half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
bits(0) → 0
bits(s(x)) → s(bits(half(s(x))))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
bits'(0') → 0'
bits'(s'(x)) → s'(bits'(half'(s'(x))))
Infered types.
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
bits'(0') → 0'
bits'(s'(x)) → s'(bits'(half'(s'(x))))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
bits' :: 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:
half', bits'
They will be analysed ascendingly in the following order:
half' < bits'
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
bits'(0') → 0'
bits'(s'(x)) → s'(bits'(half'(s'(x))))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
bits' :: 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:
half', bits'
They will be analysed ascendingly in the following order:
half' < bits'
Proved the following rewrite lemma:
half'(_gen_0':s'2(*(2, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)
Induction Base:
half'(_gen_0':s'2(*(2, 0))) →RΩ(1)
0'
Induction Step:
half'(_gen_0':s'2(*(2, +(_$n5, 1)))) →RΩ(1)
s'(half'(_gen_0':s'2(*(2, _$n5)))) →IH
s'(_gen_0':s'2(_$n5))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
bits'(0') → 0'
bits'(s'(x)) → s'(bits'(half'(s'(x))))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
half'(_gen_0':s'2(*(2, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)
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:
bits'
Could not prove a rewrite lemma for the defined symbol bits'.
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
bits'(0') → 0'
bits'(s'(x)) → s'(bits'(half'(s'(x))))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
bits' :: 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
half'(_gen_0':s'2(*(2, _n4))) → _gen_0':s'2(_n4), 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:
half'(_gen_0':s'2(*(2, _n4))) → _gen_0':s'2(_n4), rt ∈ Ω(1 + n4)