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

plus(0, x) → x
plus(s(x), y) → s(plus(p(s(x)), y))
times(0, y) → 0
times(s(x), y) → plus(y, times(p(s(x)), y))
p(s(0)) → 0
p(s(s(x))) → s(p(s(x)))
fac(0, x) → x
fac(s(x), y) → fac(p(s(x)), times(s(x), y))
factorial(x) → fac(x, s(0))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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


plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
p' :: 0':s' → 0':s'
times' :: 0':s' → 0':s' → 0':s'
fac' :: 0':s' → 0':s' → 0':s'
factorial' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Heuristically decided to analyse the following defined symbols:
plus', p', times', fac'

They will be analysed ascendingly in the following order:
p' < plus'
plus' < times'
p' < times'
p' < fac'
times' < fac'

Rules:
plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
p' :: 0':s' → 0':s'
times' :: 0':s' → 0':s' → 0':s'
fac' :: 0':s' → 0':s' → 0':s'
factorial' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
p', plus', times', fac'

They will be analysed ascendingly in the following order:
p' < plus'
plus' < times'
p' < times'
p' < fac'
times' < fac'

Proved the following rewrite lemma:
p'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)

Induction Base:
p'(_gen_0':s'2(+(1, 0))) →RΩ(1)
0'

Induction Step:
p'(_gen_0':s'2(+(1, +(_$n5, 1)))) →RΩ(1)
s'(p'(s'(_gen_0':s'2(_$n5)))) →IH
s'(_gen_0':s'2(_$n5))

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

Rules:
plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
p' :: 0':s' → 0':s'
times' :: 0':s' → 0':s' → 0':s'
fac' :: 0':s' → 0':s' → 0':s'
factorial' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
p'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
plus', times', fac'

They will be analysed ascendingly in the following order:
plus' < times'
times' < fac'

Proved the following rewrite lemma:
plus'(_gen_0':s'2(_n382), _gen_0':s'2(b)) → _gen_0':s'2(+(_n382, b)), rt ∈ Ω(1 + n382 + n3822)

Induction Base:
plus'(_gen_0':s'2(0), _gen_0':s'2(b)) →RΩ(1)
_gen_0':s'2(b)

Induction Step:
plus'(_gen_0':s'2(+(_$n383, 1)), _gen_0':s'2(_b522)) →RΩ(1)
s'(plus'(p'(s'(_gen_0':s'2(_$n383))), _gen_0':s'2(_b522))) →LΩ(1 + $n383)
s'(plus'(_gen_0':s'2(_$n383), _gen_0':s'2(_b522))) →IH
s'(_gen_0':s'2(+(_$n383, _b522)))

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

Rules:
plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
p' :: 0':s' → 0':s'
times' :: 0':s' → 0':s' → 0':s'
fac' :: 0':s' → 0':s' → 0':s'
factorial' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
p'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)
plus'(_gen_0':s'2(_n382), _gen_0':s'2(b)) → _gen_0':s'2(+(_n382, b)), rt ∈ Ω(1 + n382 + n3822)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
times', fac'

They will be analysed ascendingly in the following order:
times' < fac'

Proved the following rewrite lemma:
times'(_gen_0':s'2(_n1073), _gen_0':s'2(b)) → _gen_0':s'2(*(_n1073, b)), rt ∈ Ω(1 + b1322·n1073 + b13222·n1073 + n1073 + n10732)

Induction Base:
times'(_gen_0':s'2(0), _gen_0':s'2(b)) →RΩ(1)
0'

Induction Step:
times'(_gen_0':s'2(+(_$n1074, 1)), _gen_0':s'2(_b1322)) →RΩ(1)
plus'(_gen_0':s'2(_b1322), times'(p'(s'(_gen_0':s'2(_$n1074))), _gen_0':s'2(_b1322))) →LΩ(1 + $n1074)
plus'(_gen_0':s'2(_b1322), times'(_gen_0':s'2(_$n1074), _gen_0':s'2(_b1322))) →IH
plus'(_gen_0':s'2(_b1322), _gen_0':s'2(*(_$n1074, _b1322))) →LΩ(1 + b1322 + b13222)
_gen_0':s'2(+(_b1322, *(_$n1074, _b1322)))

We have rt ∈ Ω(n3) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n3).

Rules:
plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
p' :: 0':s' → 0':s'
times' :: 0':s' → 0':s' → 0':s'
fac' :: 0':s' → 0':s' → 0':s'
factorial' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
p'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)
plus'(_gen_0':s'2(_n382), _gen_0':s'2(b)) → _gen_0':s'2(+(_n382, b)), rt ∈ Ω(1 + n382 + n3822)
times'(_gen_0':s'2(_n1073), _gen_0':s'2(b)) → _gen_0':s'2(*(_n1073, b)), rt ∈ Ω(1 + b1322·n1073 + b13222·n1073 + n1073 + n10732)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

The following defined symbols remain to be analysed:
fac'

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

The following conjecture could not be proven:

fac'(_gen_0':s'2(_n2152), _gen_0':s'2(b)) →? _*3

Rules:
plus'(0', x) → x
plus'(s'(x), y) → s'(plus'(p'(s'(x)), y))
times'(0', y) → 0'
times'(s'(x), y) → plus'(y, times'(p'(s'(x)), y))
p'(s'(0')) → 0'
p'(s'(s'(x))) → s'(p'(s'(x)))
fac'(0', x) → x
fac'(s'(x), y) → fac'(p'(s'(x)), times'(s'(x), y))
factorial'(x) → fac'(x, s'(0'))

Types:
plus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
p' :: 0':s' → 0':s'
times' :: 0':s' → 0':s' → 0':s'
fac' :: 0':s' → 0':s' → 0':s'
factorial' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'

Lemmas:
p'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)
plus'(_gen_0':s'2(_n382), _gen_0':s'2(b)) → _gen_0':s'2(+(_n382, b)), rt ∈ Ω(1 + n382 + n3822)
times'(_gen_0':s'2(_n1073), _gen_0':s'2(b)) → _gen_0':s'2(*(_n1073, b)), rt ∈ Ω(1 + b1322·n1073 + b13222·n1073 + n1073 + n10732)

Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))

No more defined symbols left to analyse.

The lowerbound Ω(n3) was proven with the following lemma:
times'(_gen_0':s'2(_n1073), _gen_0':s'2(b)) → _gen_0':s'2(*(_n1073, b)), rt ∈ Ω(1 + b1322·n1073 + b13222·n1073 + n1073 + n10732)