f(0) → s(0)
f(s(0)) → s(0)
f(s(s(x))) → f(f(s(x)))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(0') → s'(0')
f'(s'(0')) → s'(0')
f'(s'(s'(x))) → f'(f'(s'(x)))
Infered types.
Rules:
f'(0') → s'(0')
f'(s'(0')) → s'(0')
f'(s'(s'(x))) → f'(f'(s'(x)))
Types:
f' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
f'
Rules:
f'(0') → s'(0')
f'(s'(0')) → s'(0')
f'(s'(s'(x))) → f'(f'(s'(x)))
Types:
f' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
The following defined symbols remain to be analysed:
f'
Proved the following rewrite lemma:
f'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(1), rt ∈ Ω(1 + n4)
Induction Base:
f'(_gen_0':s'2(+(1, 0))) →RΩ(1)
s'(0')
Induction Step:
f'(_gen_0':s'2(+(1, +(_$n5, 1)))) →RΩ(1)
f'(f'(s'(_gen_0':s'2(_$n5)))) →IH
f'(_gen_0':s'2(1)) →RΩ(1)
s'(0')
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(0') → s'(0')
f'(s'(0')) → s'(0')
f'(s'(s'(x))) → f'(f'(s'(x)))
Types:
f' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
f'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(1), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
f'(_gen_0':s'2(+(1, _n4))) → _gen_0':s'2(1), rt ∈ Ω(1 + n4)