(**********************************************************************)
(** * Definitions *)

(** Being an inverse *)

Definition is_inverse A B f g := (forall a:A, g (f a) = a) /\ (forall b:B, f (g b) = b).(** The diagonal over A and the one-one correspondence with A *)

#[universes(template)]
Record Delta A := { pi1:A; pi2:A; eq:pi1=pi2 }.(** The diagonal over A and the one-one correspondence with A *)

#[universes(template)]
Record Delta A := { pi1:A; pi2:A; eq:pi1=pi2 }.(** The diagonal over A and the one-one correspondence with A *)

#[universes(template)]
Record Delta A := { pi1:A; pi2:A; eq:pi1=pi2 }.(** The diagonal over A and the one-one correspondence with A *)

#[universes(template)]
Record Delta A := { pi1:A; pi2:A; eq:pi1=pi2 }.Definition delta {A} (a:A) := {| pi1 := a; pi2 := a; eq := eq_refl a |}.

Lemma diagonal_inverse1 : forall A, is_inverse (A:=A) delta pi1.