f(g(x)) → f(a(g(g(f(x))), g(f(x))))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(g'(x)) → f'(a'(g'(g'(f'(x))), g'(f'(x))))
Infered types.
Rules:
f'(g'(x)) → f'(a'(g'(g'(f'(x))), g'(f'(x))))
Types:
f' :: g':a' → g':a'
g' :: g':a' → g':a'
a' :: g':a' → g':a' → g':a'
_hole_g':a'1 :: g':a'
_gen_g':a'2 :: Nat → g':a'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(g'(x)) → f'(a'(g'(g'(f'(x))), g'(f'(x))))
Types:
f' :: g':a' → g':a'
g' :: g':a' → g':a'
a' :: g':a' → g':a' → g':a'
_hole_g':a'1 :: g':a'
_gen_g':a'2 :: Nat → g':a'
Generator Equations:
_gen_g':a'2(0) ⇔ _hole_g':a'1
_gen_g':a'2(+(x, 1)) ⇔ g'(_gen_g':a'2(x))
The following defined symbols remain to be analysed:
f'
Proved the following rewrite lemma:
f'(_gen_g':a'2(+(1, _n4))) → _*3, rt ∈ Ω(2n)
Induction Base:
f'(_gen_g':a'2(+(1, 0)))
Induction Step:
f'(_gen_g':a'2(+(1, +(_$n5, 1)))) →RΩ(1)
f'(a'(g'(g'(f'(_gen_g':a'2(+(1, _$n5))))), g'(f'(_gen_g':a'2(+(1, _$n5)))))) →IH
f'(a'(g'(g'(_*3)), g'(f'(_gen_g':a'2(+(1, _$n5)))))) →IH
f'(a'(g'(g'(_*3)), g'(_*3)))
We have rt ∈ Ω(2n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(2n)