The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y)
<(0, S(y)) → True
<(x, 0) → False
>(S(x), S(y)) → >(x, y)
>(0, y) → False
>(S(x), 0) → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2)

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x), x', Cons(x, xs), xs1, xs2)

Rewrite Strategy: INNERMOST

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
part[Ite][True][Ite][False][Ite]/1
part[Ite][True][Ite][False][Ite]/2
part[Ite][True][Ite][False][Ite]/3
part[Ite][True][Ite][False][Ite]/4

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)

The (relative) TRS S consists of the following rules:

<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Rewrite Strategy: INNERMOST

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
quicksort, part, >, app, <

They will be analysed ascendingly in the following order:
quicksort = part
> < part
app < part
< < part

(8) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
>, quicksort, part, app, <

They will be analysed ascendingly in the following order:
quicksort = part
> < part
app < part
< < part

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)

Induction Base:
>(gen_S:0'5_0(0), gen_S:0'5_0(0)) →_R^Ω(0)
False

Induction Step:
>(gen_S:0'5_0(+(n7_0, 1)), gen_S:0'5_0(+(n7_0, 1))) →_R^Ω(0)
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) →_IH
False

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(10) Complex Obligation (BEST)

(11) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
app, quicksort, part, <

They will be analysed ascendingly in the following order:
quicksort = part
app < part
< < part

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)

Induction Base:
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) →_R^Ω(1)
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)

Induction Step:
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, 1)), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) →_R^Ω(1)
Cons(0', app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b))) →_IH
Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(b, c295_0)))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
<, quicksort, part

They will be analysed ascendingly in the following order:
quicksort = part
< < part

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
<(gen_S:0'5_0(n1165_0), gen_S:0'5_0(+(1, n1165_0))) → True, rt ∈ Ω(0)

Induction Base:
<(gen_S:0'5_0(0), gen_S:0'5_0(+(1, 0))) →_R^Ω(0)
True

Induction Step:
<(gen_S:0'5_0(+(n1165_0, 1)), gen_S:0'5_0(+(1, +(n1165_0, 1)))) →_R^Ω(0)
<(gen_S:0'5_0(n1165_0), gen_S:0'5_0(+(1, n1165_0))) →_IH
True

We have rt ∈ Ω(1) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n⁰).

(16) Complex Obligation (BEST)

(17) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)
<(gen_S:0'5_0(n1165_0), gen_S:0'5_0(+(1, n1165_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
part, quicksort

They will be analysed ascendingly in the following order:
quicksort = part

(18) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol part.

(19) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)
<(gen_S:0'5_0(n1165_0), gen_S:0'5_0(+(1, n1165_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

The following defined symbols remain to be analysed:
quicksort

They will be analysed ascendingly in the following order:
quicksort = part

(20) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol quicksort.

(21) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)
<(gen_S:0'5_0(n1165_0), gen_S:0'5_0(+(1, n1165_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(22) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)

(23) BOUNDS(n^1, INF)

(24) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)
<(gen_S:0'5_0(n1165_0), gen_S:0'5_0(+(1, n1165_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(25) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)

(26) BOUNDS(n^1, INF)

(27) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(28) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
app(gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(n294_0), gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(b)) → gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(n294_0, b)), rt ∈ Ω(1 + n294₀)

(29) BOUNDS(n^1, INF)

(30) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x, Cons(x', xs)), Cons(x, Nil), Nil)
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
part(x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite](>(x', x), x', Cons(x, xs), xs1, xs2)
part(x, Nil, xs1, xs2) → app(quicksort(xs1), quicksort(xs2))
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
goal(xs) → quicksort(xs)
<(S(x), S(y)) → <(x, y)
<(0', S(y)) → True
<(x, 0') → False
>(S(x), S(y)) → >(x, y)
>(0', y) → False
>(S(x), 0') → True
part[Ite][True][Ite](True, x', Cons(x, xs), xs1, xs2) → part(x', xs, Cons(x, xs1), xs2)
part[Ite][True][Ite](False, x', Cons(x, xs), xs1, xs2) → part[Ite][True][Ite][False][Ite](<(x', x))

Types:
quicksort :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Cons :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
part :: S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
Nil :: Cons:Nil:part[Ite][True][Ite][False][Ite]
part[Ite][True][Ite] :: True:False → S:0' → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
> :: S:0' → S:0' → True:False
app :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
notEmpty :: Cons:Nil:part[Ite][True][Ite][False][Ite] → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil:part[Ite][True][Ite][False][Ite] → Cons:Nil:part[Ite][True][Ite][False][Ite]
< :: S:0' → S:0' → True:False
S :: S:0' → S:0'
0' :: S:0'
part[Ite][True][Ite][False][Ite] :: True:False → Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_Cons:Nil:part[Ite][True][Ite][False][Ite]1_0 :: Cons:Nil:part[Ite][True][Ite][False][Ite]
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0 :: Nat → Cons:Nil:part[Ite][True][Ite][False][Ite]
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(0) ⇔ Nil
gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil:part[Ite][True][Ite][False][Ite]4_0(x))
gen_S:0'5_0(0) ⇔ 0'
gen_S:0'5_0(+(x, 1)) ⇔ S(gen_S:0'5_0(x))

No more defined symbols left to analyse.

(31) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(1) was proven with the following lemma:
>(gen_S:0'5_0(n7_0), gen_S:0'5_0(n7_0)) → False, rt ∈ Ω(0)

(32) BOUNDS(1, INF)