Exercice : Autant de a que de b est un langage irrégulier

a b
Σ = {a, b}
L=={ω∈Σ∗| |ω|a=|ω|b}L
a b
n ω ∈L|ω|a=|ω|b|ω| ≥ n
ω=an.bn=a. . . . .a
| {z }
n
.b. . . . .b
| {z }
n
|ω|= 2n≥n
ω xyz |x.y| ≤ n x y
a x =aX, y =ayz=aZ1.bnY > 0y6=
k= 2 m=x.y2.z =aX.a2y.az1.bn
|m|a=X+ 2Y+Z1=X+Y+Z1
| {z }
n
+Y=n+Y|m|b=n
|ma|
|{z }
n+Y
6=|m|b
|{z}
n
Y > 0
m L L
L<={ω∈Σ∗| |ω|a<|ω|b}L>={ω∈Σ∗| |ω|a>|ω|b}
L<∪L>={ω∈Σ∗| |ω|a6=|ω|b}=L=
L<∪L>
L<L>A<
A>L<∪L>A<+A>
L<∪L>
L<L>
L<L>
L<L>m:a7→
b, b 7→ a
(1) m(L<) = L>
(2) L<=m(L<)
m(L) =⇒L
L<L<L>
L>
L<, L>
L<L>