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

f(n__f(n__a)) → f(n__g(n__f(n__a)))
f(X) → n__f(X)
a → n__a
g(X) → n__g(X)
activate(n__f(X)) → f(X)
activate(n__a) → a
activate(n__g(X)) → g(activate(X))
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'(n__f'(n__a')) → f'(n__g'(n__f'(n__a')))
f'(X) → n__f'(X)
a' → n__a'
g'(X) → n__g'(X)
activate'(n__f'(X)) → f'(X)
activate'(n__a') → a'
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Rewrite Strategy: INNERMOST

Infered types.

Rules:
f'(n__f'(n__a')) → f'(n__g'(n__f'(n__a')))
f'(X) → n__f'(X)
a' → n__a'
g'(X) → n__g'(X)
activate'(n__f'(X)) → f'(X)
activate'(n__a') → a'
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__a' :: n__a':n__f':n__g'
n__g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
a' :: n__a':n__f':n__g'
g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
activate' :: n__a':n__f':n__g' → n__a':n__f':n__g'
_hole_n__a':n__f':n__g'1 :: n__a':n__f':n__g'
_gen_n__a':n__f':n__g'2 :: Nat → n__a':n__f':n__g'

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

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

Rules:
f'(n__f'(n__a')) → f'(n__g'(n__f'(n__a')))
f'(X) → n__f'(X)
a' → n__a'
g'(X) → n__g'(X)
activate'(n__f'(X)) → f'(X)
activate'(n__a') → a'
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__a' :: n__a':n__f':n__g'
n__g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
a' :: n__a':n__f':n__g'
g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
activate' :: n__a':n__f':n__g' → n__a':n__f':n__g'
_hole_n__a':n__f':n__g'1 :: n__a':n__f':n__g'
_gen_n__a':n__f':n__g'2 :: Nat → n__a':n__f':n__g'

Generator Equations:
_gen_n__a':n__f':n__g'2(0) ⇔ n__a'
_gen_n__a':n__f':n__g'2(+(x, 1)) ⇔ n__f'(_gen_n__a':n__f':n__g'2(x))

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

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

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

Rules:
f'(n__f'(n__a')) → f'(n__g'(n__f'(n__a')))
f'(X) → n__f'(X)
a' → n__a'
g'(X) → n__g'(X)
activate'(n__f'(X)) → f'(X)
activate'(n__a') → a'
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__a' :: n__a':n__f':n__g'
n__g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
a' :: n__a':n__f':n__g'
g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
activate' :: n__a':n__f':n__g' → n__a':n__f':n__g'
_hole_n__a':n__f':n__g'1 :: n__a':n__f':n__g'
_gen_n__a':n__f':n__g'2 :: Nat → n__a':n__f':n__g'

Generator Equations:
_gen_n__a':n__f':n__g'2(0) ⇔ n__a'
_gen_n__a':n__f':n__g'2(+(x, 1)) ⇔ n__f'(_gen_n__a':n__f':n__g'2(x))

The following defined symbols remain to be analysed:
activate'

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

Rules:
f'(n__f'(n__a')) → f'(n__g'(n__f'(n__a')))
f'(X) → n__f'(X)
a' → n__a'
g'(X) → n__g'(X)
activate'(n__f'(X)) → f'(X)
activate'(n__a') → a'
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X

Types:
f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__f' :: n__a':n__f':n__g' → n__a':n__f':n__g'
n__a' :: n__a':n__f':n__g'
n__g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
a' :: n__a':n__f':n__g'
g' :: n__a':n__f':n__g' → n__a':n__f':n__g'
activate' :: n__a':n__f':n__g' → n__a':n__f':n__g'
_hole_n__a':n__f':n__g'1 :: n__a':n__f':n__g'
_gen_n__a':n__f':n__g'2 :: Nat → n__a':n__f':n__g'

Generator Equations:
_gen_n__a':n__f':n__g'2(0) ⇔ n__a'
_gen_n__a':n__f':n__g'2(+(x, 1)) ⇔ n__f'(_gen_n__a':n__f':n__g'2(x))

No more defined symbols left to analyse.