minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, s(y)) → if_quot(le(s(y), x), x, s(y))
if_quot(true, x, y) → s(quot(minus(x, y), y))
if_quot(false, x, y) → 0
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, s'(y)) → if_quot'(le'(s'(y), x), x, s'(y))
if_quot'(true', x, y) → s'(quot'(minus'(x, y), y))
if_quot'(false', x, y) → 0'
Infered types.
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, s'(y)) → if_quot'(le'(s'(y), x), x, s'(y))
if_quot'(true', x, y) → s'(quot'(minus'(x, y), y))
if_quot'(false', x, y) → 0'
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
quot' :: 0':s' → 0':s' → 0':s'
if_quot' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
minus', le', quot'
They will be analysed ascendingly in the following order:
minus' < quot'
le' < quot'
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, s'(y)) → if_quot'(le'(s'(y), x), x, s'(y))
if_quot'(true', x, y) → s'(quot'(minus'(x, y), y))
if_quot'(false', x, y) → 0'
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
quot' :: 0':s' → 0':s' → 0':s'
if_quot' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
The following defined symbols remain to be analysed:
minus', le', quot'
They will be analysed ascendingly in the following order:
minus' < quot'
le' < quot'
Proved the following rewrite lemma:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
Induction Base:
minus'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
_gen_0':s'3(0)
Induction Step:
minus'(_gen_0':s'3(+(_$n6, 1)), _gen_0':s'3(+(_$n6, 1))) →RΩ(1)
minus'(_gen_0':s'3(_$n6), _gen_0':s'3(_$n6)) →IH
_gen_0':s'3(0)
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, s'(y)) → if_quot'(le'(s'(y), x), x, s'(y))
if_quot'(true', x, y) → s'(quot'(minus'(x, y), y))
if_quot'(false', x, y) → 0'
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
quot' :: 0':s' → 0':s' → 0':s'
if_quot' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'
Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
The following defined symbols remain to be analysed:
le', quot'
They will be analysed ascendingly in the following order:
le' < quot'
Proved the following rewrite lemma:
le'(_gen_0':s'3(_n519), _gen_0':s'3(_n519)) → true', rt ∈ Ω(1 + n519)
Induction Base:
le'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
true'
Induction Step:
le'(_gen_0':s'3(+(_$n520, 1)), _gen_0':s'3(+(_$n520, 1))) →RΩ(1)
le'(_gen_0':s'3(_$n520), _gen_0':s'3(_$n520)) →IH
true'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, s'(y)) → if_quot'(le'(s'(y), x), x, s'(y))
if_quot'(true', x, y) → s'(quot'(minus'(x, y), y))
if_quot'(false', x, y) → 0'
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
quot' :: 0':s' → 0':s' → 0':s'
if_quot' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'
Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
le'(_gen_0':s'3(_n519), _gen_0':s'3(_n519)) → true', rt ∈ Ω(1 + n519)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
The following defined symbols remain to be analysed:
quot'
Could not prove a rewrite lemma for the defined symbol quot'.
Rules:
minus'(x, 0') → x
minus'(s'(x), s'(y)) → minus'(x, y)
le'(0', y) → true'
le'(s'(x), 0') → false'
le'(s'(x), s'(y)) → le'(x, y)
quot'(x, s'(y)) → if_quot'(le'(s'(y), x), x, s'(y))
if_quot'(true', x, y) → s'(quot'(minus'(x, y), y))
if_quot'(false', x, y) → 0'
Types:
minus' :: 0':s' → 0':s' → 0':s'
0' :: 0':s'
s' :: 0':s' → 0':s'
le' :: 0':s' → 0':s' → true':false'
true' :: true':false'
false' :: true':false'
quot' :: 0':s' → 0':s' → 0':s'
if_quot' :: true':false' → 0':s' → 0':s' → 0':s'
_hole_0':s'1 :: 0':s'
_hole_true':false'2 :: true':false'
_gen_0':s'3 :: Nat → 0':s'
Lemmas:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)
le'(_gen_0':s'3(_n519), _gen_0':s'3(_n519)) → true', rt ∈ Ω(1 + n519)
Generator Equations:
_gen_0':s'3(0) ⇔ 0'
_gen_0':s'3(+(x, 1)) ⇔ s'(_gen_0':s'3(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
minus'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → _gen_0':s'3(0), rt ∈ Ω(1 + n5)