| 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.
Proof.
intros x y H.
case H.
apply beq_refl.
Defined.
Definition beq_eq_not_false : forall x y:A, x = y -> false <> beq x y.
Proof.
intros x y e.
rewrite <- beq_eq_true; trivial; discriminate.
Defined.
Definition beq_false_not_eq : forall x y:A, false = beq x y -> x <> y.
Proof.
exact
(fun (x y:A) (H:false = beq x y) (e:x = y) => beq_eq_not_false x y e H).
Defined.
Definition exists_beq_eq : forall x y:A, {b : bool | b = beq x y}.
Proof.
intros.
exists (beq x y).
constructor.
Defined.
Definition not_eq_false_beq : forall x y:A, x <> y -> false = beq x y.
Proof.
intros x y H.
symmetry in |- *.
apply not_true_is_false.
intro.
apply H.
apply beq_eq.
symmetry in |- *.
assumption.
Defined.
Definition eq_dec : forall x y:A, {x = y} + {x <> y}.
Proof.
intros x y; case (exists_beq_eq x y).
intros b; case b; intro H.
left; apply beq_eq; assumption.
right; apply beq_false_not_eq; assumption.
Defined.
End Bool_eq_dec.