f(s(x), x) → f(s(x), round(s(x)))
round(0) → 0
round(0) → s(0)
round(s(0)) → s(0)
round(s(s(x))) → s(s(round(x)))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
f'(s'(x), x) → f'(s'(x), round'(s'(x)))
round'(0') → 0'
round'(0') → s'(0')
round'(s'(0')) → s'(0')
round'(s'(s'(x))) → s'(s'(round'(x)))
Infered types.
Rules:
f'(s'(x), x) → f'(s'(x), round'(s'(x)))
round'(0') → 0'
round'(0') → s'(0')
round'(s'(0')) → s'(0')
round'(s'(s'(x))) → s'(s'(round'(x)))
Types:
f' :: s':0' → s':0' → f'
s' :: s':0' → s':0'
round' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_gen_s':0'3 :: Nat → s':0'
Heuristically decided to analyse the following defined symbols:
f', round'
They will be analysed ascendingly in the following order:
round' < f'
Rules:
f'(s'(x), x) → f'(s'(x), round'(s'(x)))
round'(0') → 0'
round'(0') → s'(0')
round'(s'(0')) → s'(0')
round'(s'(s'(x))) → s'(s'(round'(x)))
Types:
f' :: s':0' → s':0' → f'
s' :: s':0' → s':0'
round' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_gen_s':0'3 :: Nat → s':0'
Generator Equations:
_gen_s':0'3(0) ⇔ 0'
_gen_s':0'3(+(x, 1)) ⇔ s'(_gen_s':0'3(x))
The following defined symbols remain to be analysed:
round', f'
They will be analysed ascendingly in the following order:
round' < f'
Proved the following rewrite lemma:
round'(_gen_s':0'3(*(2, _n5))) → _gen_s':0'3(*(2, _n5)), rt ∈ Ω(1 + n5)
Induction Base:
round'(_gen_s':0'3(*(2, 0))) →RΩ(1)
0'
Induction Step:
round'(_gen_s':0'3(*(2, +(_$n6, 1)))) →RΩ(1)
s'(s'(round'(_gen_s':0'3(*(2, _$n6))))) →IH
s'(s'(_gen_s':0'3(*(2, _$n6))))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
f'(s'(x), x) → f'(s'(x), round'(s'(x)))
round'(0') → 0'
round'(0') → s'(0')
round'(s'(0')) → s'(0')
round'(s'(s'(x))) → s'(s'(round'(x)))
Types:
f' :: s':0' → s':0' → f'
s' :: s':0' → s':0'
round' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_gen_s':0'3 :: Nat → s':0'
Lemmas:
round'(_gen_s':0'3(*(2, _n5))) → _gen_s':0'3(*(2, _n5)), rt ∈ Ω(1 + n5)
Generator Equations:
_gen_s':0'3(0) ⇔ 0'
_gen_s':0'3(+(x, 1)) ⇔ s'(_gen_s':0'3(x))
The following defined symbols remain to be analysed:
f'
Could not prove a rewrite lemma for the defined symbol f'.
Rules:
f'(s'(x), x) → f'(s'(x), round'(s'(x)))
round'(0') → 0'
round'(0') → s'(0')
round'(s'(0')) → s'(0')
round'(s'(s'(x))) → s'(s'(round'(x)))
Types:
f' :: s':0' → s':0' → f'
s' :: s':0' → s':0'
round' :: s':0' → s':0'
0' :: s':0'
_hole_f'1 :: f'
_hole_s':0'2 :: s':0'
_gen_s':0'3 :: Nat → s':0'
Lemmas:
round'(_gen_s':0'3(*(2, _n5))) → _gen_s':0'3(*(2, _n5)), rt ∈ Ω(1 + n5)
Generator Equations:
_gen_s':0'3(0) ⇔ 0'
_gen_s':0'3(+(x, 1)) ⇔ s'(_gen_s':0'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
round'(_gen_s':0'3(*(2, _n5))) → _gen_s':0'3(*(2, _n5)), rt ∈ Ω(1 + n5)