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

g(x, y) → x
g(x, y) → y
f(0, 1, x) → f(s(x), x, x)
f(x, y, s(z)) → s(f(0, 1, z))

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

g'(x, y) → x
g'(x, y) → y
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))

Rewrite Strategy: INNERMOST

Infered types.

Rules:
g'(x, y) → x
g'(x, y) → y
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))

Types:
g' :: g' → g' → g'
f' :: 0':1':s' → 0':1':s' → 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
_hole_g'1 :: g'
_hole_0':1':s'2 :: 0':1':s'
_gen_0':1':s'3 :: Nat → 0':1':s'

Heuristically decided to analyse the following defined symbols:
f'

Rules:
g'(x, y) → x
g'(x, y) → y
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))

Types:
g' :: g' → g' → g'
f' :: 0':1':s' → 0':1':s' → 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
_hole_g'1 :: g'
_hole_0':1':s'2 :: 0':1':s'
_gen_0':1':s'3 :: Nat → 0':1':s'

Generator Equations:
_gen_0':1':s'3(0) ⇔ 1'
_gen_0':1':s'3(+(x, 1)) ⇔ s'(_gen_0':1':s'3(x))

The following defined symbols remain to be analysed:
f'

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

Rules:
g'(x, y) → x
g'(x, y) → y
f'(0', 1', x) → f'(s'(x), x, x)
f'(x, y, s'(z)) → s'(f'(0', 1', z))

Types:
g' :: g' → g' → g'
f' :: 0':1':s' → 0':1':s' → 0':1':s' → 0':1':s'
0' :: 0':1':s'
1' :: 0':1':s'
s' :: 0':1':s' → 0':1':s'
_hole_g'1 :: g'
_hole_0':1':s'2 :: 0':1':s'
_gen_0':1':s'3 :: Nat → 0':1':s'

Generator Equations:
_gen_0':1':s'3(0) ⇔ 1'
_gen_0':1':s'3(+(x, 1)) ⇔ s'(_gen_0':1':s'3(x))

No more defined symbols left to analyse.