/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), ?) proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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), 292 ms] (8) BEST (9) proven lower bound (10) LowerBoundPropagationProof [FINISHED, 0 ms] (11) BOUNDS(n^1, INF) (12) typed CpxTrs (13) RewriteLemmaProof [LOWER BOUND(ID), 34 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 75 ms] (16) typed CpxTrs (17) RewriteLemmaProof [LOWER BOUND(ID), 30 ms] (18) typed CpxTrs (19) RewriteLemmaProof [LOWER BOUND(ID), 49 ms] (20) typed CpxTrs (21) RewriteLemmaProof [LOWER BOUND(ID), 21 ms] (22) 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: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0 length(cons(x, xs)) -> s(length(xs)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) get(n, nil) -> 0 get(n, cons(x, nil)) -> x get(0, cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) 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: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) S is empty. Rewrite Strategy: FULL ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: TRS: Rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s ---------------------------------------- (5) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: qs, half, length, append, filterlow, get, filterhigh, ge They will be analysed ascendingly in the following order: half < qs append < qs filterlow < qs get < qs filterhigh < qs ge < filterlow ge < filterhigh ---------------------------------------- (6) Obligation: TRS: Rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: half, qs, length, append, filterlow, get, filterhigh, ge They will be analysed ascendingly in the following order: half < qs append < qs filterlow < qs get < qs filterhigh < qs ge < filterlow ge < filterhigh ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) Induction Base: half(gen_0':s5_0(*(2, 0))) ->_R^Omega(1) 0' Induction Step: half(gen_0':s5_0(*(2, +(n7_0, 1)))) ->_R^Omega(1) s(half(gen_0':s5_0(*(2, n7_0)))) ->_IH s(gen_0':s5_0(c8_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: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: half, qs, length, append, filterlow, get, filterhigh, ge They will be analysed ascendingly in the following order: half < qs append < qs filterlow < qs get < qs filterhigh < qs ge < filterlow ge < filterhigh ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: TRS: Rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Lemmas: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: length, qs, append, filterlow, get, filterhigh, ge They will be analysed ascendingly in the following order: append < qs filterlow < qs get < qs filterhigh < qs ge < filterlow ge < filterhigh ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: length(gen_nil:cons:ys4_0(n419_0)) -> gen_0':s5_0(n419_0), rt in Omega(1 + n419_0) Induction Base: length(gen_nil:cons:ys4_0(0)) ->_R^Omega(1) 0' Induction Step: length(gen_nil:cons:ys4_0(+(n419_0, 1))) ->_R^Omega(1) s(length(gen_nil:cons:ys4_0(n419_0))) ->_IH s(gen_0':s5_0(c420_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: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Lemmas: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) length(gen_nil:cons:ys4_0(n419_0)) -> gen_0':s5_0(n419_0), rt in Omega(1 + n419_0) Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: append, qs, filterlow, get, filterhigh, ge They will be analysed ascendingly in the following order: append < qs filterlow < qs get < qs filterhigh < qs ge < filterlow ge < filterhigh ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: get(gen_0':s5_0(n739_0), gen_nil:cons:ys4_0(+(1, n739_0))) -> gen_0':s5_0(0), rt in Omega(1 + n739_0) Induction Base: get(gen_0':s5_0(0), gen_nil:cons:ys4_0(+(1, 0))) ->_R^Omega(1) 0' Induction Step: get(gen_0':s5_0(+(n739_0, 1)), gen_nil:cons:ys4_0(+(1, +(n739_0, 1)))) ->_R^Omega(1) get(gen_0':s5_0(n739_0), cons(0', gen_nil:cons:ys4_0(n739_0))) ->_IH gen_0':s5_0(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: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Lemmas: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) length(gen_nil:cons:ys4_0(n419_0)) -> gen_0':s5_0(n419_0), rt in Omega(1 + n419_0) get(gen_0':s5_0(n739_0), gen_nil:cons:ys4_0(+(1, n739_0))) -> gen_0':s5_0(0), rt in Omega(1 + n739_0) Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: ge, qs, filterlow, filterhigh They will be analysed ascendingly in the following order: filterlow < qs filterhigh < qs ge < filterlow ge < filterhigh ---------------------------------------- (17) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: