/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). (0) DCpxTrs (1) DerivationalComplexityToRuntimeComplexityProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxRelTRS (3) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 189 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), 426 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), 33 ms] (18) BOUNDS(1, INF) ---------------------------------------- (0) Obligation: The Derivational Complexity (full) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) DerivationalComplexityToRuntimeComplexityProof (BOTH BOUNDS(ID, ID)) The following rules have been added to S to convert the given derivational complexity problem to a runtime complexity problem: encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) The (relative) TRS S consists of the following rules: encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Rewrite Strategy: FULL ---------------------------------------- (3) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (4) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) The (relative) TRS S consists of the following rules: encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Rewrite Strategy: FULL ---------------------------------------- (5) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (6) Obligation: The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) The (relative) TRS S consists of the following rules: encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Rewrite Strategy: FULL ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: TRS: Rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Types: top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check nil :: sent:nil:cons:cons_top:cons_rest:cons_check cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encArg :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_nil :: sent:nil:cons:cons_top:cons_rest:cons_check encode_cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check hole_sent:nil:cons:cons_top:cons_rest:cons_check1_0 :: sent:nil:cons:cons_top:cons_rest:cons_check gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0 :: Nat -> sent:nil:cons:cons_top:cons_rest:cons_check ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: top, check, encArg They will be analysed ascendingly in the following order: check < top top < encArg check < encArg ---------------------------------------- (10) Obligation: TRS: Rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Types: top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check nil :: sent:nil:cons:cons_top:cons_rest:cons_check cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encArg :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_nil :: sent:nil:cons:cons_top:cons_rest:cons_check encode_cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check hole_sent:nil:cons:cons_top:cons_rest:cons_check1_0 :: sent:nil:cons:cons_top:cons_rest:cons_check gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0 :: Nat -> sent:nil:cons:cons_top:cons_rest:cons_check Generator Equations: gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(0) <=> nil gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(x, 1)) <=> sent(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(x)) The following defined symbols remain to be analysed: check, top, encArg They will be analysed ascendingly in the following order: check < top top < encArg check < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: check(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Induction Base: check(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(1, 0))) Induction Step: check(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1) sent(check(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(1, n4_0)))) ->_IH sent(*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: TRS: Rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Types: top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check nil :: sent:nil:cons:cons_top:cons_rest:cons_check cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encArg :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_nil :: sent:nil:cons:cons_top:cons_rest:cons_check encode_cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check hole_sent:nil:cons:cons_top:cons_rest:cons_check1_0 :: sent:nil:cons:cons_top:cons_rest:cons_check gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0 :: Nat -> sent:nil:cons:cons_top:cons_rest:cons_check Generator Equations: gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(0) <=> nil gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(x, 1)) <=> sent(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(x)) The following defined symbols remain to be analysed: check, top, encArg They will be analysed ascendingly in the following order: check < top top < encArg check < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: TRS: Rules: top(sent(x)) -> top(check(rest(x))) rest(nil) -> sent(nil) rest(cons(x, y)) -> sent(y) check(sent(x)) -> sent(check(x)) check(rest(x)) -> rest(check(x)) check(cons(x, y)) -> cons(check(x), y) check(cons(x, y)) -> cons(x, check(y)) check(cons(x, y)) -> cons(x, y) encArg(sent(x_1)) -> sent(encArg(x_1)) encArg(nil) -> nil encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(cons_top(x_1)) -> top(encArg(x_1)) encArg(cons_rest(x_1)) -> rest(encArg(x_1)) encArg(cons_check(x_1)) -> check(encArg(x_1)) encode_top(x_1) -> top(encArg(x_1)) encode_sent(x_1) -> sent(encArg(x_1)) encode_check(x_1) -> check(encArg(x_1)) encode_rest(x_1) -> rest(encArg(x_1)) encode_nil -> nil encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) Types: top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check nil :: sent:nil:cons:cons_top:cons_rest:cons_check cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encArg :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check cons_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_top :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_sent :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_check :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_rest :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check encode_nil :: sent:nil:cons:cons_top:cons_rest:cons_check encode_cons :: sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check -> sent:nil:cons:cons_top:cons_rest:cons_check hole_sent:nil:cons:cons_top:cons_rest:cons_check1_0 :: sent:nil:cons:cons_top:cons_rest:cons_check gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0 :: Nat -> sent:nil:cons:cons_top:cons_rest:cons_check Lemmas: check(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0) Generator Equations: gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(0) <=> nil gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(x, 1)) <=> sent(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(x)) The following defined symbols remain to be analysed: top, encArg They will be analysed ascendingly in the following order: top < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(n4552_0)) -> gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(n4552_0), rt in Omega(0) Induction Base: encArg(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(0)) ->_R^Omega(0) nil Induction Step: encArg(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(+(n4552_0, 1))) ->_R^Omega(0) sent(encArg(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(n4552_0))) ->_IH sent(gen_sent:nil:cons:cons_top:cons_rest:cons_check2_0(c4553_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)