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

f(0) → 1
f(s(x)) → g(x, s(x))
g(0, y) → y
g(s(x), y) → g(x, +(y, s(x)))
+(x, 0) → x
+(x, s(y)) → s(+(x, y))
g(s(x), y) → g(x, s(+(y, x)))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(0') → 1'
f'(s'(x)) → g'(x, s'(x))
g'(0', y) → y
g'(s'(x), y) → g'(x, +'(y, s'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
g'(s'(x), y) → g'(x, s'(+'(y, x)))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(0') → 1'
f'(s'(x)) → g'(x, s'(x))
g'(0', y) → y
g'(s'(x), y) → g'(x, +'(y, s'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
g'(s'(x), y) → g'(x, s'(+'(y, x)))

Types:
f' :: 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
g' :: 0':1':s' → 0':1':s' → 0':1':s'
+' :: 0':1':s' → 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'

Heuristically decided to analyse the following defined symbols:
g', +'

They will be analysed ascendingly in the following order:
+' < g'

Rules:
f'(0') → 1'
f'(s'(x)) → g'(x, s'(x))
g'(0', y) → y
g'(s'(x), y) → g'(x, +'(y, s'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
g'(s'(x), y) → g'(x, s'(+'(y, x)))

Types:
f' :: 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
g' :: 0':1':s' → 0':1':s' → 0':1':s'
+' :: 0':1':s' → 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'

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

The following defined symbols remain to be analysed:
+', g'

They will be analysed ascendingly in the following order:
+' < g'

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

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

Induction Step:
+'(_gen_0':1':s'2(_a137), _gen_0':1':s'2(+(_\$n5, 1))) →RΩ(1)
s'(+'(_gen_0':1':s'2(_a137), _gen_0':1':s'2(_\$n5))) →IH
s'(_gen_0':1':s'2(+(_\$n5, _a137)))

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

Rules:
f'(0') → 1'
f'(s'(x)) → g'(x, s'(x))
g'(0', y) → y
g'(s'(x), y) → g'(x, +'(y, s'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
g'(s'(x), y) → g'(x, s'(+'(y, x)))

Types:
f' :: 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
g' :: 0':1':s' → 0':1':s' → 0':1':s'
+' :: 0':1':s' → 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'

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

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

The following defined symbols remain to be analysed:
g'

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

The following conjecture could not be proven:

g'(_gen_0':1':s'2(_n502), _gen_0':1':s'2(b)) →? _*3

Rules:
f'(0') → 1'
f'(s'(x)) → g'(x, s'(x))
g'(0', y) → y
g'(s'(x), y) → g'(x, +'(y, s'(x)))
+'(x, 0') → x
+'(x, s'(y)) → s'(+'(x, y))
g'(s'(x), y) → g'(x, s'(+'(y, x)))

Types:
f' :: 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
g' :: 0':1':s' → 0':1':s' → 0':1':s'
+' :: 0':1':s' → 0':1':s' → 0':1':s'
_hole_0':1':s'1 :: 0':1':s'
_gen_0':1':s'2 :: Nat → 0':1':s'

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

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
+'(_gen_0':1':s'2(a), _gen_0':1':s'2(_n4)) → _gen_0':1':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)