f(s(0), g(x)) → f(x, g(x))
g(s(x)) → g(x)
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(s'(0'), g'(x)) → f'(x, g'(x))
g'(s'(x)) → g'(x)
Infered types.
Rules:
f'(s'(0'), g'(x)) → f'(x, g'(x))
g'(s'(x)) → g'(x)
Types:
f' :: 0':s' → g' → f'
s' :: 0':s' → 0':s'
0' :: 0':s'
g' :: 0':s' → g'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_hole_g'3 :: g'
_gen_0':s'4 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
f', g'
They will be analysed ascendingly in the following order:
g' < f'
Rules:
f'(s'(0'), g'(x)) → f'(x, g'(x))
g'(s'(x)) → g'(x)
Types:
f' :: 0':s' → g' → f'
s' :: 0':s' → 0':s'
0' :: 0':s'
g' :: 0':s' → g'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_hole_g'3 :: g'
_gen_0':s'4 :: Nat → 0':s'
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
The following defined symbols remain to be analysed:
g', f'
They will be analysed ascendingly in the following order:
g' < f'
Proved the following rewrite lemma:
g'(_gen_0':s'4(+(1, _n6))) → _*5, rt ∈ Ω(n6)
Induction Base:
g'(_gen_0':s'4(+(1, 0)))
Induction Step:
g'(_gen_0':s'4(+(1, +(_$n7, 1)))) →RΩ(1)
g'(_gen_0':s'4(+(1, _$n7))) →IH
_*5
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(s'(0'), g'(x)) → f'(x, g'(x))
g'(s'(x)) → g'(x)
Types:
f' :: 0':s' → g' → f'
s' :: 0':s' → 0':s'
0' :: 0':s'
g' :: 0':s' → g'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_hole_g'3 :: g'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
g'(_gen_0':s'4(+(1, _n6))) → _*5, rt ∈ Ω(n6)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(s'(0'), g'(x)) → f'(x, g'(x))
g'(s'(x)) → g'(x)
Types:
f' :: 0':s' → g' → f'
s' :: 0':s' → 0':s'
0' :: 0':s'
g' :: 0':s' → g'
_hole_f'1 :: f'
_hole_0':s'2 :: 0':s'
_hole_g'3 :: g'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
g'(_gen_0':s'4(+(1, _n6))) → _*5, rt ∈ Ω(n6)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
g'(_gen_0':s'4(+(1, _n6))) → _*5, rt ∈ Ω(n6)