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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: b':a' → b':a' → f'
a' :: b':a' → b':a'
b' :: b':a' → b':a'
_hole_f'1 :: f'
_hole_b':a'2 :: b':a'
_gen_b':a'3 :: Nat → b':a'

Heuristically decided to analyse the following defined symbols:
f'

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

Types:
f' :: b':a' → b':a' → f'
a' :: b':a' → b':a'
b' :: b':a' → b':a'
_hole_f'1 :: f'
_hole_b':a'2 :: b':a'
_gen_b':a'3 :: Nat → b':a'

Generator Equations:
_gen_b':a'3(0) ⇔ _hole_b':a'2
_gen_b':a'3(+(x, 1)) ⇔ a'(_gen_b':a'3(x))

The following defined symbols remain to be analysed:
f'

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

The following conjecture could not be proven:

f'(_gen_b':a'3(+(1, _n5)), _gen_b':a'3(b)) →? _*4

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

Types:
f' :: b':a' → b':a' → f'
a' :: b':a' → b':a'
b' :: b':a' → b':a'
_hole_f'1 :: f'
_hole_b':a'2 :: b':a'
_gen_b':a'3 :: Nat → b':a'

Generator Equations:
_gen_b':a'3(0) ⇔ _hole_b':a'2
_gen_b':a'3(+(x, 1)) ⇔ a'(_gen_b':a'3(x))

No more defined symbols left to analyse.