### (0) Obligation:

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

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
le(s(x), s(y)) →+ le(x, y)
gives rise to a decreasing loop by considering the right hand sides subterm at position [].
The pumping substitution is [x / s(x), y / s(y)].
The result substitution is [ ].

### (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:

le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

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

Heuristically decided to analyse the following defined symbols:
le, app, low, high, quicksort

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

### (8) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
le, app, low, high, quicksort

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

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

Proved the following rewrite lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Induction Base:
le(gen_0':s4_0(0), gen_0':s4_0(0)) →RΩ(1)
true

Induction Step:
le(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(1)
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →IH
true

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

### (11) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
app, low, high, quicksort

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

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

Proved the following rewrite lemma:

Induction Base:

Induction Step:

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

### (14) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
low, high, quicksort

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

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

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

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

### (17) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

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

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

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

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

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

### (20) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

The following defined symbols remain to be analysed:
quicksort

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

Proved the following rewrite lemma:

Induction Base:
nil

Induction Step:

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (23) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (24) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (26) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (27) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (29) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (32) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (35) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))

No more defined symbols left to analyse.

### (36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

### (38) Obligation:

TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
app(nil, y) → y
low(n, nil) → nil
if_low(false, n, add(m, x)) → low(n, x)
high(n, nil) → nil
if_high(true, n, add(m, x)) → high(n, x)
isempty(nil) → true
if_qs(true, x, n, y) → nil
if_qs(false, x, n, y) → app(quicksort(x), add(n, quicksort(y)))

Types:
le :: 0':s → 0':s → true:false
0' :: 0':s
true :: true:false
s :: 0':s → 0':s
false :: true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
gen_0':s4_0 :: Nat → 0':s

Lemmas:
le(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → true, rt ∈ Ω(1 + n70)

Generator Equations:
gen_0':s4_0(0) ⇔ 0'
gen_0':s4_0(+(x, 1)) ⇔ s(gen_0':s4_0(x))