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

f(x, x) → a
f(g(x), y) → 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'(x, x) → a'
f'(g'(x), y) → f'(x, y)

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)

Types:
f' :: g' → g' → a'
a' :: a'
g' :: g' → g'
_hole_a'1 :: a'
_hole_g'2 :: g'
_gen_g'3 :: Nat → g'

Heuristically decided to analyse the following defined symbols:
f'

Rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)

Types:
f' :: g' → g' → a'
a' :: a'
g' :: g' → g'
_hole_a'1 :: a'
_hole_g'2 :: g'
_gen_g'3 :: Nat → g'

Generator Equations:
_gen_g'3(0) ⇔ _hole_g'2
_gen_g'3(+(x, 1)) ⇔ g'(_gen_g'3(x))

The following defined symbols remain to be analysed:
f'

Proved the following rewrite lemma:
f'(_gen_g'3(+(1, _n5)), _gen_g'3(b)) → _*4, rt ∈ Ω(n5)

Induction Base:
f'(_gen_g'3(+(1, 0)), _gen_g'3(b))

Induction Step:
f'(_gen_g'3(+(1, +(_\$n6, 1))), _gen_g'3(_b325)) →RΩ(1)
f'(_gen_g'3(+(1, _\$n6)), _gen_g'3(_b325)) →IH
_*4

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

Rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)

Types:
f' :: g' → g' → a'
a' :: a'
g' :: g' → g'
_hole_a'1 :: a'
_hole_g'2 :: g'
_gen_g'3 :: Nat → g'

Lemmas:
f'(_gen_g'3(+(1, _n5)), _gen_g'3(b)) → _*4, rt ∈ Ω(n5)

Generator Equations:
_gen_g'3(0) ⇔ _hole_g'2
_gen_g'3(+(x, 1)) ⇔ g'(_gen_g'3(x))

No more defined symbols left to analyse.

The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_g'3(+(1, _n5)), _gen_g'3(b)) → _*4, rt ∈ Ω(n5)