/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 250 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), 75 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 45 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
        (18) 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(nil) -> nil
   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)))))
   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))
   last(nil) -> 0
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(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(nil) -> nil
   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)))))
   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))
   last(nil) -> 0'
   last(cons(x, nil)) -> x
   last(cons(x, cons(y, xs))) -> last(cons(y, xs))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
qsort(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
qsort, append, filterlow, last, filterhigh, ge

They will be analysed ascendingly in the following order:
append < qsort
filterlow < qsort
last < qsort
filterhigh < qsort
ge < filterlow
ge < filterhigh

----------------------------------------

(6)
Obligation:
TRS:
Rules:
qsort(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
append, qsort, filterlow, last, filterhigh, ge

They will be analysed ascendingly in the following order:
append < qsort
filterlow < qsort
last < qsort
filterhigh < qsort
ge < filterlow
ge < filterhigh

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0)

Induction Base:
last(gen_nil:cons:ys4_0(+(1, 0))) ->_R^Omega(1)
0'

Induction Step:
last(gen_nil:cons:ys4_0(+(1, +(n17_0, 1)))) ->_R^Omega(1)
last(cons(0', gen_nil:cons:ys4_0(n17_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).
----------------------------------------

(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
qsort(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
last, qsort, filterlow, filterhigh, ge

They will be analysed ascendingly in the following order:
filterlow < qsort
last < qsort
filterhigh < qsort
ge < filterlow
ge < filterhigh

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
TRS:
Rules:
qsort(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), 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:
ge, qsort, filterlow, filterhigh

They will be analysed ascendingly in the following order:
filterlow < qsort
filterhigh < qsort
ge < filterlow
ge < filterhigh

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_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(+(n363_0, 1)), gen_0':s5_0(+(n363_0, 1))) ->_R^Omega(1)
ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) ->_IH
true

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(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0)
ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_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

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
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)

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(+(n667_0, 1))) ->_R^Omega(1)
if1(ge(gen_0':s5_0(0), 0'), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n667_0)) ->_L^Omega(1)
if1(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n667_0)) ->_R^Omega(1)
filterlow(gen_0':s5_0(0), gen_nil:cons:ys4_0(n667_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:
TRS:
Rules:
qsort(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0)
ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_0)
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)


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

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
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)

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(+(n1204_0, 1))) ->_R^Omega(1)
if2(ge(0', gen_0':s5_0(0)), gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n1204_0)) ->_L^Omega(1)
if2(true, gen_0':s5_0(0), 0', gen_nil:cons:ys4_0(n1204_0)) ->_R^Omega(1)
filterhigh(gen_0':s5_0(0), gen_nil:cons:ys4_0(n1204_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).
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(18)
Obligation:
TRS:
Rules:
qsort(nil) -> nil
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)))))
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))
last(nil) -> 0'
last(cons(x, nil)) -> x
last(cons(x, cons(y, xs))) -> last(cons(y, xs))

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
last :: 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:
last(gen_nil:cons:ys4_0(+(1, n17_0))) -> gen_0':s5_0(0), rt in Omega(1 + n17_0)
ge(gen_0':s5_0(n363_0), gen_0':s5_0(n363_0)) -> true, rt in Omega(1 + n363_0)
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)
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)


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