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