WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox/benchmark/theBenchmark.trs # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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) SlicingProof [LOWER BOUND(ID), 0 ms] (4) CpxTRS (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) typed CpxTrs (7) OrderProof [LOWER BOUND(ID), 17 ms] (8) typed CpxTrs (9) RewriteLemmaProof [LOWER BOUND(ID), 10.1 s] (10) BEST (11) proven lower bound (12) LowerBoundPropagationProof [FINISHED, 0 ms] (13) BOUNDS(n^1, INF) (14) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) a__sqr(0) -> 0 a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0) -> 0 a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0, X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0, X) -> nil a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) a__half(0) -> 0 a__half(s(0)) -> 0 a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) a__sqr(0') -> 0' a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0', X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0', X) -> nil a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) a__half(0') -> 0' a__half(s(0')) -> 0' a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) SlicingProof (LOWER BOUND(ID)) Sliced the following arguments: cons/1 ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: a__terms(N) -> cons(recip(a__sqr(mark(N)))) a__sqr(0') -> 0' a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0', X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0', X) -> nil a__first(s(X), cons(Y)) -> cons(mark(Y)) a__half(0') -> 0' a__half(s(0')) -> 0' a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1)) -> cons(mark(X1)) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Innermost TRS: Rules: a__terms(N) -> cons(recip(a__sqr(mark(N)))) a__sqr(0') -> 0' a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0', X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0', X) -> nil a__first(s(X), cons(Y)) -> cons(mark(Y)) a__half(0') -> 0' a__half(s(0')) -> 0' a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1)) -> cons(mark(X1)) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) Types: a__terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half cons :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half recip :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half mark :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half 0' :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half s :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half nil :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half hole_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half1_0 :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0 :: Nat -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half ---------------------------------------- (7) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: a__terms, a__sqr, mark, a__add, a__dbl, a__half They will be analysed ascendingly in the following order: a__terms = a__sqr a__terms = mark a__terms = a__add a__terms = a__dbl a__terms = a__half a__sqr = mark a__sqr = a__add a__sqr = a__dbl a__sqr = a__half mark = a__add mark = a__dbl mark = a__half a__add = a__dbl a__add = a__half a__dbl = a__half ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a__terms(N) -> cons(recip(a__sqr(mark(N)))) a__sqr(0') -> 0' a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0', X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0', X) -> nil a__first(s(X), cons(Y)) -> cons(mark(Y)) a__half(0') -> 0' a__half(s(0')) -> 0' a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1)) -> cons(mark(X1)) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) Types: a__terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half cons :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half recip :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half mark :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half 0' :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half s :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half nil :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half hole_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half1_0 :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0 :: Nat -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half Generator Equations: gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(0) <=> 0' gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(+(x, 1)) <=> cons(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(x)) The following defined symbols remain to be analysed: a__sqr, a__terms, mark, a__add, a__dbl, a__half They will be analysed ascendingly in the following order: a__terms = a__sqr a__terms = mark a__terms = a__add a__terms = a__dbl a__terms = a__half a__sqr = mark a__sqr = a__add a__sqr = a__dbl a__sqr = a__half mark = a__add mark = a__dbl mark = a__half a__add = a__dbl a__add = a__half a__dbl = a__half ---------------------------------------- (9) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(n91279_0)) -> gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(n91279_0), rt in Omega(1 + n91279_0) Induction Base: mark(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(0)) ->_R^Omega(1) 0' Induction Step: mark(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(+(n91279_0, 1))) ->_R^Omega(1) cons(mark(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(n91279_0))) ->_IH cons(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(c91280_0)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (10) Complex Obligation (BEST) ---------------------------------------- (11) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: a__terms(N) -> cons(recip(a__sqr(mark(N)))) a__sqr(0') -> 0' a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0', X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0', X) -> nil a__first(s(X), cons(Y)) -> cons(mark(Y)) a__half(0') -> 0' a__half(s(0')) -> 0' a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1)) -> cons(mark(X1)) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) Types: a__terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half cons :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half recip :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half mark :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half 0' :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half s :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half nil :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half hole_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half1_0 :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0 :: Nat -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half Generator Equations: gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(0) <=> 0' gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(+(x, 1)) <=> cons(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(x)) The following defined symbols remain to be analysed: mark, a__terms, a__dbl, a__half They will be analysed ascendingly in the following order: a__terms = a__sqr a__terms = mark a__terms = a__add a__terms = a__dbl a__terms = a__half a__sqr = mark a__sqr = a__add a__sqr = a__dbl a__sqr = a__half mark = a__add mark = a__dbl mark = a__half a__add = a__dbl a__add = a__half a__dbl = a__half ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: Innermost TRS: Rules: a__terms(N) -> cons(recip(a__sqr(mark(N)))) a__sqr(0') -> 0' a__sqr(s(X)) -> s(a__add(a__sqr(mark(X)), a__dbl(mark(X)))) a__dbl(0') -> 0' a__dbl(s(X)) -> s(s(a__dbl(mark(X)))) a__add(0', X) -> mark(X) a__add(s(X), Y) -> s(a__add(mark(X), mark(Y))) a__first(0', X) -> nil a__first(s(X), cons(Y)) -> cons(mark(Y)) a__half(0') -> 0' a__half(s(0')) -> 0' a__half(s(s(X))) -> s(a__half(mark(X))) a__half(dbl(X)) -> mark(X) mark(terms(X)) -> a__terms(mark(X)) mark(sqr(X)) -> a__sqr(mark(X)) mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) mark(dbl(X)) -> a__dbl(mark(X)) mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) mark(half(X)) -> a__half(mark(X)) mark(cons(X1)) -> cons(mark(X1)) mark(recip(X)) -> recip(mark(X)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(nil) -> nil a__terms(X) -> terms(X) a__sqr(X) -> sqr(X) a__add(X1, X2) -> add(X1, X2) a__dbl(X) -> dbl(X) a__first(X1, X2) -> first(X1, X2) a__half(X) -> half(X) Types: a__terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half cons :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half recip :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half mark :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half 0' :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half s :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half nil :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half a__half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half dbl :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half terms :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half sqr :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half add :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half first :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half half :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half hole_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half1_0 :: recip:cons:0':s:nil:dbl:terms:sqr:add:first:half gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0 :: Nat -> recip:cons:0':s:nil:dbl:terms:sqr:add:first:half Lemmas: mark(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(n91279_0)) -> gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(n91279_0), rt in Omega(1 + n91279_0) Generator Equations: gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(0) <=> 0' gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(+(x, 1)) <=> cons(gen_recip:cons:0':s:nil:dbl:terms:sqr:add:first:half2_0(x)) The following defined symbols remain to be analysed: a__terms, a__sqr, a__add, a__dbl, a__half They will be analysed ascendingly in the following order: a__terms = a__sqr a__terms = mark a__terms = a__add a__terms = a__dbl a__terms = a__half a__sqr = mark a__sqr = a__add a__sqr = a__dbl a__sqr = a__half mark = a__add mark = a__dbl mark = a__half a__add = a__dbl a__add = a__half a__dbl = a__half