# Library Coq.Bool.BoolEq

Properties of a boolean equality
Require Export Bool.
Section Bool_eq_dec.
Variable A : Set.
Variable beq : A -> A -> bool.
Variable beq_refl : forall x:A, true = beq x x.
Variable beq_eq : forall x y:A, true = beq x y -> x = y.
Definition beq_eq_true : forall x y:A, x = y -> true = beq x y.
Definition beq_eq_not_false : forall x y:A, x = y -> false <> beq x y.
Definition beq_false_not_eq : forall x y:A, false = beq x y -> x <> y.
Definition exists_beq_eq : forall x y:A, {b : bool | b = beq x y}.
Definition not_eq_false_beq : forall x y:A, x <> y -> false = beq x y.
Definition eq_dec : forall x y:A, {x = y} + {x <> y}.
End Bool_eq_dec.