diff options
author | letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2006-04-25 22:52:18 +0000 |
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committer | letouzey <letouzey@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2006-04-25 22:52:18 +0000 |
commit | b2e0ec34cdd95017ce9ecec99b3431494b452d6c (patch) | |
tree | ccbc1c35da20152063cc5d094cc2e1b256fa4cf0 /theories/Arith/Compare_dec.v | |
parent | 0274e797162cd62fe9145436a66027a61d727d64 (diff) |
Un gros coup de lifting pour IntMap:
* le type ad des adresses est maintenant un alias vers
le N de NArith, qui lui est isomorphe.
* toutes les operations sur ces adresses (p.ex. un xor
bit a bit) sont maintenant dans de nouveaux fichiers du
repertoire NArith.
* Intmap utilise maintenant le meme type option que le
reste du monde
* etc etc...
Tout ceci ne preserve pas forcement la compatibilite. Les 4 contribs
utilisant Intmap sont adaptees en consequence. Me demander si besoin
ma moulinette d'adaptation (incomplete).
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@8733 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Arith/Compare_dec.v')
-rw-r--r-- | theories/Arith/Compare_dec.v | 136 |
1 files changed, 136 insertions, 0 deletions
diff --git a/theories/Arith/Compare_dec.v b/theories/Arith/Compare_dec.v index 8e1866538..2fa5f7ba4 100644 --- a/theories/Arith/Compare_dec.v +++ b/theories/Arith/Compare_dec.v @@ -105,3 +105,139 @@ Qed. Theorem not_lt : forall n m, ~ n < m -> n >= m. 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 + 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. +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. +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. +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. +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. + swap H. + destruct (nat_compare_gt n m); 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. + swap H. + destruct (nat_compare_lt n m); auto. +Qed. + +(** A boolean version of [le] over [nat]. *) + +Fixpoint leb (m:nat) : nat -> bool := + match m with + | O => fun _:nat => true + | S m' => + fun n:nat => match n with + | O => false + | S n' => leb m' n' + end + end. + +Lemma leb_correct : forall m n:nat, 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. +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. +Proof. + intros. + generalize (leb_complete n m). + destruct (leb n m); auto. + intros. + elim (lt_irrefl _ (lt_le_trans _ _ _ H (H0 (refl_equal true)))). +Qed. + +Lemma leb_complete_conv : forall m n:nat, 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. +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. + |