half(0) → 0
half(s(0)) → 0
half(s(s(x))) → s(half(x))
lastbit(0) → 0
lastbit(s(0)) → s(0)
lastbit(s(s(x))) → lastbit(x)
zero(0) → true
zero(s(x)) → false
conv(x) → conviter(x, cons(0, nil))
conviter(x, l) → if(zero(x), x, l)
if(true, x, l) → l
if(false, x, l) → conviter(half(x), cons(lastbit(x), l))
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))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0', nil'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x), l))
Sliced the following arguments:
cons'/1
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))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x)))
Infered types.
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x)))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lastbit' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
conv' :: 0':s' → cons'
conviter' :: 0':s' → cons' → cons'
cons' :: 0':s' → cons'
if' :: true':false' → 0':s' → cons' → cons'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_hole_cons'3 :: cons'
_gen_0':s'4 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
half', lastbit', conviter'
They will be analysed ascendingly in the following order:
half' < conviter'
lastbit' < conviter'
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x)))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lastbit' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
conv' :: 0':s' → cons'
conviter' :: 0':s' → cons' → cons'
cons' :: 0':s' → cons'
if' :: true':false' → 0':s' → cons' → cons'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_hole_cons'3 :: cons'
_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:
half', lastbit', conviter'
They will be analysed ascendingly in the following order:
half' < conviter'
lastbit' < conviter'
Proved the following rewrite lemma:
half'(_gen_0':s'4(*(2, _n6))) → _gen_0':s'4(_n6), rt ∈ Ω(1 + n6)
Induction Base:
half'(_gen_0':s'4(*(2, 0))) →RΩ(1)
0'
Induction Step:
half'(_gen_0':s'4(*(2, +(_$n7, 1)))) →RΩ(1)
s'(half'(_gen_0':s'4(*(2, _$n7)))) →IH
s'(_gen_0':s'4(_$n7))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x)))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lastbit' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
conv' :: 0':s' → cons'
conviter' :: 0':s' → cons' → cons'
cons' :: 0':s' → cons'
if' :: true':false' → 0':s' → cons' → cons'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_hole_cons'3 :: cons'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
half'(_gen_0':s'4(*(2, _n6))) → _gen_0':s'4(_n6), rt ∈ Ω(1 + n6)
Generator Equations:
_gen_0':s'4(0) ⇔ 0'
_gen_0':s'4(+(x, 1)) ⇔ s'(_gen_0':s'4(x))
The following defined symbols remain to be analysed:
lastbit', conviter'
They will be analysed ascendingly in the following order:
lastbit' < conviter'
Proved the following rewrite lemma:
lastbit'(_gen_0':s'4(*(2, _n465))) → _gen_0':s'4(0), rt ∈ Ω(1 + n465)
Induction Base:
lastbit'(_gen_0':s'4(*(2, 0))) →RΩ(1)
0'
Induction Step:
lastbit'(_gen_0':s'4(*(2, +(_$n466, 1)))) →RΩ(1)
lastbit'(_gen_0':s'4(*(2, _$n466))) →IH
_gen_0':s'4(0)
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))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x)))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lastbit' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
conv' :: 0':s' → cons'
conviter' :: 0':s' → cons' → cons'
cons' :: 0':s' → cons'
if' :: true':false' → 0':s' → cons' → cons'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_hole_cons'3 :: cons'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
half'(_gen_0':s'4(*(2, _n6))) → _gen_0':s'4(_n6), rt ∈ Ω(1 + n6)
lastbit'(_gen_0':s'4(*(2, _n465))) → _gen_0':s'4(0), rt ∈ Ω(1 + n465)
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:
conviter'
Could not prove a rewrite lemma for the defined symbol conviter'.
Rules:
half'(0') → 0'
half'(s'(0')) → 0'
half'(s'(s'(x))) → s'(half'(x))
lastbit'(0') → 0'
lastbit'(s'(0')) → s'(0')
lastbit'(s'(s'(x))) → lastbit'(x)
zero'(0') → true'
zero'(s'(x)) → false'
conv'(x) → conviter'(x, cons'(0'))
conviter'(x, l) → if'(zero'(x), x, l)
if'(true', x, l) → l
if'(false', x, l) → conviter'(half'(x), cons'(lastbit'(x)))
Types:
half' :: 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
lastbit' :: 0':s' → 0':s'
zero' :: 0':s' → true':false'
true' :: true':false'
false' :: true':false'
conv' :: 0':s' → cons'
conviter' :: 0':s' → cons' → cons'
cons' :: 0':s' → cons'
if' :: true':false' → 0':s' → cons' → cons'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_hole_cons'3 :: cons'
_gen_0':s'4 :: Nat → 0':s'
Lemmas:
half'(_gen_0':s'4(*(2, _n6))) → _gen_0':s'4(_n6), rt ∈ Ω(1 + n6)
lastbit'(_gen_0':s'4(*(2, _n465))) → _gen_0':s'4(0), rt ∈ Ω(1 + n465)
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:
half'(_gen_0':s'4(*(2, _n6))) → _gen_0':s'4(_n6), rt ∈ Ω(1 + n6)