f(s(s(s(s(s(s(s(s(x)))))))), y, y) → f(id(s(s(s(s(s(s(s(s(x))))))))), y, y)
id(s(x)) → s(id(x))
id(0) → 0
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'
Infered types.
Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'
Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'
Heuristically decided to analyse the following defined symbols:
f', id'
They will be analysed ascendingly in the following order:
id' < f'
Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'
Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'
Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))
The following defined symbols remain to be analysed:
id', f'
They will be analysed ascendingly in the following order:
id' < f'
Proved the following rewrite lemma:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)
Induction Base:
id'(_gen_s':0'4(0)) →RΩ(1)
0'
Induction Step:
id'(_gen_s':0'4(+(_$n7, 1))) →RΩ(1)
s'(id'(_gen_s':0'4(_$n7))) →IH
s'(_gen_s':0'4(_$n7))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'
Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'
Lemmas:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)
Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(s'(s'(s'(s'(s'(s'(s'(s'(x)))))))), y, y) → f'(id'(s'(s'(s'(s'(s'(s'(s'(s'(x))))))))), y, y)
id'(s'(x)) → s'(id'(x))
id'(0') → 0'
Types:
f' :: s':0' → a → a → f'
s' :: s':0' → s':0'
id' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_hole_a3 :: a
_gen_s':0'4 :: Nat → s':0'
Lemmas:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)
Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
id'(_gen_s':0'4(_n6)) → _gen_s':0'4(_n6), rt ∈ Ω(1 + n6)