WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (4) typed CpxTrs (5) OrderProof [LOWER BOUND(ID), 0 ms] (6) typed CpxTrs (7) RewriteLemmaProof [LOWER BOUND(ID), 31.0 s] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 25.5 s] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 29.5 s] (16) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0, cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0) -> 01 a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0, cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0) -> 01 a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0) -> 0 mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01) -> 01 mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) Types: a__dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 0' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote nil :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote cons :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote mark :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 01' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote hole_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote1_0 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0 :: Nat -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a__sel, mark, a__dbl1, a__sel1, a__quote They will be analysed ascendingly in the following order: a__sel = mark a__sel = a__dbl1 a__sel = a__sel1 a__sel = a__quote mark = a__dbl1 mark = a__sel1 mark = a__quote a__dbl1 = a__sel1 a__dbl1 = a__quote a__sel1 = a__quote ---------------------------------------- (6) Obligation: TRS: Rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) Types: a__dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 0' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote nil :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote cons :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote mark :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 01' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote hole_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote1_0 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0 :: Nat -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote Generator Equations: gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(0) <=> 0' gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(x, 1)) <=> s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(x)) The following defined symbols remain to be analysed: mark, a__sel, a__dbl1, a__sel1, a__quote They will be analysed ascendingly in the following order: a__sel = mark a__sel = a__dbl1 a__sel = a__sel1 a__sel = a__quote mark = a__dbl1 mark = a__sel1 mark = a__quote a__dbl1 = a__sel1 a__dbl1 = a__quote a__sel1 = a__quote ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n79114_0))) -> *3_0, rt in Omega(n79114_0) Induction Base: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, 0))) Induction Step: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, +(n79114_0, 1)))) ->_R^Omega(1) s1(s1(a__dbl1(mark(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n79114_0)))))) ->_R^Omega(1) s1(s1(a__dbl1(s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(n79114_0))))) ->_IH s1(s1(*3_0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (8) Complex Obligation (BEST) ---------------------------------------- (9) Obligation: Proved the lower bound n^1 for the following obligation: TRS: Rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) Types: a__dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 0' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote nil :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote cons :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote mark :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 01' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote hole_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote1_0 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0 :: Nat -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote Generator Equations: gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(0) <=> 0' gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(x, 1)) <=> s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(x)) The following defined symbols remain to be analysed: a__dbl1, a__sel1, a__quote They will be analysed ascendingly in the following order: a__sel = mark a__sel = a__dbl1 a__sel = a__sel1 a__sel = a__quote mark = a__dbl1 mark = a__sel1 mark = a__quote a__dbl1 = a__sel1 a__dbl1 = a__quote a__sel1 = a__quote ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) Types: a__dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 0' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote nil :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote cons :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote mark :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 01' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote hole_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote1_0 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0 :: Nat -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote Lemmas: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n79114_0))) -> *3_0, rt in Omega(n79114_0) Generator Equations: gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(0) <=> 0' gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(x, 1)) <=> s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(x)) The following defined symbols remain to be analysed: a__sel1, a__sel, mark, a__quote They will be analysed ascendingly in the following order: a__sel = mark a__sel = a__dbl1 a__sel = a__sel1 a__sel = a__quote mark = a__dbl1 