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

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

Rewrite Strategy: INNERMOST

Infered types.

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

Types:
f' :: c':s' → c':s' → c':s'
c' :: c':s' → c':s'
s' :: c':s' → c':s'
_hole_c':s'1 :: c':s'
_gen_c':s'2 :: Nat → c':s'

Heuristically decided to analyse the following defined symbols:
f'

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

Types:
f' :: c':s' → c':s' → c':s'
c' :: c':s' → c':s'
s' :: c':s' → c':s'
_hole_c':s'1 :: c':s'
_gen_c':s'2 :: Nat → c':s'

Generator Equations:
_gen_c':s'2(0) ⇔ _hole_c':s'1
_gen_c':s'2(+(x, 1)) ⇔ c'(_gen_c':s'2(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_c':s'2(a), _gen_c':s'2(+(1, _n4))) →? _*3

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

Types:
f' :: c':s' → c':s' → c':s'
c' :: c':s' → c':s'
s' :: c':s' → c':s'
_hole_c':s'1 :: c':s'
_gen_c':s'2 :: Nat → c':s'

Generator Equations:
_gen_c':s'2(0) ⇔ _hole_c':s'1
_gen_c':s'2(+(x, 1)) ⇔ c'(_gen_c':s'2(x))

No more defined symbols left to analyse.