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

f(f(0, x), 1) → f(g(f(x, x)), x)
f(g(x), y) → g(f(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:

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

Rewrite Strategy: INNERMOST

Infered types.

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

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

Heuristically decided to analyse the following defined symbols:
f'

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

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

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

The following defined symbols remain to be analysed:
f'

Proved the following rewrite lemma:
f'(_gen_0':1':g'2(+(1, _n4)), _gen_0':1':g'2(b)) → _*3, rt ∈ Ω(n4)

Induction Base:
f'(_gen_0':1':g'2(+(1, 0)), _gen_0':1':g'2(b))

Induction Step:
f'(_gen_0':1':g'2(+(1, +(_\$n5, 1))), _gen_0':1':g'2(_b555)) →RΩ(1)
g'(f'(_gen_0':1':g'2(+(1, _\$n5)), _gen_0':1':g'2(_b555))) →IH
g'(_*3)

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

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

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

Lemmas:
f'(_gen_0':1':g'2(+(1, _n4)), _gen_0':1':g'2(b)) → _*3, rt ∈ Ω(n4)

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

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_0':1':g'2(+(1, _n4)), _gen_0':1':g'2(b)) → _*3, rt ∈ Ω(n4)