times(x, 0) → 0
times(x, s(y)) → plus(times(x, y), x)
plus(x, 0) → x
plus(0, x) → x
plus(x, s(y)) → s(plus(x, y))
plus(s(x), y) → s(plus(x, y))
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
times'(x, 0') → 0'
times'(x, s'(y)) → plus'(times'(x, y), x)
plus'(x, 0') → x
plus'(0', x) → x
plus'(x, s'(y)) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(x, y))
Infered types.
Rules:
times'(x, 0') → 0'
times'(x, s'(y)) → plus'(times'(x, y), x)
plus'(x, 0') → x
plus'(0', x) → x
plus'(x, s'(y)) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(x, y))
Types:
times' :: 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:
times', plus'
They will be analysed ascendingly in the following order:
plus' < times'
Rules:
times'(x, 0') → 0'
times'(x, s'(y)) → plus'(times'(x, y), x)
plus'(x, 0') → x
plus'(0', x) → x
plus'(x, s'(y)) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(x, y))
Types:
times' :: 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', times'
They will be analysed ascendingly in the following order:
plus' < times'
Proved the following rewrite lemma:
plus'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
Induction Base:
plus'(_gen_0':s'2(a), _gen_0':s'2(0)) →RΩ(1)
_gen_0':s'2(a)
Induction Step:
plus'(_gen_0':s'2(_a181), _gen_0':s'2(+(_$n5, 1))) →RΩ(1)
s'(plus'(_gen_0':s'2(_a181), _gen_0':s'2(_$n5))) →IH
s'(_gen_0':s'2(+(_$n5, _a181)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
times'(x, 0') → 0'
times'(x, s'(y)) → plus'(times'(x, y), x)
plus'(x, 0') → x
plus'(0', x) → x
plus'(x, s'(y)) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(x, y))
Types:
times' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
plus' :: 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_gen_0':s'2 :: Nat → 0':s'
Lemmas:
plus'(_gen_0':s'2(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
Generator Equations:
_gen_0':s'2(0) ⇔ 0'
_gen_0':s'2(+(x, 1)) ⇔ s'(_gen_0':s'2(x))
The following defined symbols remain to be analysed:
times'
Proved the following rewrite lemma:
times'(_gen_0':s'2(a), _gen_0':s'2(_n533)) → _gen_0':s'2(*(_n533, a)), rt ∈ Ω(1 + a793·n533 + n533)
Induction Base:
times'(_gen_0':s'2(a), _gen_0':s'2(0)) →RΩ(1)
0'
Induction Step:
times'(_gen_0':s'2(_a793), _gen_0':s'2(+(_$n534, 1))) →RΩ(1)
plus'(times'(_gen_0':s'2(_a793), _gen_0':s'2(_$n534)), _gen_0':s'2(_a793)) →IH
plus'(_gen_0':s'2(*(_$n534, _a793)), _gen_0':s'2(_a793)) →LΩ(1 + a793)
_gen_0':s'2(+(_a793, *(_$n534, _a793)))
We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).
Rules:
times'(x, 0') → 0'
times'(x, s'(y)) → plus'(times'(x, y), x)
plus'(x, 0') → x
plus'(0', x) → x
plus'(x, s'(y)) → s'(plus'(x, y))
plus'(s'(x), y) → s'(plus'(x, y))
Types:
times' :: 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(a), _gen_0':s'2(_n4)) → _gen_0':s'2(+(_n4, a)), rt ∈ Ω(1 + n4)
times'(_gen_0':s'2(a), _gen_0':s'2(_n533)) → _gen_0':s'2(*(_n533, a)), rt ∈ Ω(1 + a793·n533 + n533)
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(n2) was proven with the following lemma:
times'(_gen_0':s'2(a), _gen_0':s'2(_n533)) → _gen_0':s'2(*(_n533, a)), rt ∈ Ω(1 + a793·n533 + n533)