Section group_trivialities.
Variable U : Type.
Variable Gr : Group U.
Let G : Ensemble U := G_ U Gr.
Let star : U -> U -> U := star_ U Gr.
Let inv : U -> U := inv_ U Gr.
Let e : U := e_ U Gr.
Definition G0 : forall a b : U, In U G a -> In U G b -> In U G (star a b) :=
G0_ U Gr.
Definition G1 : forall a b c : U, star a (star b c) = star (star a b) c :=
G1_ U Gr.
Definition G2a : In U G e := G2a_ U Gr.
Definition G2b : forall a : U, star e a = a := G2b_ U Gr.
Definition G2c : forall a : U, star a e = a := G2c_ U Gr.
Definition G3a : forall a : U, In U G a -> In U G (inv a) := G3a_ U Gr.
Definition G3b : forall a : U, star a (inv a) = e := G3b_ U Gr.
Definition G3c : forall a : U, star (inv a) a = e := G3c_ U Gr.
Hint Resolve G1.
Hint Resolve G2a G2b G2c.
Hint Resolve G3a G3b G3c.
Hint Resolve G0.
intros a b; try assumption.
rewrite (G1 (inv a) a b); auto.
rewrite G3c; auto.
Qed.
intros a b; try assumption.
rewrite <- (G1 b a (inv a)); auto.
rewrite (G3b a); auto.
Qed.
intros a b H'1.
cut (star (star b a) (inv a) = inv a).
rewrite <- (G1 b a (inv a)); auto.
rewrite (G3b a); auto.
rewrite (G2c b); auto.
rewrite H'1.
rewrite (G2b (inv a)); auto.
Qed.
apply resolve; auto.
Qed.
intros a b.
apply resolve.
rewrite <- (G1 (inv b) (inv a) (star a b)).
rewrite (G1 (inv a) a b).
rewrite (G3c a).
rewrite (G2b b); auto.
Qed.
intros a b H'.
cut (star (inv a) (star a b) = b).
rewrite H'.
rewrite (G3c a); auto.
rewrite (G1 (inv a) a b).
rewrite (G3c a); auto.
Qed.

Theorem inv_involution : forall a : U, a = inv (inv a).