f(X, n__g(X), Y) → f(activate(Y), activate(Y), activate(Y))
g(b) → c
b → c
g(X) → n__g(X)
activate(n__g(X)) → g(activate(X))
activate(X) → X
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b' → c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X
Infered types.
Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b' → c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X
Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'
Heuristically decided to analyse the following defined symbols:
f', activate'
They will be analysed ascendingly in the following order:
activate' < f'
Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b' → c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X
Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'
Generator Equations:
_gen_n__g':c'3(0) ⇔ c'
_gen_n__g':c'3(+(x, 1)) ⇔ n__g'(_gen_n__g':c'3(x))
The following defined symbols remain to be analysed:
activate', f'
They will be analysed ascendingly in the following order:
activate' < f'
Proved the following rewrite lemma:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Induction Base:
activate'(_gen_n__g':c'3(+(1, 0)))
Induction Step:
activate'(_gen_n__g':c'3(+(1, +(_$n6, 1)))) →RΩ(1)
g'(activate'(_gen_n__g':c'3(+(1, _$n6)))) →IH
g'(_*4)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b' → c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X
Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'
Lemmas:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_n__g':c'3(0) ⇔ c'
_gen_n__g':c'3(+(x, 1)) ⇔ n__g'(_gen_n__g':c'3(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(X, n__g'(X), Y) → f'(activate'(Y), activate'(Y), activate'(Y))
g'(b') → c'
b' → c'
g'(X) → n__g'(X)
activate'(n__g'(X)) → g'(activate'(X))
activate'(X) → X
Types:
f' :: n__g':c' → n__g':c' → n__g':c' → f'
n__g' :: n__g':c' → n__g':c'
activate' :: n__g':c' → n__g':c'
g' :: n__g':c' → n__g':c'
b' :: n__g':c'
c' :: n__g':c'
_hole_f'1 :: f'
_hole_n__g':c'2 :: n__g':c'
_gen_n__g':c'3 :: Nat → n__g':c'
Lemmas:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_n__g':c'3(0) ⇔ c'
_gen_n__g':c'3(+(x, 1)) ⇔ n__g'(_gen_n__g':c'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
activate'(_gen_n__g':c'3(+(1, _n5))) → _*4, rt ∈ Ω(n5)