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

f(g(X), Y) → f(X, n__f(g(X), activate(Y)))
f(X1, X2) → n__f(X1, X2)
activate(n__f(X1, X2)) → f(X1, X2)
activate(X) → X

Rewrite Strategy: INNERMOST

Renamed function symbols to avoid clashes with predefined symbol.

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

f'(g'(X), Y) → f'(X, n__f'(g'(X), activate'(Y)))
f'(X1, X2) → n__f'(X1, X2)
activate'(n__f'(X1, X2)) → f'(X1, X2)
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(g'(X), Y) → f'(X, n__f'(g'(X), activate'(Y)))
f'(X1, X2) → n__f'(X1, X2)
activate'(n__f'(X1, X2)) → f'(X1, X2)
activate'(X) → X

Types:
f' :: g' → n__f' → n__f'
g' :: g' → g'
n__f' :: g' → n__f' → n__f'
activate' :: n__f' → n__f'
_hole_n__f'1 :: n__f'
_hole_g'2 :: g'
_gen_n__f'3 :: Nat → n__f'
_gen_g'4 :: Nat → g'

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

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

Rules:
f'(g'(X), Y) → f'(X, n__f'(g'(X), activate'(Y)))
f'(X1, X2) → n__f'(X1, X2)
activate'(n__f'(X1, X2)) → f'(X1, X2)
activate'(X) → X

Types:
f' :: g' → n__f' → n__f'
g' :: g' → g'
n__f' :: g' → n__f' → n__f'
activate' :: n__f' → n__f'
_hole_n__f'1 :: n__f'
_hole_g'2 :: g'
_gen_n__f'3 :: Nat → n__f'
_gen_g'4 :: Nat → g'

Generator Equations:
_gen_n__f'3(0) ⇔ _hole_n__f'1
_gen_n__f'3(+(x, 1)) ⇔ n__f'(_hole_g'2, _gen_n__f'3(x))
_gen_g'4(0) ⇔ _hole_g'2
_gen_g'4(+(x, 1)) ⇔ g'(_gen_g'4(x))

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

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

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

Rules:
f'(g'(X), Y) → f'(X, n__f'(g'(X), activate'(Y)))
f'(X1, X2) → n__f'(X1, X2)
activate'(n__f'(X1, X2)) → f'(X1, X2)
activate'(X) → X

Types:
f' :: g' → n__f' → n__f'
g' :: g' → g'
n__f' :: g' → n__f' → n__f'
activate' :: n__f' → n__f'
_hole_n__f'1 :: n__f'
_hole_g'2 :: g'
_gen_n__f'3 :: Nat → n__f'
_gen_g'4 :: Nat → g'

Generator Equations:
_gen_n__f'3(0) ⇔ _hole_n__f'1
_gen_n__f'3(+(x, 1)) ⇔ n__f'(_hole_g'2, _gen_n__f'3(x))
_gen_g'4(0) ⇔ _hole_g'2
_gen_g'4(+(x, 1)) ⇔ g'(_gen_g'4(x))

The following defined symbols remain to be analysed:
f'

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

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

Rules:
f'(g'(X), Y) → f'(X, n__f'(g'(X), activate'(Y)))
f'(X1, X2) → n__f'(X1, X2)
activate'(n__f'(X1, X2)) → f'(X1, X2)
activate'(X) → X

Types:
f' :: g' → n__f' → n__f'
g' :: g' → g'
n__f' :: g' → n__f' → n__f'
activate' :: n__f' → n__f'
_hole_n__f'1 :: n__f'
_hole_g'2 :: g'
_gen_n__f'3 :: Nat → n__f'
_gen_g'4 :: Nat → g'

Generator Equations:
_gen_n__f'3(0) ⇔ _hole_n__f'1
_gen_n__f'3(+(x, 1)) ⇔ n__f'(_hole_g'2, _gen_n__f'3(x))
_gen_g'4(0) ⇔ _hole_g'2
_gen_g'4(+(x, 1)) ⇔ g'(_gen_g'4(x))

No more defined symbols left to analyse.