From Coq Require Import Bool.
From MetaCoq.Template Require Import MCPrelude.
Require Import ssreflect.
From Equations Require Import Equations.

Inductive reflectT (A : Type) : bool → Type :=
| ReflectT : A → reflectT A true
| ReflectF : (A → False) → reflectT A false.

Lemma reflectT_reflect (A : Prop) b : reflectT A b → reflect A b.
Proof.
  destruct 1; now constructor.
Qed.

Lemma reflect_reflectT (A : Prop) b : reflect A b → reflectT A b.
Proof.
  destruct 1; now constructor.
Qed.

Lemma equiv_reflectT P (b : bool) : (P → b) → (b → P) → reflectT P b.
Proof.
  intros. destruct b; constructor; auto.
  intros p; specialize (H p). discriminate.
Qed.

Lemma elimT {T} {b} : reflectT T b → b → T.
Proof. intros [] ⇒ //. Qed.
Coercion elimT : reflectT >-> Funclass.

Lemma introT {T} {b} : reflectT T b → T → b.
Proof. intros [] ⇒ //. Qed.

Hint View for move/ introT|2.

Lemma reflectT_subrelation {A} {R} {r : A → A → bool} : (∀ x y, reflectT (R x y) (r x y)) → CRelationClasses.subrelation R r.
Proof.
  intros. intros x y h. destruct (X x y); auto.
Qed.

Lemma reflectT_subrelation' {A} {R} {r : A → A → bool} : (∀ x y, reflectT (R x y) (r x y)) → CRelationClasses.subrelation r R.
Proof.
  intros. intros x y h. destruct (X x y); auto. discriminate.
Qed.

Lemma reflectT_pred {A} {p : A → bool} : ∀ x, reflectT (p x) (p x).
Proof.
  intros x. now apply equiv_reflectT.
Qed.

Lemma reflectT_pred2 {A B} {p : A → B → bool} : ∀ x y, reflectT (p x y) (p x y).
Proof.
  intros x y. now apply equiv_reflectT.
Qed.