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

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

Rewrite Strategy: INNERMOST

Infered types.

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

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

Heuristically decided to analyse the following defined symbols:
f'

Generator Equations:
_gen_b':a':c'3(0) ⇔ _hole_b':a':c'2
_gen_b':a':c'3(+(x, 1)) ⇔ a'(_gen_b':a':c'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':c'3(+(1, _n5)), _gen_b':a':c'3(b)) →^? _*4

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

No more defined symbols left to analyse.