WORST_CASE(Omega(n^1),?) proof of /export/starexec/sandbox2/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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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), 13 ms] (8) typed CpxTrs (9) RewriteLemmaProof [LOWER BOUND(ID), 297 ms] (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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes 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__filter(cons(X, Y), 0', M) -> cons(0', filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0', Y)) -> cons(0', sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes 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__filter(cons(X), 0', M) -> cons(0') a__filter(cons(X), s(N), M) -> cons(mark(X)) a__sieve(cons(0')) -> cons(0') a__sieve(cons(s(N))) -> cons(s(mark(N))) a__nats(N) -> cons(mark(N)) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1)) -> cons(mark(X1)) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Innermost TRS: Rules: a__filter(cons(X), 0', M) -> cons(0') a__filter(cons(X), s(N), M) -> cons(mark(X)) a__sieve(cons(0')) -> cons(0') a__sieve(cons(s(N))) -> cons(s(mark(N))) a__nats(N) -> cons(mark(N)) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1)) -> cons(mark(X1)) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes Types: a__filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes cons :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes 0' :: cons:0':s:filter:sieve:nats:zprimes s :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes mark :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__zprimes :: cons:0':s:filter:sieve:nats:zprimes filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes zprimes :: cons:0':s:filter:sieve:nats:zprimes hole_cons:0':s:filter:sieve:nats:zprimes1_0 :: cons:0':s:filter:sieve:nats:zprimes gen_cons:0':s:filter:sieve:nats:zprimes2_0 :: Nat -> cons:0':s:filter:sieve:nats:zprimes ---------------------------------------- (7) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: mark, a__sieve, a__nats, a__zprimes They will be analysed ascendingly in the following order: mark = a__sieve mark = a__nats mark = a__zprimes a__sieve = a__nats a__sieve = a__zprimes a__nats = a__zprimes ---------------------------------------- (8) Obligation: Innermost TRS: Rules: a__filter(cons(X), 0', M) -> cons(0') a__filter(cons(X), s(N), M) -> cons(mark(X)) a__sieve(cons(0')) -> cons(0') a__sieve(cons(s(N))) -> cons(s(mark(N))) a__nats(N) -> cons(mark(N)) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1)) -> cons(mark(X1)) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes Types: a__filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes cons :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes 0' :: cons:0':s:filter:sieve:nats:zprimes s :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes mark :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__zprimes :: cons:0':s:filter:sieve:nats:zprimes filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes zprimes :: cons:0':s:filter:sieve:nats:zprimes hole_cons:0':s:filter:sieve:nats:zprimes1_0 :: cons:0':s:filter:sieve:nats:zprimes gen_cons:0':s:filter:sieve:nats:zprimes2_0 :: Nat -> cons:0':s:filter:sieve:nats:zprimes Generator Equations: gen_cons:0':s:filter:sieve:nats:zprimes2_0(0) <=> 0' gen_cons:0':s:filter:sieve:nats:zprimes2_0(+(x, 1)) <=> cons(gen_cons:0':s:filter:sieve:nats:zprimes2_0(x)) The following defined symbols remain to be analysed: a__sieve, mark, a__nats, a__zprimes They will be analysed ascendingly in the following order: mark = a__sieve mark = a__nats mark = a__zprimes a__sieve = a__nats a__sieve = a__zprimes a__nats = a__zprimes ---------------------------------------- (9) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: mark(gen_cons:0':s:filter:sieve:nats:zprimes2_0(n15_0)) -> gen_cons:0':s:filter:sieve:nats:zprimes2_0(n15_0), rt in Omega(1 + n15_0) Induction Base: mark(gen_cons:0':s:filter:sieve:nats:zprimes2_0(0)) ->_R^Omega(1) 0' Induction Step: mark(gen_cons:0':s:filter:sieve:nats:zprimes2_0(+(n15_0, 1))) ->_R^Omega(1) cons(mark(gen_cons:0':s:filter:sieve:nats:zprimes2_0(n15_0))) ->_IH cons(gen_cons:0':s:filter:sieve:nats:zprimes2_0(c16_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__filter(cons(X), 0', M) -> cons(0') a__filter(cons(X), s(N), M) -> cons(mark(X)) a__sieve(cons(0')) -> cons(0') a__sieve(cons(s(N))) -> cons(s(mark(N))) a__nats(N) -> cons(mark(N)) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1)) -> cons(mark(X1)) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes Types: a__filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes cons :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes 0' :: cons:0':s:filter:sieve:nats:zprimes s :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes mark :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__zprimes :: cons:0':s:filter:sieve:nats:zprimes filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes zprimes :: cons:0':s:filter:sieve:nats:zprimes hole_cons:0':s:filter:sieve:nats:zprimes1_0 :: cons:0':s:filter:sieve:nats:zprimes gen_cons:0':s:filter:sieve:nats:zprimes2_0 :: Nat -> cons:0':s:filter:sieve:nats:zprimes Generator Equations: gen_cons:0':s:filter:sieve:nats:zprimes2_0(0) <=> 0' gen_cons:0':s:filter:sieve:nats:zprimes2_0(+(x, 1)) <=> cons(gen_cons:0':s:filter:sieve:nats:zprimes2_0(x)) The following defined symbols remain to be analysed: mark, a__nats, a__zprimes They will be analysed ascendingly in the following order: mark = a__sieve mark = a__nats mark = a__zprimes a__sieve = a__nats a__sieve = a__zprimes a__nats = a__zprimes ---------------------------------------- (12) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (13) BOUNDS(n^1, INF) ---------------------------------------- (14) Obligation: Innermost TRS: Rules: a__filter(cons(X), 0', M) -> cons(0') a__filter(cons(X), s(N), M) -> cons(mark(X)) a__sieve(cons(0')) -> cons(0') a__sieve(cons(s(N))) -> cons(s(mark(N))) a__nats(N) -> cons(mark(N)) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1)) -> cons(mark(X1)) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes Types: a__filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes cons :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes 0' :: cons:0':s:filter:sieve:nats:zprimes s :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes mark :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__zprimes :: cons:0':s:filter:sieve:nats:zprimes filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes zprimes :: cons:0':s:filter:sieve:nats:zprimes hole_cons:0':s:filter:sieve:nats:zprimes1_0 :: cons:0':s:filter:sieve:nats:zprimes gen_cons:0':s:filter:sieve:nats:zprimes2_0 :: Nat -> cons:0':s:filter:sieve:nats:zprimes Lemmas: mark(gen_cons:0':s:filter:sieve:nats:zprimes2_0(n15_0)) -> gen_cons:0':s:filter:sieve:nats:zprimes2_0(n15_0), rt in Omega(1 + n15_0) Generator Equations: gen_cons:0':s:filter:sieve:nats:zprimes2_0(0) <=> 0' gen_cons:0':s:filter:sieve:nats:zprimes2_0(+(x, 1)) <=> cons(gen_cons:0':s:filter:sieve:nats:zprimes2_0(x)) The following defined symbols remain to be analysed: a__nats, a__sieve, a__zprimes They will be analysed ascendingly in the following order: mark = a__sieve mark = a__nats mark = a__zprimes a__sieve = a__nats a__sieve = a__zprimes a__nats = a__zprimes