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

minus(minus(x, y), z) → minus(x, plus(y, z))
minus(0, y) → 0
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
plus(s(x), y) → s(plus(y, x))
zero(s(x)) → false
zero(0) → true
p(s(x)) → x
p(0) → 0
div(x, y) → quot(x, y, 0)
quot(s(x), s(y), z) → quot(minus(p(ack(0, x)), y), s(y), s(z))
quot(0, s(y), z) → z
ack(0, x) → s(x)
ack(0, x) → plus(x, s(0))
ack(s(x), 0) → ack(x, s(0))
ack(s(x), s(y)) → ack(x, ack(s(x), y))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Types:
minus' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
zero' :: 0':s' → false':true'
false' :: false':true'
true' :: false':true'
p' :: 0':s' → 0':s'
div' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
ack' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_false':true'2 :: false':true'
_gen_0':s'3 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
minus', plus', quot', ack'

They will be analysed ascendingly in the following order:
plus' < minus'
minus' < quot'
plus' < ack'
ack' < quot'

Rules:
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Types:
minus' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
zero' :: 0':s' → false':true'
false' :: false':true'
true' :: false':true'
p' :: 0':s' → 0':s'
div' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
ack' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_false':true'2 :: false':true'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
plus', minus', quot', ack'

They will be analysed ascendingly in the following order:
plus' < minus'
minus' < quot'
plus' < ack'
ack' < quot'

Could not prove a rewrite lemma for the defined symbol plus'.

The following conjecture could not be proven:

plus'(_gen_0':s'3(_n5), _gen_0':s'3(b)) →? _gen_0':s'3(+(_n5, b))

Rules:
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Types:
minus' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
zero' :: 0':s' → false':true'
false' :: false':true'
true' :: false':true'
p' :: 0':s' → 0':s'
div' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
ack' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_false':true'2 :: false':true'
_gen_0':s'3 :: Nat → 0':s'

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
minus', quot', ack'

They will be analysed ascendingly in the following order:
minus' < quot'
ack' < quot'

Proved the following rewrite lemma:
minus'(_gen_0':s'3(_n3558), _gen_0':s'3(_n3558)) → _gen_0':s'3(0), rt ∈ Ω(1 + n3558)

Induction Base:
minus'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
0'

Induction Step:
minus'(_gen_0':s'3(+(_\$n3559, 1)), _gen_0':s'3(+(_\$n3559, 1))) →RΩ(1)
minus'(_gen_0':s'3(_\$n3559), _gen_0':s'3(_\$n3559)) →IH
_gen_0':s'3(0)

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

Rules:
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Types:
minus' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
zero' :: 0':s' → false':true'
false' :: false':true'
true' :: false':true'
p' :: 0':s' → 0':s'
div' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
ack' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_false':true'2 :: false':true'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n3558), _gen_0':s'3(_n3558)) → _gen_0':s'3(0), rt ∈ Ω(1 + n3558)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
ack', quot'

They will be analysed ascendingly in the following order:
ack' < quot'

Proved the following rewrite lemma:
ack'(_gen_0':s'3(1), _gen_0':s'3(+(1, _n4617))) → _*4, rt ∈ Ω(n4617)

Induction Base:
ack'(_gen_0':s'3(1), _gen_0':s'3(+(1, 0)))

Induction Step:
ack'(_gen_0':s'3(1), _gen_0':s'3(+(1, +(_\$n4618, 1)))) →RΩ(1)
ack'(_gen_0':s'3(0), ack'(s'(_gen_0':s'3(0)), _gen_0':s'3(+(1, _\$n4618)))) →IH
ack'(_gen_0':s'3(0), _*4)

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

Rules:
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Types:
minus' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
zero' :: 0':s' → false':true'
false' :: false':true'
true' :: false':true'
p' :: 0':s' → 0':s'
div' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
ack' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_false':true'2 :: false':true'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n3558), _gen_0':s'3(_n3558)) → _gen_0':s'3(0), rt ∈ Ω(1 + n3558)
ack'(_gen_0':s'3(1), _gen_0':s'3(+(1, _n4617))) → _*4, rt ∈ Ω(n4617)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

The following defined symbols remain to be analysed:
quot'

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

Rules:
minus'(minus'(x, y), z) → minus'(x, plus'(y, z))
minus'(0', y) → 0'
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
plus'(0', y) → y
plus'(s'(x), y) → plus'(x, s'(y))
plus'(s'(x), y) → s'(plus'(y, x))
zero'(s'(x)) → false'
zero'(0') → true'
p'(s'(x)) → x
p'(0') → 0'
div'(x, y) → quot'(x, y, 0')
quot'(s'(x), s'(y), z) → quot'(minus'(p'(ack'(0', x)), y), s'(y), s'(z))
quot'(0', s'(y), z) → z
ack'(0', x) → s'(x)
ack'(0', x) → plus'(x, s'(0'))
ack'(s'(x), 0') → ack'(x, s'(0'))
ack'(s'(x), s'(y)) → ack'(x, ack'(s'(x), y))

Types:
minus' :: 0':s' → 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
zero' :: 0':s' → false':true'
false' :: false':true'
true' :: false':true'
p' :: 0':s' → 0':s'
div' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
ack' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_false':true'2 :: false':true'
_gen_0':s'3 :: Nat → 0':s'

Lemmas:
minus'(_gen_0':s'3(_n3558), _gen_0':s'3(_n3558)) → _gen_0':s'3(0), rt ∈ Ω(1 + n3558)
ack'(_gen_0':s'3(1), _gen_0':s'3(+(1, _n4617))) → _*4, rt ∈ Ω(n4617)

Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
minus'(_gen_0':s'3(_n3558), _gen_0':s'3(_n3558)) → _gen_0':s'3(0), rt ∈ Ω(1 + n3558)