### (0) Obligation:

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

minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
quot(0, s(y)) → 0
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
minus(s(x), s(y)) →+ minus(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:

minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

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

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

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

### (8) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
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:
minus, quot, le, app, low, high, quicksort

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

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

Proved the following rewrite lemma:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)

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

Induction Step:
minus(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) →RΩ(1)
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) →IH
gen_0':s4_0(0)

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

### (11) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), 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:
quot, le, app, low, high, quicksort

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

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

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

### (13) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), 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:
le, app, low, high, quicksort

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

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

Proved the following rewrite lemma:
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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

Induction Step:
le(gen_0':s4_0(+(n521_0, 1)), gen_0':s4_0(+(n521_0, 1))) →RΩ(1)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) →IH
true

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

### (16) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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

### (17) 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).

### (19) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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

### (20) 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).

### (22) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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

### (23) 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).

### (25) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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

### (26) 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).

### (28) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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.

### (29) LowerBoundsProof (EQUIVALENT transformation)

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

### (31) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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.

### (32) LowerBoundsProof (EQUIVALENT transformation)

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

### (34) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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.

### (35) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)

### (37) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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.

### (38) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)

### (40) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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.

### (41) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)

### (43) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)
le(gen_0':s4_0(n521_0), gen_0':s4_0(n521_0)) → true, rt ∈ Ω(1 + n5210)

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.

### (44) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n1) was proven with the following lemma:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)

### (46) Obligation:

TRS:
Rules:
minus(x, 0') → x
minus(s(x), s(y)) → minus(x, y)
quot(0', s(y)) → 0'
quot(s(x), s(y)) → s(quot(minus(x, y), s(y)))
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)
quicksort(nil) → nil

Types:
minus :: 0':s → 0':s → 0':s
0' :: 0':s
s :: 0':s → 0':s
quot :: 0':s → 0':s → 0':s
le :: 0':s → 0':s → true:false
true :: true:false
false :: true:false
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s4_0 :: Nat → 0':s

Lemmas:
minus(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) → gen_0':s4_0(0), rt ∈ Ω(1 + n70)

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