(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
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(*                           Group Theory in Coq                            *)
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(*                                Coq V5.10                                 *)
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(*			         Gilles Kahn 				    *)
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(*                                  INRIA                                   *)
(*                             Sophia-Antipolis                             *)
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(*				 January 1996				    *)
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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.

Theorem triv1 : forall a b : U, star (inv a) (star a b) = b.
intros a b; try assumption.