quot(0, s(y), s(z)) → 0
quot(s(x), s(y), z) → quot(x, y, z)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
quot(x, 0, s(z)) → s(quot(x, plus(z, s(0)), s(z)))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
quot'(0', s'(y), s'(z)) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
quot'(x, 0', s'(z)) → s'(quot'(x, plus'(z, s'(0')), s'(z)))
Infered types.
Rules:
quot'(0', s'(y), s'(z)) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
quot'(x, 0', s'(z)) → s'(quot'(x, plus'(z, s'(0')), s'(z)))
Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
plus' :: 0':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:
quot', plus'
They will be analysed ascendingly in the following order:
plus' < quot'
Rules:
quot'(0', s'(y), s'(z)) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
quot'(x, 0', s'(z)) → s'(quot'(x, plus'(z, s'(0')), s'(z)))
Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
plus' :: 0':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:
plus', quot'
They will be analysed ascendingly in the following order:
plus' < quot'
Proved the following rewrite lemma:
plus'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)
Induction Base:
plus'(_gen_0':s'2(0), _gen_0':s'2(b)) →RΩ(1)
_gen_0':s'2(b)
Induction Step:
plus'(_gen_0':s'2(+(_$n5, 1)), _gen_0':s'2(_b137)) →RΩ(1)
s'(plus'(_gen_0':s'2(_$n5), _gen_0':s'2(_b137))) →IH
s'(_gen_0':s'2(+(_$n5, _b137)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
quot'(0', s'(y), s'(z)) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
quot'(x, 0', s'(z)) → s'(quot'(x, plus'(z, s'(0')), s'(z)))
Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
The following defined symbols remain to be analysed:
quot'
Proved the following rewrite lemma:
quot'(_gen_0':s'2(_n486), _gen_0':s'2(+(1, _n486)), _gen_0':s'2(1)) → _gen_0':s'2(0), rt ∈ Ω(1 + n486)
Induction Base:
quot'(_gen_0':s'2(0), _gen_0':s'2(+(1, 0)), _gen_0':s'2(1)) →RΩ(1)
0'
Induction Step:
quot'(_gen_0':s'2(+(_$n487, 1)), _gen_0':s'2(+(1, +(_$n487, 1))), _gen_0':s'2(1)) →RΩ(1)
quot'(_gen_0':s'2(_$n487), _gen_0':s'2(+(1, _$n487)), _gen_0':s'2(1)) →IH
_gen_0':s'2(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
quot'(0', s'(y), s'(z)) → 0'
quot'(s'(x), s'(y), z) → quot'(x, y, z)
plus'(0', y) → y
plus'(s'(x), y) → s'(plus'(x, y))
quot'(x, 0', s'(z)) → s'(quot'(x, plus'(z, s'(0')), s'(z)))
Types:
quot' :: 0':s' → 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)
quot'(_gen_0':s'2(_n486), _gen_0':s'2(+(1, _n486)), _gen_0':s'2(1)) → _gen_0':s'2(0), rt ∈ Ω(1 + n486)
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:
plus'(_gen_0':s'2(_n4), _gen_0':s'2(b)) → _gen_0':s'2(+(_n4, b)), rt ∈ Ω(1 + n4)