mark = a__sel1 mark = a__quote a__dbl1 = a__sel1 a__dbl1 = a__quote a__sel1 = a__quote ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a__quote(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n1977816_0))) -> *3_0, rt in Omega(n1977816_0) Induction Base: a__quote(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, 0))) Induction Step: a__quote(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, +(n1977816_0, 1)))) ->_R^Omega(1) s1(a__quote(mark(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n1977816_0))))) ->_R^Omega(1) s1(a__quote(s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(n1977816_0)))) ->_IH s1(*3_0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (14) Obligation: TRS: Rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) Types: a__dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 0' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote nil :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote cons :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote mark :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 01' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote hole_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote1_0 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0 :: Nat -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote Lemmas: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n79114_0))) -> *3_0, rt in Omega(n79114_0) a__quote(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n1977816_0))) -> *3_0, rt in Omega(n1977816_0) Generator Equations: gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(0) <=> 0' gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(x, 1)) <=> s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(x)) The following defined symbols remain to be analysed: mark, a__sel, a__dbl1, a__sel1 They will be analysed ascendingly in the following order: a__sel = mark a__sel = a__dbl1 a__sel = a__sel1 a__sel = a__quote mark = a__dbl1 mark = a__sel1 mark = a__quote a__dbl1 = a__sel1 a__dbl1 = a__quote a__sel1 = a__quote ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n3958185_0))) -> *3_0, rt in Omega(n3958185_0) Induction Base: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, 0))) Induction Step: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, +(n3958185_0, 1)))) ->_R^Omega(1) s1(s1(a__dbl1(mark(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n3958185_0)))))) ->_R^Omega(1) s1(s1(a__dbl1(s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(n3958185_0))))) ->_IH s1(s1(*3_0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (16) Obligation: TRS: Rules: a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(dbl(X))) a__dbls(nil) -> nil a__dbls(cons(X, Y)) -> cons(dbl(X), dbls(Y)) a__sel(0', cons(X, Y)) -> mark(X) a__sel(s(X), cons(Y, Z)) -> a__sel(mark(X), mark(Z)) a__indx(nil, X) -> nil a__indx(cons(X, Y), Z) -> cons(sel(X, Z), indx(Y, Z)) a__from(X) -> cons(X, from(s(X))) a__dbl1(0') -> 01' a__dbl1(s(X)) -> s1(s1(a__dbl1(mark(X)))) a__sel1(0', cons(X, Y)) -> mark(X) a__sel1(s(X), cons(Y, Z)) -> a__sel1(mark(X), mark(Z)) a__quote(0') -> 01' a__quote(s(X)) -> s1(a__quote(mark(X))) a__quote(dbl(X)) -> a__dbl1(mark(X)) a__quote(sel(X, Y)) -> a__sel1(mark(X), mark(Y)) mark(dbl(X)) -> a__dbl(mark(X)) mark(dbls(X)) -> a__dbls(mark(X)) mark(sel(X1, X2)) -> a__sel(mark(X1), mark(X2)) mark(indx(X1, X2)) -> a__indx(mark(X1), X2) mark(from(X)) -> a__from(X) mark(dbl1(X)) -> a__dbl1(mark(X)) mark(sel1(X1, X2)) -> a__sel1(mark(X1), mark(X2)) mark(quote(X)) -> a__quote(mark(X)) mark(0') -> 0' mark(s(X)) -> s(X) mark(nil) -> nil mark(cons(X1, X2)) -> cons(X1, X2) mark(01') -> 01' mark(s1(X)) -> s1(mark(X)) a__dbl(X) -> dbl(X) a__dbls(X) -> dbls(X) a__sel(X1, X2) -> sel(X1, X2) a__indx(X1, X2) -> indx(X1, X2) a__from(X) -> from(X) a__dbl1(X) -> dbl1(X) a__sel1(X1, X2) -> sel1(X1, X2) a__quote(X) -> quote(X) Types: a__dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 0' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote nil :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote cons :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbls :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote mark :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote indx :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote from :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote 01' :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote s1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote a__quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote dbl1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote sel1 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote quote :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote hole_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote1_0 :: 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0 :: Nat -> 0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote Lemmas: a__dbl1(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n3958185_0))) -> *3_0, rt in Omega(n3958185_0) a__quote(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(1, n1977816_0))) -> *3_0, rt in Omega(n1977816_0) Generator Equations: gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(0) <=> 0' gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(+(x, 1)) <=> s(gen_0':s:dbl:nil:cons:dbls:sel:indx:from:01':s1:dbl1:sel1:quote2_0(x)) The following defined symbols remain to be analysed: a__sel1 They will be analysed ascendingly in the following order: a__sel = mark a__sel = a__dbl1 a__sel = a__sel1 a__sel = a__quote mark = a__dbl1 mark = a__sel1 mark = a__quote a__dbl1 = a__sel1 a__dbl1 = a__quote a__sel1 = a__quote