From Hammer Require Import Hammer.














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. hammer_hook "BoolEq" "BoolEq.beq_eq_true".  
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. hammer_hook "BoolEq" "BoolEq.beq_eq_not_false".  
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. hammer_hook "BoolEq" "BoolEq.beq_false_not_eq".  
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. hammer_hook "BoolEq" "BoolEq.exists_beq_eq".  
intros.
exists (beq x y).
constructor.
Defined.

Definition not_eq_false_beq : forall x y:A, x <> y -> false = beq x y.
Proof. hammer_hook "BoolEq" "BoolEq.not_eq_false_beq".  
intros x y H.
symmetry .
apply not_true_is_false.
intro.
apply H.
apply beq_eq.
symmetry .
assumption.
Defined.

Definition eq_dec : forall x y:A, {x = y} + {x <> y}.
Proof. hammer_hook "BoolEq" "BoolEq.eq_dec".  
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.
