f(x, x) → a
f(g(x), y) → f(x, y)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)
Infered types.
Rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)
Types:
f' :: g' → g' → a'
a' :: a'
g' :: g' → g'
_hole_a'1 :: a'
_hole_g'2 :: g'
_gen_g'3 :: Nat → g'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)
Types:
f' :: g' → g' → a'
a' :: a'
g' :: g' → g'
_hole_a'1 :: a'
_hole_g'2 :: g'
_gen_g'3 :: Nat → g'
Generator Equations:
_gen_g'3(0) ⇔ _hole_g'2
_gen_g'3(+(x, 1)) ⇔ g'(_gen_g'3(x))
The following defined symbols remain to be analysed:
f'
Proved the following rewrite lemma:
f'(_gen_g'3(+(1, _n5)), _gen_g'3(b)) → _*4, rt ∈ Ω(n5)
Induction Base:
f'(_gen_g'3(+(1, 0)), _gen_g'3(b))
Induction Step:
f'(_gen_g'3(+(1, +(_$n6, 1))), _gen_g'3(_b325)) →RΩ(1)
f'(_gen_g'3(+(1, _$n6)), _gen_g'3(_b325)) →IH
_*4
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(x, x) → a'
f'(g'(x), y) → f'(x, y)
Types:
f' :: g' → g' → a'
a' :: a'
g' :: g' → g'
_hole_a'1 :: a'
_hole_g'2 :: g'
_gen_g'3 :: Nat → g'
Lemmas:
f'(_gen_g'3(+(1, _n5)), _gen_g'3(b)) → _*4, rt ∈ Ω(n5)
Generator Equations:
_gen_g'3(0) ⇔ _hole_g'2
_gen_g'3(+(x, 1)) ⇔ g'(_gen_g'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_g'3(+(1, _n5)), _gen_g'3(b)) → _*4, rt ∈ Ω(n5)