311.48/291.49 WORST_CASE(Omega(n^1), ?) 311.48/291.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.xml 311.48/291.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 311.48/291.49 311.48/291.49 311.48/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.48/291.49 311.48/291.49 (0) CpxTRS 311.48/291.49 (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] 311.48/291.49 (2) CpxTRS 311.48/291.49 (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] 311.48/291.49 (4) typed CpxTrs 311.48/291.49 (5) OrderProof [LOWER BOUND(ID), 0 ms] 311.48/291.49 (6) typed CpxTrs 311.48/291.49 (7) RewriteLemmaProof [LOWER BOUND(ID), 217 ms] 311.48/291.49 (8) BEST 311.48/291.49 (9) proven lower bound 311.48/291.49 (10) LowerBoundPropagationProof [FINISHED, 0 ms] 311.48/291.49 (11) BOUNDS(n^1, INF) 311.48/291.49 (12) typed CpxTrs 311.48/291.49 (13) RewriteLemmaProof [LOWER BOUND(ID), 86 ms] 311.48/291.49 (14) typed CpxTrs 311.48/291.49 (15) RewriteLemmaProof [LOWER BOUND(ID), 47 ms] 311.48/291.49 (16) typed CpxTrs 311.48/291.49 (17) RewriteLemmaProof [LOWER BOUND(ID), 0 ms] 311.48/291.49 (18) typed CpxTrs 311.48/291.49 311.48/291.49 311.48/291.49 ---------------------------------------- 311.48/291.49 311.48/291.49 (0) 311.48/291.49 Obligation: 311.48/291.49 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.48/291.49 311.48/291.49 311.48/291.49 The TRS R consists of the following rules: 311.48/291.49 311.48/291.49 qsort(nil) -> nil 311.48/291.49 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.49 filterlow(n, nil) -> nil 311.48/291.49 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.49 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.49 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.49 filterhigh(n, nil) -> nil 311.48/291.49 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.49 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.49 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.49 ge(x, 0) -> true 311.48/291.49 ge(0, s(x)) -> false 311.48/291.49 ge(s(x), s(y)) -> ge(x, y) 311.48/291.49 append(nil, ys) -> ys 311.48/291.49 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.49 last(nil) -> 0 311.48/291.49 last(cons(x, nil)) -> x 311.48/291.49 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.49 311.48/291.49 S is empty. 311.48/291.49 Rewrite Strategy: FULL 311.48/291.49 ---------------------------------------- 311.48/291.49 311.48/291.49 (1) RenamingProof (BOTH BOUNDS(ID, ID)) 311.48/291.49 Renamed function symbols to avoid clashes with predefined symbol. 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (2) 311.48/291.50 Obligation: 311.48/291.50 The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF). 311.48/291.50 311.48/291.50 311.48/291.50 The TRS R consists of the following rules: 311.48/291.50 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 S is empty. 311.48/291.50 Rewrite Strategy: FULL 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (3) TypeInferenceProof (BOTH BOUNDS(ID, ID)) 311.48/291.50 Infered types. 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (4) 311.48/291.50 Obligation: 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (5) OrderProof (LOWER BOUND(ID)) 311.48/291.50 Heuristically decided to analyse the following defined symbols: 311.48/291.50 qsort, append, filterlow, last, filterhigh, ge 311.48/291.50 311.48/291.50 They will be analysed ascendingly in the following order: 311.48/291.50 append < qsort 311.48/291.50 filterlow < qsort 311.48/291.50 last < qsort 311.48/291.50 filterhigh < qsort 311.48/291.50 ge < filterlow 311.48/291.50 ge < filterhigh 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (6) 311.48/291.50 Obligation: 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 311.48/291.50 Generator Equations: 311.48/291.50 gen_nil:cons:ys4_0(0) <=> nil 311.48/291.50 gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) 311.48/291.50 gen_0':s5_0(0) <=> 0' 311.48/291.50 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 311.48/291.50 311.48/291.50 311.48/291.50 The following defined symbols remain to be analysed: 311.48/291.50 append, qsort, filterlow, last, filterhigh, ge 311.48/291.50 311.48/291.50 They will be analysed ascendingly in the following order: 311.48/291.50 append < qsort 311.48/291.50 filterlow < qsort 311.48/291.50 last < qsort 311.48/291.50 filterhigh < qsort 311.48/291.50 ge < filterlow 311.48/291.50 ge < filterhigh 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (7) RewriteLemmaProof (LOWER BOUND(ID)) 311.48/291.50 Proved the following rewrite lemma: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0) 311.48/291.50 311.48/291.50 Induction Base: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, 0))) ->_R^Omega(1) 311.48/291.50 0' 311.48/291.50 