1 files changed, 142 insertions, 88 deletions

diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v
index e6cb5be4..8fc92579 100644
--- a/theories/Arith/Compare_dec.v
+++ b/theories/Arith/Compare_dec.v
@@ -6,7 +6,7 @@
 (*         *       GNU Lesser General Public License Version 2.1        *)
 (************************************************************************)
 
-(*i $Id: Compare_dec.v 10295 2007-11-06 22:46:21Z letouzey $ i*)
+(*i $Id$ i*)
 
 Require Import Le.
 Require Import Lt.
@@ -18,20 +18,24 @@ Open Local Scope nat_scope.
 Implicit Types m n x y : nat.
 
 Definition zerop n : {n = 0} + {0 < n}.
+Proof.
   destruct n; auto with arith.
 Defined.
 
-Definition lt_eq_lt_dec n m : {n < m} + {n = m} + {m < n}.
-  induction n; simple destruct m; auto with arith.
-  intros m0; elim (IHn m0); auto with arith.
-  induction 1; auto with arith.
+Definition lt_eq_lt_dec : forall n m, {n < m} + {n = m} + {m < n}.
+Proof.
+  induction n; destruct m; auto with arith.
+  destruct (IHn m) as [H|H]; auto with arith.
+  destruct H; auto with arith.
 Defined.
 
-Definition gt_eq_gt_dec n m : {m > n} + {n = m} + {n > m}.
-  exact lt_eq_lt_dec.
+Definition gt_eq_gt_dec : forall n m, {m > n} + {n = m} + {n > m}.
+Proof.
+  intros; apply lt_eq_lt_dec; assumption.
 Defined.
 
-Definition le_lt_dec n m : {n <= m} + {m < n}.
+Definition le_lt_dec : forall n m, {n <= m} + {m < n}.
+Proof.
   induction n.
   auto with arith.
   destruct m.
@@ -40,43 +44,68 @@ Definition le_lt_dec n m : {n <= m} + {m < n}.
 Defined.
 
 Definition le_le_S_dec n m : {n <= m} + {S m <= n}.
-  exact le_lt_dec.
+Proof.
+  intros; exact (le_lt_dec n m).
 Defined.
 
 Definition le_ge_dec n m : {n <= m} + {n >= m}.
+Proof.
   intros; elim (le_lt_dec n m); auto with arith.
 Defined.
 
 Definition le_gt_dec n m : {n <= m} + {n > m}.
-  exact le_lt_dec.
+Proof.
+  intros; exact (le_lt_dec n m).
 Defined.
 
 Definition le_lt_eq_dec n m : n <= m -> {n < m} + {n = m}.
-  intros; elim (lt_eq_lt_dec n m); auto with arith.
+Proof.
+  intros; destruct (lt_eq_lt_dec n m); auto with arith.
   intros; absurd (m < n); auto with arith.
 Defined.
 
+Theorem le_dec : forall n m, {n <= m} + {~ n <= m}.
+Proof.
+  intros n m. destruct (le_gt_dec n m).
+   auto with arith.
+   right. apply gt_not_le. assumption.
+Defined.
+
+Theorem lt_dec : forall n m, {n < m} + {~ n < m}.
+Proof.
+  intros; apply le_dec.
+Defined.
+
+Theorem gt_dec : forall n m, {n > m} + {~ n > m}.
+Proof.
+  intros; apply lt_dec.
+Defined.
+
+Theorem ge_dec : forall n m, {n >= m} + {~ n >= m}.
+Proof.
+  intros; apply le_dec.
+Defined.
+
 (** Proofs of decidability *)
 
 Theorem dec_le : forall n m, decidable (n <= m).
 Proof.
-  intros x y; unfold decidable in |- *; elim (le_gt_dec x y);
-    [ auto with arith | intro; right; apply gt_not_le; assumption ].
+  intros n m; destruct (le_dec n m); unfold decidable; auto.
 Qed.
 
 Theorem dec_lt : forall n m, decidable (n < m).
 Proof.
-  intros x y; unfold lt in |- *; apply dec_le.
+  intros; apply dec_le.
 Qed.
 
 Theorem dec_gt : forall n m, decidable (n > m).
 Proof.
-  intros x y; unfold gt in |- *; apply dec_lt.
+  intros; apply dec_lt.
 Qed.
 
 Theorem dec_ge : forall n m, decidable (n >= m).
 Proof.
-  intros x y; unfold ge in |- *; apply dec_le.
+  intros; apply dec_le.
 Qed.
 
 Theorem not_eq : forall n m, n <> m -> n < m \/ m < n.
@@ -107,86 +136,111 @@ Qed.
 
