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

f(c(s(x), y)) → f(c(x, s(y)))
f(c(s(x), s(y))) → g(c(x, y))
g(c(x, s(y))) → g(c(s(x), y))
g(c(s(x), s(y))) → f(c(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'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(c'(s'(x), s'(y))) → g'(c'(x, y))
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
g'(c'(s'(x), s'(y))) → f'(c'(x, y))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(c'(s'(x), s'(y))) → g'(c'(x, y))
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
g'(c'(s'(x), s'(y))) → f'(c'(x, y))

Types:
f' :: c' → f':g'
c' :: s' → s' → c'
s' :: s' → s'
g' :: c' → f':g'
_hole_f':g'1 :: f':g'
_hole_c'2 :: c'
_hole_s'3 :: s'
_gen_s'4 :: Nat → s'

Heuristically decided to analyse the following defined symbols:
f', g'

They will be analysed ascendingly in the following order:
f' = g'

Rules:
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(c'(s'(x), s'(y))) → g'(c'(x, y))
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
g'(c'(s'(x), s'(y))) → f'(c'(x, y))

Types:
f' :: c' → f':g'
c' :: s' → s' → c'
s' :: s' → s'
g' :: c' → f':g'
_hole_f':g'1 :: f':g'
_hole_c'2 :: c'
_hole_s'3 :: s'
_gen_s'4 :: Nat → s'

Generator Equations:
_gen_s'4(0) ⇔ _hole_s'3
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

The following defined symbols remain to be analysed:
g', f'

They will be analysed ascendingly in the following order:
f' = g'

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

Rules:
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(c'(s'(x), s'(y))) → g'(c'(x, y))
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
g'(c'(s'(x), s'(y))) → f'(c'(x, y))

Types:
f' :: c' → f':g'
c' :: s' → s' → c'
s' :: s' → s'
g' :: c' → f':g'
_hole_f':g'1 :: f':g'
_hole_c'2 :: c'
_hole_s'3 :: s'
_gen_s'4 :: Nat → s'

Generator Equations:
_gen_s'4(0) ⇔ _hole_s'3
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

The following defined symbols remain to be analysed:
f'

They will be analysed ascendingly in the following order:
f' = g'

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

Rules:
f'(c'(s'(x), y)) → f'(c'(x, s'(y)))
f'(c'(s'(x), s'(y))) → g'(c'(x, y))
g'(c'(x, s'(y))) → g'(c'(s'(x), y))
g'(c'(s'(x), s'(y))) → f'(c'(x, y))

Types:
f' :: c' → f':g'
c' :: s' → s' → c'
s' :: s' → s'
g' :: c' → f':g'
_hole_f':g'1 :: f':g'
_hole_c'2 :: c'
_hole_s'3 :: s'
_gen_s'4 :: Nat → s'

Generator Equations:
_gen_s'4(0) ⇔ _hole_s'3
_gen_s'4(+(x, 1)) ⇔ s'(_gen_s'4(x))

No more defined symbols left to analyse.