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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: d':c' → b':a' → f'
a' :: b':a' → b':a'
b' :: b':a' → b':a'
c' :: d':c' → d':c'
d' :: d':c' → d':c'
_hole_f'1 :: f'
_hole_d':c'2 :: d':c'
_hole_b':a'3 :: b':a'
_gen_d':c'4 :: Nat → d':c'
_gen_b':a'5 :: Nat → b':a'

Heuristically decided to analyse the following defined symbols:
f'

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

Types:
f' :: d':c' → b':a' → f'
a' :: b':a' → b':a'
b' :: b':a' → b':a'
c' :: d':c' → d':c'
d' :: d':c' → d':c'
_hole_f'1 :: f'
_hole_d':c'2 :: d':c'
_hole_b':a'3 :: b':a'
_gen_d':c'4 :: Nat → d':c'
_gen_b':a'5 :: Nat → b':a'

Generator Equations:
_gen_d':c'4(0) ⇔ _hole_d':c'2
_gen_d':c'4(+(x, 1)) ⇔ c'(_gen_d':c'4(x))
_gen_b':a'5(0) ⇔ _hole_b':a'3
_gen_b':a'5(+(x, 1)) ⇔ a'(_gen_b':a'5(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_d':c'4(+(1, _n7)), _gen_b':a'5(b)) →? _*6

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

Types:
f' :: d':c' → b':a' → f'
a' :: b':a' → b':a'
b' :: b':a' → b':a'
c' :: d':c' → d':c'
d' :: d':c' → d':c'
_hole_f'1 :: f'
_hole_d':c'2 :: d':c'
_hole_b':a'3 :: b':a'
_gen_d':c'4 :: Nat → d':c'
_gen_b':a'5 :: Nat → b':a'

Generator Equations:
_gen_d':c'4(0) ⇔ _hole_d':c'2
_gen_d':c'4(+(x, 1)) ⇔ c'(_gen_d':c'4(x))
_gen_b':a'5(0) ⇔ _hole_b':a'3
_gen_b':a'5(+(x, 1)) ⇔ a'(_gen_b':a'5(x))

No more defined symbols left to analyse.