f(g(x)) → g(g(f(x)))
f(g(x)) → g(g(g(x)))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(g'(x)) → g'(g'(f'(x)))
f'(g'(x)) → g'(g'(g'(x)))
Infered types.
Rules:
f'(g'(x)) → g'(g'(f'(x)))
f'(g'(x)) → g'(g'(g'(x)))
Types:
f' :: g' → g'
g' :: g' → g'
_hole_g'1 :: g'
_gen_g'2 :: Nat → g'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(g'(x)) → g'(g'(f'(x)))
f'(g'(x)) → g'(g'(g'(x)))
Types:
f' :: g' → g'
g' :: g' → g'
_hole_g'1 :: g'
_gen_g'2 :: Nat → g'
Generator Equations:
_gen_g'2(0) ⇔ _hole_g'1
_gen_g'2(+(x, 1)) ⇔ g'(_gen_g'2(x))
The following defined symbols remain to be analysed:
f'
Proved the following rewrite lemma:
f'(_gen_g'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Induction Base:
f'(_gen_g'2(+(1, 0)))
Induction Step:
f'(_gen_g'2(+(1, +(_$n5, 1)))) →RΩ(1)
g'(g'(f'(_gen_g'2(+(1, _$n5))))) →IH
g'(g'(_*3))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(g'(x)) → g'(g'(f'(x)))
f'(g'(x)) → g'(g'(g'(x)))
Types:
f' :: g' → g'
g' :: g' → g'
_hole_g'1 :: g'
_gen_g'2 :: Nat → g'
Lemmas:
f'(_gen_g'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)
Generator Equations:
_gen_g'2(0) ⇔ _hole_g'1
_gen_g'2(+(x, 1)) ⇔ g'(_gen_g'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_g'2(+(1, _n4))) → _*3, rt ∈ Ω(n4)