 Theorem not_lt : forall n m, ~ n < m -> n >= m.
 Proof.
-  intros x y H; exact (not_gt y x H). 
+  intros x y H; exact (not_gt y x H).
 Qed.
 
 
 (** A ternary comparison function in the spirit of [Zcompare]. *)
 
-Definition nat_compare (n m:nat) :=
-  match lt_eq_lt_dec n m with 
-    | inleft (left _) => Lt
-    | inleft (right _) => Eq
-    | inright _ => Gt
+Fixpoint nat_compare n m :=
+  match n, m with
+   | O, O => Eq
+   | O, S _ => Lt
+   | S _, O => Gt
+   | S n', S m' => nat_compare n' m'
   end.
 
 Lemma nat_compare_S : forall n m, nat_compare (S n) (S m) = nat_compare n m.
 Proof.
-  unfold nat_compare; intros.
-  simpl; destruct (lt_eq_lt_dec n m) as [[H|H]|H]; simpl; auto.
+ reflexivity.
+Qed.
+
+Lemma nat_compare_eq_iff : forall n m, nat_compare n m = Eq <-> n = m.
+Proof.
+  induction n; destruct m; simpl; split; auto; try discriminate;
+   destruct (IHn m); auto.
 Qed.
 
 Lemma nat_compare_eq : forall n m, nat_compare n m = Eq -> n = m.
 Proof.
-  induction n; destruct m; simpl; auto.
-  unfold nat_compare; destruct (lt_eq_lt_dec 0 (S m)) as [[H|H]|H]; 
-    auto; intros; try discriminate.
-  unfold nat_compare; destruct (lt_eq_lt_dec (S n) 0) as [[H|H]|H]; 
-    auto; intros; try discriminate.
-  rewrite nat_compare_S; auto.
+  intros; apply -> nat_compare_eq_iff; auto.
 Qed.
 
 Lemma nat_compare_lt : forall n m, n<m <-> nat_compare n m = Lt.
 Proof.
-  induction n; destruct m; simpl.
-  unfold nat_compare; simpl; intuition; [inversion H | discriminate H].
-  split; auto with arith.
-  split; [inversion 1 |].
-  unfold nat_compare; destruct (lt_eq_lt_dec (S n) 0) as [[H|H]|H]; 
-    auto; intros; try discriminate.
-  rewrite nat_compare_S.
-  generalize (IHn m); clear IHn; intuition.
+  induction n; destruct m; simpl; split; auto with arith;
+   try solve [inversion 1].
+  destruct (IHn m); auto with arith.
+  destruct (IHn m); auto with arith.
 Qed.
 
 Lemma nat_compare_gt : forall n m, n>m <-> nat_compare n m = Gt.
 Proof.
-  induction n; destruct m; simpl.
-  unfold nat_compare; simpl; intuition; [inversion H | discriminate H].
-  split; [inversion 1 |].
-  unfold nat_compare; destruct (lt_eq_lt_dec 0 (S m)) as [[H|H]|H]; 
-    auto; intros; try discriminate.
-  split; auto with arith.
-  rewrite nat_compare_S.
-  generalize (IHn m); clear IHn; intuition.
+  induction n; destruct m; simpl; split; auto with arith;
+   try solve [inversion 1].
+  destruct (IHn m); auto with arith.
+  destruct (IHn m); auto with arith.
 Qed.
 
 Lemma nat_compare_le : forall n m, n<=m <-> nat_compare n m <> Gt.
 Proof.
   split.
-  intros.
-  intro.
-  destruct (nat_compare_gt n m).
-  generalize (le_lt_trans _ _ _ H (H2 H0)).
-  exact (lt_irrefl n).
-  intros.
-  apply not_gt.
-  contradict H.
-  destruct (nat_compare_gt n m); auto.
-Qed.  
+  intros LE; contradict LE.
+   apply lt_not_le. apply <- nat_compare_gt; auto.
+  intros NGT. apply not_lt. contradict NGT.
+   apply -> nat_compare_gt; auto.
+Qed.
 
