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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Derivational Complexity (innermost) of the given DCpxTrs could be proven to be BOUNDS(n^1, INF).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), 224 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), 719 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), 40 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: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) 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(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) ---------------------------------------- (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: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) 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: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) 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: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) The (relative) TRS S consists of the following rules: encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Innermost TRS: Rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) Types: circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst id :: cons:lift:id:cons_circ:cons_subst:cons_msubst subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encArg :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_id :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst hole_cons:lift:id:cons_circ:cons_subst:cons_msubst1_0 :: cons:lift:id:cons_circ:cons_subst:cons_msubst gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0 :: Nat -> cons:lift:id:cons_circ:cons_subst:cons_msubst ---------------------------------------- (9) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: circ, msubst, encArg They will be analysed ascendingly in the following order: circ = msubst circ < encArg msubst < encArg ---------------------------------------- (10) Obligation: Innermost TRS: Rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) Types: circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst id :: cons:lift:id:cons_circ:cons_subst:cons_msubst subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encArg :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_id :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst hole_cons:lift:id:cons_circ:cons_subst:cons_msubst1_0 :: cons:lift:id:cons_circ:cons_subst:cons_msubst gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0 :: Nat -> cons:lift:id:cons_circ:cons_subst:cons_msubst Generator Equations: gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(0) <=> lift gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(x, 1)) <=> cons(lift, gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(x)) The following defined symbols remain to be analysed: msubst, circ, encArg They will be analysed ascendingly in the following order: circ = msubst circ < encArg msubst < encArg ---------------------------------------- (11) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: circ(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, n15_0)), gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, n15_0))) -> *3_0, rt in Omega(n15_0) Induction Base: circ(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, 0)), gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, 0))) Induction Step: circ(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, +(n15_0, 1))), gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, +(n15_0, 1)))) ->_R^Omega(1) cons(lift, circ(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, n15_0)), gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, n15_0)))) ->_IH cons(lift, *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: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) Types: circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst id :: cons:lift:id:cons_circ:cons_subst:cons_msubst subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encArg :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_id :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst hole_cons:lift:id:cons_circ:cons_subst:cons_msubst1_0 :: cons:lift:id:cons_circ:cons_subst:cons_msubst gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0 :: Nat -> cons:lift:id:cons_circ:cons_subst:cons_msubst Generator Equations: gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(0) <=> lift gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(x, 1)) <=> cons(lift, gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(x)) The following defined symbols remain to be analysed: circ, encArg They will be analysed ascendingly in the following order: circ = msubst circ < encArg msubst < encArg ---------------------------------------- (14) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (15) BOUNDS(n^1, INF) ---------------------------------------- (16) Obligation: Innermost TRS: Rules: circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) circ(circ(s, t), u) -> circ(s, circ(t, u)) circ(s, id) -> s circ(id, s) -> s circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u) subst(a, id) -> a msubst(a, id) -> a msubst(msubst(a, s), t) -> msubst(a, circ(s, t)) encArg(cons(x_1, x_2)) -> cons(encArg(x_1), encArg(x_2)) encArg(lift) -> lift encArg(id) -> id encArg(cons_circ(x_1, x_2)) -> circ(encArg(x_1), encArg(x_2)) encArg(cons_subst(x_1, x_2)) -> subst(encArg(x_1), encArg(x_2)) encArg(cons_msubst(x_1, x_2)) -> msubst(encArg(x_1), encArg(x_2)) encode_circ(x_1, x_2) -> circ(encArg(x_1), encArg(x_2)) encode_cons(x_1, x_2) -> cons(encArg(x_1), encArg(x_2)) encode_msubst(x_1, x_2) -> msubst(encArg(x_1), encArg(x_2)) encode_lift -> lift encode_id -> id encode_subst(x_1, x_2) -> subst(encArg(x_1), encArg(x_2)) Types: circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst id :: cons:lift:id:cons_circ:cons_subst:cons_msubst subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encArg :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst cons_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_circ :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_cons :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_msubst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst encode_lift :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_id :: cons:lift:id:cons_circ:cons_subst:cons_msubst encode_subst :: cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst -> cons:lift:id:cons_circ:cons_subst:cons_msubst hole_cons:lift:id:cons_circ:cons_subst:cons_msubst1_0 :: cons:lift:id:cons_circ:cons_subst:cons_msubst gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0 :: Nat -> cons:lift:id:cons_circ:cons_subst:cons_msubst Lemmas: circ(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, n15_0)), gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(1, n15_0))) -> *3_0, rt in Omega(n15_0) Generator Equations: gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(0) <=> lift gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(x, 1)) <=> cons(lift, gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(x)) The following defined symbols remain to be analysed: msubst, encArg They will be analysed ascendingly in the following order: circ = msubst circ < encArg msubst < encArg ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: encArg(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(n8804_0)) -> gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(n8804_0), rt in Omega(0) Induction Base: encArg(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(0)) ->_R^Omega(0) lift Induction Step: encArg(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(+(n8804_0, 1))) ->_R^Omega(0) cons(encArg(lift), encArg(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(n8804_0))) ->_R^Omega(0) cons(lift, encArg(gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(n8804_0))) ->_IH cons(lift, gen_cons:lift:id:cons_circ:cons_subst:cons_msubst2_0(c8805_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)