Require Export Bool.
Require Import Orders.
Import BoolNotations.

Lemma le_refl : forall b, b <= b.

Lemma le_trans : forall b1 b2 b3,
  b1 <= b2 -> b2 <= b3 -> b1 <= b3.

Lemma le_true : forall b, b <= true.

Lemma false_le : forall b, false <= b.

#[global]
Instance le_compat : Proper (eq ==> eq ==> iff) Bool.le.

Lemma lt_irrefl : forall b, ~ b < b.

Lemma lt_trans : forall b1 b2 b3,
  b1 < b2 -> b2 < b3 -> b1 < b3.

#[global]
Instance lt_compat : Proper (eq ==> eq ==> iff) Bool.lt.

Lemma lt_trichotomy : forall b1 b2, { b1 < b2 } + { b1 = b2 } + { b2 < b1 }.

Lemma lt_total : forall b1 b2, b1 < b2 \/ b1 = b2 \/ b2 < b1.

Lemma lt_le_incl : forall b1 b2, b1 < b2 -> b1 <= b2.

Lemma le_lteq_dec : forall b1 b2, b1 <= b2 -> { b1 < b2 } + { b1 = b2 }.

Lemma le_lteq : forall b1 b2, b1 <= b2 <-> b1 < b2 \/ b1 = b2.

#[global]
Instance le_preorder : PreOrder Bool.le.

#[global]
Instance lt_strorder : StrictOrder Bool.lt.

Module BoolOrd <: UsualDecidableTypeFull <: OrderedTypeFull <: TotalOrder.
  Definition t := bool.
  Definition eq := @eq bool.
  Definition eq_equiv := @eq_equivalence bool.
  Definition lt := Bool.lt.
  Definition lt_strorder := lt_strorder.
  Definition lt_compat := lt_compat.
  Definition le := Bool.le.
  Definition le_lteq := le_lteq.
  Definition lt_total := lt_total.
  Definition compare := Bool.compare.
  Definition compare_spec := compare_spec.
  Definition eq_dec := bool_dec.
  Definition eq_refl := @eq_Reflexive bool.
  Definition eq_sym := @eq_Symmetric bool.
  Definition eq_trans := @eq_Transitive bool.
  Definition eqb := eqb.
  Definition eqb_eq := eqb_true_iff.
End BoolOrd.