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', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
partLt(S(y39712_1), Cons(0, xs)) →⁺ Cons(0, partLt(S(y39712_1), xs))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [xs / Cons(0, xs)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

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', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

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

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

They will be analysed ascendingly in the following order:
partLt < quicksort
partGt < quicksort
app < quicksort
< < partLt
> < partGt

(8) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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, partLt, partGt, >, app

They will be analysed ascendingly in the following order:
partLt < quicksort
partGt < quicksort
app < quicksort
< < partLt
> < partGt

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

Proved the following rewrite lemma:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_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(+(n7_0, 1)), gen_S:0'5_0(+(1, +(n7_0, 1)))) →_R^Ω(0)
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) →_IH
True

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', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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:
partLt, quicksort, partGt, >, app

They will be analysed ascendingly in the following order:
partLt < quicksort
partGt < quicksort
app < quicksort
> < partGt

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

Proved the following rewrite lemma:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

Induction Base:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(0)) →_R^Ω(1)
Nil

Induction Step:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(+(n324_0, 1))) →_R^Ω(1)
partLt[Ite][True][Ite](<(0', gen_S:0'5_0(0)), gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n324_0))) →_R^Ω(0)
partLt[Ite][True][Ite](False, gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n324_0))) →_R^Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) →_IH
gen_Cons:Nil4_0(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', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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, partGt, app

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

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

Proved the following rewrite lemma:
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_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(+(n937_0, 1)), gen_S:0'5_0(+(n937_0, 1))) →_R^Ω(0)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) →_IH
False

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', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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:
partGt, quicksort, app

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

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n1266₀)

Induction Base:
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(0)) →_R^Ω(1)
Nil

Induction Step:
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(+(n1266_0, 1))) →_R^Ω(1)
partGt[Ite][True][Ite](>(0', gen_S:0'5_0(0)), gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n1266_0))) →_L^Ω(0)
partGt[Ite][True][Ite](False, gen_S:0'5_0(0), Cons(0', gen_Cons:Nil4_0(n1266_0))) →_R^Ω(0)
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) →_IH
gen_Cons:Nil4_0(0)

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

(19) Complex Obligation (BEST)

(20) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n1266₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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

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

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
app(gen_Cons:Nil4_0(n2032_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n2032_0, b)), rt ∈ Ω(1 + n2032₀)

Induction Base:
app(gen_Cons:Nil4_0(0), gen_Cons:Nil4_0(b)) →_R^Ω(1)
gen_Cons:Nil4_0(b)

Induction Step:
app(gen_Cons:Nil4_0(+(n2032_0, 1)), gen_Cons:Nil4_0(b)) →_R^Ω(1)
Cons(0', app(gen_Cons:Nil4_0(n2032_0), gen_Cons:Nil4_0(b))) →_IH
Cons(0', gen_Cons:Nil4_0(+(b, c2033_0)))

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

(22) Complex Obligation (BEST)

(23) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n1266₀)
app(gen_Cons:Nil4_0(n2032_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n2032_0, b)), rt ∈ Ω(1 + n2032₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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

(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

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

(25) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n1266₀)
app(gen_Cons:Nil4_0(n2032_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n2032_0, b)), rt ∈ Ω(1 + n2032₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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.

(26) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

(27) BOUNDS(n^1, INF)

(28) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n1266₀)
app(gen_Cons:Nil4_0(n2032_0), gen_Cons:Nil4_0(b)) → gen_Cons:Nil4_0(+(n2032_0, b)), rt ∈ Ω(1 + n2032₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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.

(29) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

(30) BOUNDS(n^1, INF)

(31) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)
partGt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n1266_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n1266₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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.

(32) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

(33) BOUNDS(n^1, INF)

(34) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)
>(gen_S:0'5_0(n937_0), gen_S:0'5_0(n937_0)) → False, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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.

(35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

(36) BOUNDS(n^1, INF)

(37) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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.

(38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
partLt(gen_S:0'5_0(0), gen_Cons:Nil4_0(n324_0)) → gen_Cons:Nil4_0(0), rt ∈ Ω(1 + n324₀)

(39) BOUNDS(n^1, INF)

(40) Obligation:

Innermost TRS:
Rules:
quicksort(Cons(x, Cons(x', xs))) → part(x, Cons(x', xs))
quicksort(Cons(x, Nil)) → Cons(x, Nil)
quicksort(Nil) → Nil
partLt(x', Cons(x, xs)) → partLt[Ite][True][Ite](<(x, x'), x', Cons(x, xs))
partLt(x, Nil) → Nil
partGt(x', Cons(x, xs)) → partGt[Ite][True][Ite](>(x, x'), x', Cons(x, xs))
partGt(x, Nil) → Nil
app(Cons(x, xs), ys) → Cons(x, app(xs, ys))
app(Nil, ys) → ys
notEmpty(Cons(x, xs)) → True
notEmpty(Nil) → False
part(x, xs) → app(quicksort(partLt(x, xs)), Cons(x, quicksort(partGt(x, xs))))
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
partLt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partLt(x', xs))
partGt[Ite][True][Ite](True, x', Cons(x, xs)) → Cons(x, partGt(x', xs))
partLt[Ite][True][Ite](False, x', Cons(x, xs)) → partLt(x', xs)
partGt[Ite][True][Ite](False, x', Cons(x, xs)) → partGt(x', xs)

Types:
quicksort :: Cons:Nil → Cons:Nil
Cons :: S:0' → Cons:Nil → Cons:Nil
part :: S:0' → Cons:Nil → Cons:Nil
Nil :: Cons:Nil
partLt :: S:0' → Cons:Nil → Cons:Nil
partLt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
< :: S:0' → S:0' → True:False
partGt :: S:0' → Cons:Nil → Cons:Nil
partGt[Ite][True][Ite] :: True:False → S:0' → Cons:Nil → Cons:Nil
> :: S:0' → S:0' → True:False
app :: Cons:Nil → Cons:Nil → Cons:Nil
notEmpty :: Cons:Nil → True:False
True :: True:False
False :: True:False
goal :: Cons:Nil → Cons:Nil
S :: S:0' → S:0'
0' :: S:0'
hole_Cons:Nil1_0 :: Cons:Nil
hole_S:0'2_0 :: S:0'
hole_True:False3_0 :: True:False
gen_Cons:Nil4_0 :: Nat → Cons:Nil
gen_S:0'5_0 :: Nat → S:0'

Lemmas:
<(gen_S:0'5_0(n7_0), gen_S:0'5_0(+(1, n7_0))) → True, rt ∈ Ω(0)

Generator Equations:
gen_Cons:Nil4_0(0) ⇔ Nil
gen_Cons:Nil4_0(+(x, 1)) ⇔ Cons(0', gen_Cons:Nil4_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.

(41) 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(+(1, n7_0))) → True, rt ∈ Ω(0)

(42) BOUNDS(1, INF)