lt(x, 0) → false
lt(0, s(y)) → true
lt(s(x), s(y)) → lt(x, y)
plus(x, 0) → x
plus(x, s(y)) → s(plus(x, y))
quot(x, s(y)) → help(x, s(y), 0)
help(x, s(y), c) → if(lt(c, x), x, s(y), c)
if(true, x, s(y), c) → s(help(x, s(y), plus(c, s(y))))
if(false, x, s(y), c) → 0
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'
Infered types.
Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'
Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'
Heuristically decided to analyse the following defined symbols:
lt', plus', help'
They will be analysed ascendingly in the following order:
lt' < help'
plus' < help'
Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'
Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_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:
lt', plus', help'
They will be analysed ascendingly in the following order:
lt' < help'
plus' < help'
Proved the following rewrite lemma:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)
Induction Base:
lt'(_gen_0':s'3(0), _gen_0':s'3(0)) →RΩ(1)
false'
Induction Step:
lt'(_gen_0':s'3(+(_$n6, 1)), _gen_0':s'3(+(_$n6, 1))) →RΩ(1)
lt'(_gen_0':s'3(_$n6), _gen_0':s'3(_$n6)) →IH
false'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'
Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'
Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', 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:
plus', help'
They will be analysed ascendingly in the following order:
plus' < help'
Proved the following rewrite lemma:
plus'(_gen_0':s'3(a), _gen_0':s'3(_n584)) → _gen_0':s'3(+(_n584, a)), rt ∈ Ω(1 + n584)
Induction Base:
plus'(_gen_0':s'3(a), _gen_0':s'3(0)) →RΩ(1)
_gen_0':s'3(a)
Induction Step:
plus'(_gen_0':s'3(_a717), _gen_0':s'3(+(_$n585, 1))) →RΩ(1)
s'(plus'(_gen_0':s'3(_a717), _gen_0':s'3(_$n585))) →IH
s'(_gen_0':s'3(+(_$n585, _a717)))
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'
Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'
Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)
plus'(_gen_0':s'3(a), _gen_0':s'3(_n584)) → _gen_0':s'3(+(_n584, a)), rt ∈ Ω(1 + n584)
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:
help'
Could not prove a rewrite lemma for the defined symbol help'.
Rules:
lt'(x, 0') → false'
lt'(0', s'(y)) → true'
lt'(s'(x), s'(y)) → lt'(x, y)
plus'(x, 0') → x
plus'(x, s'(y)) → s'(plus'(x, y))
quot'(x, s'(y)) → help'(x, s'(y), 0')
help'(x, s'(y), c) → if'(lt'(c, x), x, s'(y), c)
if'(true', x, s'(y), c) → s'(help'(x, s'(y), plus'(c, s'(y))))
if'(false', x, s'(y), c) → 0'
Types:
lt' :: 0':s' → 0':s' → false':true'
0' :: 0':s'
false' :: false':true'
s' :: 0':s' → 0':s'
true' :: false':true'
plus' :: 0':s' → 0':s' → 0':s'
quot' :: 0':s' → 0':s' → 0':s'
help' :: 0':s' → 0':s' → 0':s' → 0':s'
if' :: false':true' → 0':s' → 0':s' → 0':s' → 0':s'
_hole_false':true'1 :: false':true'
_hole_0':s'2 :: 0':s'
_gen_0':s'3 :: Nat → 0':s'
Lemmas:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)
plus'(_gen_0':s'3(a), _gen_0':s'3(_n584)) → _gen_0':s'3(+(_n584, a)), rt ∈ Ω(1 + n584)
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:
lt'(_gen_0':s'3(_n5), _gen_0':s'3(_n5)) → false', rt ∈ Ω(1 + n5)