 Lemma nat_compare_ge : forall n m, n>=m <-> nat_compare n m <> Lt.
 Proof.
   split.
-  intros.
-  intro.
-  destruct (nat_compare_lt n m).
-  generalize (le_lt_trans _ _ _ H (H2 H0)).
-  exact (lt_irrefl m).
-  intros.
-  apply not_lt.
-  contradict H.
-  destruct (nat_compare_lt n m); auto.
-Qed.  
+  intros GE; contradict GE.
+   apply lt_not_le. apply <- nat_compare_lt; auto.
+  intros NLT. apply not_lt. contradict NLT.
+   apply -> nat_compare_lt; auto.
+Qed.
+
+Lemma nat_compare_spec : forall x y, CompSpec eq lt x y (nat_compare x y).
+Proof.
+ intros.
+ destruct (nat_compare x y) as [ ]_eqn; constructor.
+ apply nat_compare_eq; auto.
+ apply <- nat_compare_lt; auto.
+ apply <- nat_compare_gt; auto.
+Qed.
+
+
+(** Some projections of the above equivalences. *)
+
+Lemma nat_compare_Lt_lt : forall n m, nat_compare n m = Lt -> n<m.
+Proof.
+  intros; apply <- nat_compare_lt; auto.
+Qed.
+
+Lemma nat_compare_Gt_gt : forall n m, nat_compare n m = Gt -> n>m.
+Proof.
+  intros; apply <- nat_compare_gt; auto.
+Qed.
+
+(** A previous definition of [nat_compare] in terms of [lt_eq_lt_dec].
+    The new version avoids the creation of proof parts. *)
+
+Definition nat_compare_alt (n m:nat) :=
+  match lt_eq_lt_dec n m with
+    | inleft (left _) => Lt
+    | inleft (right _) => Eq
+    | inright _ => Gt
+  end.
+
+Lemma nat_compare_equiv: forall n m,
+ nat_compare n m = nat_compare_alt n m.
+Proof.
+  intros; unfold nat_compare_alt; destruct lt_eq_lt_dec as [[LT|EQ]|GT].
+  apply -> nat_compare_lt; auto.
+  apply <- nat_compare_eq_iff; auto.
+  apply -> nat_compare_gt; auto.
+Qed.
+
 
 (** A boolean version of [le] over [nat]. *)
 
@@ -200,48 +254,48 @@ Fixpoint leb (m:nat) : nat -> bool :=
                    end
   end.
 
-Lemma leb_correct : forall m n:nat, m <= n -> leb m n = true.
+Lemma leb_correct : forall m n, m <= n -> leb m n = true.
 Proof.
   induction m as [| m IHm]. trivial.
   destruct n. intro H. elim (le_Sn_O _ H).
   intros. simpl in |- *. apply IHm. apply le_S_n. assumption.
 Qed.
 
-Lemma leb_complete : forall m n:nat, leb m n = true -> m <= n.
+Lemma leb_complete : forall m n, leb m n = true -> m <= n.
 Proof.
   induction m. trivial with arith.
   destruct n. intro H. discriminate H.
   auto with arith.
 Qed.
 
-Lemma leb_correct_conv : forall m n:nat, m < n -> leb n m = false.
+Lemma leb_iff : forall m n, leb m n = true <-> m <= n.
 Proof.
-  intros. 
+  split; auto using leb_correct, leb_complete.
+Qed.
+
+Lemma leb_correct_conv : forall m n, m < n -> leb n m = false.
+Proof.
+  intros.
   generalize (leb_complete n m).
   destruct (leb n m); auto.
-  intros.
-  elim (lt_irrefl _ (lt_le_trans _ _ _ H (H0 (refl_equal true)))).
+  intros; elim (lt_not_le m n); auto.
 Qed.
 
-Lemma leb_complete_conv : forall m n:nat, leb n m = false -> m < n.
+Lemma leb_complete_conv : forall m n, leb n m = false -> m < n.
 Proof.
-  intros. elim (le_or_lt n m). intro. conditional trivial rewrite leb_correct in H. discriminate H.
-  trivial.
+  intros m n EQ. apply not_le.
+  intro LE. apply leb_correct in LE. rewrite LE in EQ; discriminate.
+Qed.
+
+Lemma leb_iff_conv : forall m n, leb n m = false <-> m < n.
+Proof.
+  split; auto using leb_complete_conv, leb_correct_conv.
 Qed.
 
 Lemma leb_compare : forall n m, leb n m = true <-> nat_compare n m <> Gt.
 Proof.
- induction n; destruct m; simpl.
- unfold nat_compare; simpl.
- intuition; discriminate.
- split; auto with arith.
- unfold nat_compare; destruct (lt_eq_lt_dec 0 (S m)) as [[H|H]|H]; 
-  intuition; try discriminate.
- inversion H.
- split; try (intros; discriminate).
- unfold nat_compare; destruct (lt_eq_lt_dec (S n) 0) as [[H|H]|H]; 
-  intuition; try discriminate.
- inversion H.
- rewrite nat_compare_S; auto.
-Qed. 
+ split; intros.
+ apply -> nat_compare_le. auto using leb_complete.
+ apply leb_correct. apply <- nat_compare_le; auto.
+Qed.