311.48/291.50 Induction Step: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, +(n17_0, 1)))) ->_R^Omega(1) 311.48/291.50 last(cons(0', gen_nil:cons:ys4_0(n17_0))) ->_IH 311.48/291.50 gen_0':s5_0(0) 311.48/291.50 311.48/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (8) 311.48/291.50 Complex Obligation (BEST) 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (9) 311.48/291.50 Obligation: 311.48/291.50 Proved the lower bound n^1 for the following obligation: 311.48/291.50 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 311.48/291.50 Generator Equations: 311.48/291.50 gen_nil:cons:ys4_0(0) <=> nil 311.48/291.50 gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) 311.48/291.50 gen_0':s5_0(0) <=> 0' 311.48/291.50 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 311.48/291.50 311.48/291.50 311.48/291.50 The following defined symbols remain to be analysed: 311.48/291.50 last, qsort, filterlow, filterhigh, ge 311.48/291.50 311.48/291.50 They will be analysed ascendingly in the following order: 311.48/291.50 filterlow < qsort 311.48/291.50 last < qsort 311.48/291.50 filterhigh < qsort 311.48/291.50 ge < filterlow 311.48/291.50 ge < filterhigh 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (10) LowerBoundPropagationProof (FINISHED) 311.48/291.50 Propagated lower bound. 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (11) 311.48/291.50 BOUNDS(n^1, INF) 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (12) 311.48/291.50 Obligation: 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 311.48/291.50 Lemmas: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0) 311.48/291.50 311.48/291.50 311.48/291.50 Generator Equations: 311.48/291.50 gen_nil:cons:ys4_0(0) <=> nil 311.48/291.50 gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) 311.48/291.50 gen_0':s5_0(0) <=> 0' 311.48/291.50 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 311.48/291.50 311.48/291.50 311.48/291.50 The following defined symbols remain to be analysed: 311.48/291.50 ge, qsort, filterlow, filterhigh 311.48/291.50 311.48/291.50 They will be analysed ascendingly in the following order: 311.48/291.50 filterlow < qsort 311.48/291.50 filterhigh < qsort 311.48/291.50 ge < filterlow 311.48/291.50 ge < filterhigh 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (13) RewriteLemmaProof (LOWER BOUND(ID)) 311.48/291.50 Proved the following rewrite lemma: 311.48/291.50 ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_0) 311.48/291.50 311.48/291.50 Induction Base: 311.48/291.50 ge(gen_0':s5_0(0), gen_0':s5_0(0)) ->_R^Omega(1) 311.48/291.50 true 311.48/291.50 311.48/291.50 Induction Step: 311.48/291.50 ge(gen_0':s5_0(+(n363_0, 1)), gen_0':s5_0(+(n363_0, 1))) ->_R^Omega(1) 311.48/291.50 ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) ->_IH 311.48/291.50 true 311.48/291.50 311.48/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (14) 311.48/291.50 Obligation: 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 311.48/291.50 Lemmas: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0) 311.48/291.50 ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_0) 311.48/291.50 311.48/291.50 311.48/291.50 Generator Equations: 311.48/291.50 gen_nil:cons:ys4_0(0) <=> nil 311.48/291.50 gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) 311.48/291.50 gen_0':s5_0(0) <=> 0' 311.48/291.50 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 311.48/291.50 311.48/291.50 311.48/291.50 The following defined symbols remain to be analysed: 311.48/291.50 filterlow, qsort, filterhigh 311.48/291.50 311.48/291.50 They will be analysed ascendingly in the following order: 311.48/291.50 filterlow < qsort 311.48/291.50 filterhigh < qsort 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (15) RewriteLemmaProof (LOWER BOUND(ID)) 311.48/291.50 Proved the following rewrite lemma: 311.48/291.50 filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n667_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n667_0) 311.48/291.50 311.48/291.50 Induction Base: 311.48/291.50 filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(0)) ->_R^Omega(1) 311.48/291.50 nil 311.48/291.50 311.48/291.50 Induction Step: 311.48/291.50 filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(+(n667_0, 1))) ->_R^Omega(1) 311.48/291.50 if1(ge(gen_0':s5_0(0), 0'), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n667_0)) ->_L^Omega(1) 311.48/291.50 if1(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n667_0)) ->_R^Omega(1) 311.48/291.50 filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n667_0)) ->_IH 311.48/291.50 gen_nil:cons:ys4_0(0) 311.48/291.50 311.48/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (16) 311.48/291.50 Obligation: 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 