cond(true, x, y) → cond(gr(x, y), p(x), s(y))
gr(0, x) → false
gr(s(x), 0) → true
gr(s(x), s(y)) → gr(x, y)
p(0) → 0
p(s(x)) → x
Renamed function symbols to avoid clashes with predefined symbol.
Runtime Complexity TRS:
The TRS R consists of the following rules:
cond'(true', x, y) → cond'(gr'(x, y), p'(x), s'(y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
Infered types.
Rules:
cond'(true', x, y) → cond'(gr'(x, y), p'(x), s'(y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → s':0' → s':0' → cond'
true' :: true':false'
gr' :: s':0' → s':0' → true':false'
p' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'
Heuristically decided to analyse the following defined symbols:
cond', gr'
They will be analysed ascendingly in the following order:
gr' < cond'
Rules:
cond'(true', x, y) → cond'(gr'(x, y), p'(x), s'(y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → s':0' → s':0' → cond'
true' :: true':false'
gr' :: s':0' → s':0' → true':false'
p' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'
Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))
The following defined symbols remain to be analysed:
gr', cond'
They will be analysed ascendingly in the following order:
gr' < cond'
Proved the following rewrite lemma:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)
Induction Base:
gr'(_gen_s':0'4(0), _gen_s':0'4(0)) →RΩ(1)
false'
Induction Step:
gr'(_gen_s':0'4(+(_$n7, 1)), _gen_s':0'4(+(_$n7, 1))) →RΩ(1)
gr'(_gen_s':0'4(_$n7), _gen_s':0'4(_$n7)) →IH
false'
We have rt ∈ Ω(n) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
Rules:
cond'(true', x, y) → cond'(gr'(x, y), p'(x), s'(y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → s':0' → s':0' → cond'
true' :: true':false'
gr' :: s':0' → s':0' → true':false'
p' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'
Lemmas:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)
Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))
The following defined symbols remain to be analysed:
cond'
Could not prove a rewrite lemma for the defined symbol cond'.
Rules:
cond'(true', x, y) → cond'(gr'(x, y), p'(x), s'(y))
gr'(0', x) → false'
gr'(s'(x), 0') → true'
gr'(s'(x), s'(y)) → gr'(x, y)
p'(0') → 0'
p'(s'(x)) → x
Types:
cond' :: true':false' → s':0' → s':0' → cond'
true' :: true':false'
gr' :: s':0' → s':0' → true':false'
p' :: s':0' → s':0'
s' :: s':0' → s':0'
0' :: s':0'
false' :: true':false'
_hole_cond'1 :: cond'
_hole_true':false'2 :: true':false'
_hole_s':0'3 :: s':0'
_gen_s':0'4 :: Nat → s':0'
Lemmas:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)
Generator Equations:
_gen_s':0'4(0) ⇔ 0'
_gen_s':0'4(+(x, 1)) ⇔ s'(_gen_s':0'4(x))
No more defined symbols left to analyse.
The lowerbound Ω(n) was proven with the following lemma:
gr'(_gen_s':0'4(_n6), _gen_s':0'4(_n6)) → false', rt ∈ Ω(1 + n6)