Section Examples.
Let ex1 : forall x y z : Q, (x+y)*z ==  (x*z)+(y*z).
intros.
ring.
Qed.
Let ex2 : forall x y : Q, x+y == y+x.
intros.
ring.
Qed.
Let ex3 : forall x y z : Q, (x+y)+z == x+(y+z).
intros.
ring.
Qed.
Let ex4 : (inject_Z 1)+(inject_Z 1)==(inject_Z 2).
ring.
Qed.
Let ex5 : 1+1 == 2#1.
ring.
Qed.
Let ex6 : (1#1)+(1#1) == 2#1.
ring.
Qed.
Let ex7 : forall x : Q, x-x== 0.
intro.
ring.
Qed.
Let ex8 : forall x : Q, x^1 == x.
intro.
ring.
Qed.
Let ex9 : forall x : Q, x^0 == 1.
intro.
ring.
Qed.
Let ex10 : forall x y : Q, ~(y==0) -> (x/y)*y == x.
intros.
field.
auto.
Qed.
End Examples.

Lemma Qopp_plus : forall a b,  -(a+b) == -a + -b.