Merge PR #12162: Fixing #12161: rename Bool.leb into Bool.le

Ack-by: Zimmi48
Reviewed-by: anton-trunov

1 files changed, 19 insertions, 23 deletions

diff --git a/theories/Bool/BoolOrder.v b/theories/Bool/BoolOrder.v
index 61aab607a9..aaa7321bfc 100644
--- a/theories/Bool/BoolOrder.v
+++ b/theories/Bool/BoolOrder.v
@@ -14,69 +14,65 @@
 
 Require Export Bool.
 Require Import Orders.
-
-Local Notation le := Bool.leb.
-Local Notation lt := Bool.ltb.
-Local Notation compare := Bool.compareb.
-Local Notation compare_spec := Bool.compareb_spec.
+Import BoolNotations.
 
 (** * Order [le] *)
 
-Lemma le_refl : forall b, le b b.
+Lemma le_refl : forall b, b <= b.
 Proof. destr_bool. Qed.
 
 Lemma le_trans : forall b1 b2 b3,
-  le b1 b2 -> le b2 b3 -> le b1 b3.
+  b1 <= b2 -> b2 <= b3 -> b1 <= b3.
 Proof. destr_bool. Qed.
 
-Lemma le_true : forall b, le b true.
+Lemma le_true : forall b, b <= true.
 Proof. destr_bool. Qed.
 
-Lemma false_le : forall b, le false b.
+Lemma false_le : forall b, false <= b.
 Proof. intros; constructor. Qed.
 
-Instance le_compat : Proper (eq ==> eq ==> iff) le.
+Instance le_compat : Proper (eq ==> eq ==> iff) Bool.le.
 Proof. intuition. Qed.
 
 (** * Strict order [lt] *)
 
-Lemma lt_irrefl : forall b, ~ lt b b.
+Lemma lt_irrefl : forall b, ~ b < b.
 Proof. destr_bool; auto. Qed.
 
 Lemma lt_trans : forall b1 b2 b3,
-  lt b1 b2 -> lt b2 b3 -> lt b1 b3.
+  b1 < b2 -> b2 < b3 -> b1 < b3.
 Proof. destr_bool; auto. Qed.
 
-Instance lt_compat : Proper (eq ==> eq ==> iff) lt.
+Instance lt_compat : Proper (eq ==> eq ==> iff) Bool.lt.
 Proof. intuition. Qed.
 
-Lemma lt_trichotomy : forall b1 b2, { lt b1 b2 } + { b1 = b2 } + { lt b2 b1 }.
+Lemma lt_trichotomy : forall b1 b2, { b1 < b2 } + { b1 = b2 } + { b2 < b1 }.
 Proof. destr_bool; auto. Qed.
 
-Lemma lt_total : forall b1 b2, lt b1 b2 \/ b1 = b2 \/ lt b2 b1.
+Lemma lt_total : forall b1 b2, b1 < b2 \/ b1 = b2 \/ b2 < b1.
 Proof. destr_bool; auto. Qed.
 
-Lemma lt_le_incl : forall b1 b2, lt b1 b2 -> le b1 b2.
+Lemma lt_le_incl : forall b1 b2, b1 < b2 -> b1 <= b2.
 Proof. destr_bool; auto. Qed.
 
-Lemma le_lteq_dec : forall b1 b2, le b1 b2 -> { lt b1 b2 } + { b1 = b2 }.
+Lemma le_lteq_dec : forall b1 b2, b1 <= b2 -> { b1 < b2 } + { b1 = b2 }.
 Proof. destr_bool; auto. Qed.
 
-Lemma le_lteq : forall b1 b2, le b1 b2 <-> lt b1 b2 \/ b1 = b2.
+Lemma le_lteq : forall b1 b2, b1 <= b2 <-> b1 < b2 \/ b1 = b2.
 Proof. destr_bool; intuition. Qed.
 
 
 (** * Order structures *)
 
 (* Class structure *)
-Instance le_preorder : PreOrder le.
+Instance le_preorder : PreOrder Bool.le.
 Proof.
 split.
 - intros b; apply le_refl.
 - intros b1 b2 b3; apply le_trans.
 Qed.
 
-Instance lt_strorder : StrictOrder lt.
+Instance lt_strorder : StrictOrder Bool.lt.
 Proof.
 split.
 - intros b; apply lt_irrefl.
@@ -88,13 +84,13 @@ Module BoolOrd <: UsualDecidableTypeFull <: OrderedTypeFull <: TotalOrder.
   Definition t := bool.
   Definition eq := @eq bool.
   Definition eq_equiv := @eq_equivalence bool.
-  Definition lt := lt.
+  Definition lt := Bool.lt.
   Definition lt_strorder := lt_strorder.
   Definition lt_compat := lt_compat.
-  Definition le := le.
+  Definition le := Bool.le.
   Definition le_lteq := le_lteq.
   Definition lt_total := lt_total.
-  Definition compare := compare.
+  Definition compare := Bool.compare.
   Definition compare_spec := compare_spec.
   Definition eq_dec := bool_dec.
   Definition eq_refl := @eq_Reflexive bool.