in Omega(0) Induction Base: encArg(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(0)) ->_R^Omega(0) nil Induction Step: encArg(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(+(n1198_3, 1))) ->_R^Omega(0) cons(encArg(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n1198_3)), encArg(nil)) ->_IH cons(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(c1199_3), encArg(nil)) ->_R^Omega(0) cons(gen_s:from:cons:nil:0':length:length1:cons_a__from:cons_a__length:cons_a__length1:cons_mark2_3(n1198_3), nil) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (18) BOUNDS(1, INF)