Library Coq.Bool.BoolEq

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.