ge(gen_0':s5_0(n1773_0), gen_0':s5_0(n1773_0)) -> true, rt in Omega(1 + n1773_0) Induction Base: ge(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1) true Induction Step: ge(gen_0':s5_0(+(n1773_0, 1)), gen_0':s5_0(+(n1773_0, 1))) ->_R^Omega(1) ge(gen_0':s5_0(n1773_0), gen_0':s5_0(n1773_0)) ->_IH true We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (18) Obligation: TRS: Rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Lemmas: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) length(gen_nil:cons:ys4_0(n419_0)) -> gen_0':s5_0(n419_0), rt in Omega(1 + n419_0) get(gen_0':s5_0(n739_0), gen_nil:cons:ys4_0(+(1, n739_0))) -> gen_0':s5_0(0), rt in Omega(1 + n739_0) ge(gen_0':s5_0(n1773_0), gen_0':s5_0(n1773_0)) -> true, rt in Omega(1 + n1773_0) Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: filterlow, qs, filterhigh They will be analysed ascendingly in the following order: filterlow < qs filterhigh < qs ---------------------------------------- (19) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2131_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n2131_0) Induction Base: filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(0)) ->_R^Omega(1) nil Induction Step: filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(+(n2131_0, 1))) ->_R^Omega(1) if1(ge(gen_0':s5_0(0), 0'), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n2131_0)) ->_L^Omega(1) if1(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n2131_0)) ->_R^Omega(1) filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2131_0)) ->_IH gen_nil:cons:ys4_0(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (20) Obligation: TRS: Rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Lemmas: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) length(gen_nil:cons:ys4_0(n419_0)) -> gen_0':s5_0(n419_0), rt in Omega(1 + n419_0) get(gen_0':s5_0(n739_0), gen_nil:cons:ys4_0(+(1, n739_0))) -> gen_0':s5_0(0), rt in Omega(1 + n739_0) ge(gen_0':s5_0(n1773_0), gen_0':s5_0(n1773_0)) -> true, rt in Omega(1 + n1773_0) filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2131_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n2131_0) Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: filterhigh, qs They will be analysed ascendingly in the following order: filterhigh < qs ---------------------------------------- (21) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2821_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n2821_0) Induction Base: filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(0)) ->_R^Omega(1) nil Induction Step: filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(+(n2821_0, 1))) ->_R^Omega(1) if2(ge(0', gen_0':s5_0(0)), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n2821_0)) ->_L^Omega(1) if2(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n2821_0)) ->_R^Omega(1) filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2821_0)) ->_IH gen_nil:cons:ys4_0(0) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (22) Obligation: TRS: Rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) Types: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0':s -> nil:cons:ys -> nil:cons:ys half :: 0':s -> 0':s length :: nil:cons:ys -> 0':s nil :: nil:cons:ys cons :: 0':s -> nil:cons:ys -> nil:cons:ys append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys get :: 0':s -> nil:cons:ys -> 0':s filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys ge :: 0':s -> 0':s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 0' :: 0':s s :: 0':s -> 0':s ys :: nil:cons:ys hole_nil:cons:ys1_0 :: nil:cons:ys hole_0':s2_0 :: 0':s hole_true:false3_0 :: true:false gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys gen_0':s5_0 :: Nat -> 0':s Lemmas: half(gen_0':s5_0(*(2, n7_0))) -> gen_0':s5_0(n7_0), rt in Omega(1 + n7_0) length(gen_nil:cons:ys4_0(n419_0)) -> gen_0':s5_0(n419_0), rt in Omega(1 + n419_0) get(gen_0':s5_0(n739_0), gen_nil:cons:ys4_0(+(1, n739_0))) -> gen_0':s5_0(0), rt in Omega(1 + n739_0) ge(gen_0':s5_0(n1773_0), gen_0':s5_0(n1773_0)) -> true, rt in Omega(1 + n1773_0) filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2131_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n2131_0) filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n2821_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n2821_0) Generator Equations: gen_nil:cons:ys4_0(0) <=> nil gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) gen_0':s5_0(0) <=> 0' gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) The following defined symbols remain to be analysed: qs