311.48/291.50 Lemmas: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0) 311.48/291.50 ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_0) 311.48/291.50 filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n667_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n667_0) 311.48/291.50 311.48/291.50 311.48/291.50 Generator Equations: 311.48/291.50 gen_nil:cons:ys4_0(0) <=> nil 311.48/291.50 gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) 311.48/291.50 gen_0':s5_0(0) <=> 0' 311.48/291.50 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 311.48/291.50 311.48/291.50 311.48/291.50 The following defined symbols remain to be analysed: 311.48/291.50 filterhigh, qsort 311.48/291.50 311.48/291.50 They will be analysed ascendingly in the following order: 311.48/291.50 filterhigh < qsort 311.48/291.50 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (17) RewriteLemmaProof (LOWER BOUND(ID)) 311.48/291.50 Proved the following rewrite lemma: 311.48/291.50 filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n1204_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n1204_0) 311.48/291.50 311.48/291.50 Induction Base: 311.48/291.50 filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(0)) ->_R^Omega(1) 311.48/291.50 nil 311.48/291.50 311.48/291.50 Induction Step: 311.48/291.50 filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(+(n1204_0, 1))) ->_R^Omega(1) 311.48/291.50 if2(ge(0', gen_0':s5_0(0)), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n1204_0)) ->_L^Omega(1) 311.48/291.50 if2(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n1204_0)) ->_R^Omega(1) 311.48/291.50 filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n1204_0)) ->_IH 311.48/291.50 gen_nil:cons:ys4_0(0) 311.48/291.50 311.48/291.50 We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). 311.48/291.50 ---------------------------------------- 311.48/291.50 311.48/291.50 (18) 311.48/291.50 Obligation: 311.48/291.50 TRS: 311.48/291.50 Rules: 311.48/291.50 qsort(nil) -> nil 311.48/291.50 qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs))))) 311.48/291.50 filterlow(n, nil) -> nil 311.48/291.50 filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) 311.48/291.50 if1(true, n, x, xs) -> filterlow(n, xs) 311.48/291.50 if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) 311.48/291.50 filterhigh(n, nil) -> nil 311.48/291.50 filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) 311.48/291.50 if2(true, n, x, xs) -> filterhigh(n, xs) 311.48/291.50 if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) 311.48/291.50 ge(x, 0') -> true 311.48/291.50 ge(0', s(x)) -> false 311.48/291.50 ge(s(x), s(y)) -> ge(x, y) 311.48/291.50 append(nil, ys) -> ys 311.48/291.50 append(cons(x, xs), ys) -> cons(x, append(xs, ys)) 311.48/291.50 last(nil) -> 0' 311.48/291.50 last(cons(x, nil)) -> x 311.48/291.50 last(cons(x, cons(y, xs))) -> last(cons(y, xs)) 311.48/291.50 311.48/291.50 Types: 311.48/291.50 qsort :: nil:cons:ys -> nil:cons:ys 311.48/291.50 nil :: nil:cons:ys 311.48/291.50 cons :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 append :: nil:cons:ys -> nil:cons:ys -> nil:cons:ys 311.48/291.50 filterlow :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 last :: nil:cons:ys -> 0':s 311.48/291.50 filterhigh :: 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 if1 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 ge :: 0':s -> 0':s -> true:false 311.48/291.50 true :: true:false 311.48/291.50 false :: true:false 311.48/291.50 if2 :: true:false -> 0':s -> 0':s -> nil:cons:ys -> nil:cons:ys 311.48/291.50 0' :: 0':s 311.48/291.50 s :: 0':s -> 0':s 311.48/291.50 ys :: nil:cons:ys 311.48/291.50 hole_nil:cons:ys1_0 :: nil:cons:ys 311.48/291.50 hole_0':s2_0 :: 0':s 311.48/291.50 hole_true:false3_0 :: true:false 311.48/291.50 gen_nil:cons:ys4_0 :: Nat -> nil:cons:ys 311.48/291.50 gen_0':s5_0 :: Nat -> 0':s 311.48/291.50 311.48/291.50 311.48/291.50 Lemmas: 311.48/291.50 last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0) 311.48/291.50 ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_0) 311.48/291.50 filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n667_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n667_0) 311.48/291.50 filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n1204_0)) -> gen_nil:cons:ys4_0(0), rt in Omega(1 + n1204_0) 311.48/291.50 311.48/291.50 311.48/291.50 Generator Equations: 311.48/291.50 gen_nil:cons:ys4_0(0) <=> nil 311.48/291.50 gen_nil:cons:ys4_0(+(x, 1)) <=> cons(0', gen_nil:cons:ys4_0(x)) 311.48/291.50 gen_0':s5_0(0) <=> 0' 311.48/291.50 gen_0':s5_0(+(x, 1)) <=> s(gen_0':s5_0(x)) 311.48/291.50 311.48/291.50 311.48/291.50 The following defined symbols remain to be analysed: 311.48/291.50 qsort 311.57/291.53 EOF