/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).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) 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), 287 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), 29 ms] (14) typed CpxTrs (15) RewriteLemmaProof [LOWER BOUND(ID), 22 ms] (16) 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: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) 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: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (4) Obligation: Innermost TRS: Rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) Types: qsort :: nil:cons:ys -> nil:cons:ys 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 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: qsort, append, filterlow, filterhigh, ge They will be analysed ascendingly in the following order: append < qsort filterlow < qsort filterhigh < qsort ge < filterlow ge < filterhigh ---------------------------------------- (6) Obligation: Innermost TRS: Rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) Types: qsort :: nil:cons:ys -> nil:cons:ys 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 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: append, qsort, filterlow, filterhigh, ge They will be analysed ascendingly in the following order: append < qsort filterlow < qsort filterhigh < qsort ge < filterlow ge < filterhigh ---------------------------------------- (7) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: ge(gen_0':s5_0(n17_0), gen_0':s5_0(n17_0)) -> true, rt in Omega(1 + n17_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(+(n17_0, 1)), gen_0':s5_0(+(n17_0, 1))) ->_R^Omega(1) ge(gen_0':s5_0(n17_0), gen_0':s5_0(n17_0)) ->_IH true 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: Innermost TRS: Rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) Types: qsort :: nil:cons:ys -> nil:cons:ys 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 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: ge, qsort, filterlow, filterhigh They will be analysed ascendingly in the following order: filterlow < qsort filterhigh < qsort ge < filterlow ge < filterhigh ---------------------------------------- (10) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (11) BOUNDS(n^1, INF) ---------------------------------------- (12) Obligation: Innermost TRS: Rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) Types: qsort :: nil:cons:ys -> nil:cons:ys 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 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: ge(gen_0':s5_0(n17_0), gen_0':s5_0(n17_0)) -> true, rt in Omega(1 + n17_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, qsort, filterhigh They will be analysed ascendingly in the following order: filterlow < qsort filterhigh < qsort ---------------------------------------- (13) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n297_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n297_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(+(n297_0, 1))) ->_R^Omega(1) if1(ge(gen_0':s5_0(0), 0'), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n297_0)) ->_L^Omega(1) if1(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n297_0)) ->_R^Omega(1) filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n297_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). ---------------------------------------- (14) Obligation: Innermost TRS: Rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) Types: qsort :: nil:cons:ys -> nil:cons:ys 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 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: ge(gen_0':s5_0(n17_0), gen_0':s5_0(n17_0)) -> true, rt in Omega(1 + n17_0) filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n297_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n297_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, qsort They will be analysed ascendingly in the following order: filterhigh < qsort ---------------------------------------- (15) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n874_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n874_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(+(n874_0, 1))) ->_R^Omega(1) if2(ge(0', gen_0':s5_0(0)), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n874_0)) ->_L^Omega(1) if2(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n874_0)) ->_R^Omega(1) filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n874_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). ---------------------------------------- (16) Obligation: Innermost TRS: Rules: qsort(nil) -> nil qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, 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)) Types: qsort :: nil:cons:ys -> nil:cons:ys 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 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: ge(gen_0':s5_0(n17_0), gen_0':s5_0(n17_0)) -> true, rt in Omega(1 + n17_0) filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n297_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n297_0) filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n874_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n874_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: qsort