From 65ceda87c740b9f5a81ebf5a7873036c081b402c Mon Sep 17 00:00:00 2001
From: letouzey
Date: Mon, 29 Oct 2007 23:52:01 +0000
Subject: Revision of the FSetWeak Interface, so that it becomes a precise
 subtype of the FSet Interface (and idem for FMapWeak / FMap).

1) No eq_dec is officially required in FSetWeakInterface.S.E
(EqualityType instead of DecidableType). But of course,
implementations still needs this eq_dec. 

2) elements_3 differs in FSet and FSetWeak (sort vs. nodup). In
FSetWeak we rename it into elements_3w, whereas in FSet we
artificially add elements_3w along to the original elements_3.

Initial steps toward factorization of FSetFacts and FSetWeakFacts, 
and so on...

Even if it's not required, FSetWeakList provides a eq_dec on sets, 
allowing weak sets of weak sets. 





git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@10271 85f007b7-540e-0410-9357-904b9bb8a0f7
---
 theories/FSets/FSetWeakProperties.v | 56 ++++++++++++++++++++++++-------------
 1 file changed, 37 insertions(+), 19 deletions(-)

(limited to 'theories/FSets/FSetWeakProperties.v')

diff --git a/theories/FSets/FSetWeakProperties.v b/theories/FSets/FSetWeakProperties.v
index 07e087241f..fa04b4fbfb 100644
--- a/theories/FSets/FSetWeakProperties.v
+++ b/theories/FSets/FSetWeakProperties.v
@@ -27,19 +27,20 @@ Unset Strict Implicit.
 Hint Unfold transpose compat_op.
 Hint Extern 1 (Setoid_Theory _ _) => constructor; congruence.
 
-Module Properties (M: S).
+Module Properties 
+ (M:FSetWeakInterface.S)
+ (D:DecidableType with Definition t:=M.E.t 
+                  with Definition eq:=M.E.eq).
   Import M.E.
   Import M.
+  Module FM := Facts M D.
+  Import FM.
   Import Logic. (* to unmask [eq] *)  
   Import Peano. (* to unmask [lt] *)
-  
-   (** Results about lists without duplicates *)
 
-  Module FM := Facts M.
-  Import FM.
+  (** Results about lists without duplicates *)
 
-  Definition Add (x : elt) (s s' : t) :=
-    forall y : elt, In y s' <-> E.eq x y \/ In y s.
+  Definition Add x s s' := forall y, In y s' <-> E.eq x y \/ In y s.
 
   Lemma In_dec : forall x s, {In x s} + {~ In x s}.
   Proof.
@@ -52,21 +53,21 @@ Module Properties (M: S).
 
   Lemma equal_refl : forall s, s[=]s.
   Proof.
-  unfold Equal; intuition.
+  exact eq_refl.
   Qed.
 
   Lemma equal_sym : forall s s', s[=]s' -> s'[=]s.
   Proof.
-  unfold Equal; intros.
-  rewrite H; intuition.
+  exact eq_sym.
   Qed.
 
   Lemma equal_trans : forall s1 s2 s3, s1[=]s2 -> s2[=]s3 -> s1[=]s3.
   Proof.
-  unfold Equal; intros.
-  rewrite H; exact (H0 a).
+  exact eq_trans.
   Qed.
 
+  Hint Immediate equal_refl equal_sym : set.
+
   Variable s s' s'' s1 s2 s3 : t.
   Variable x x' : elt.
 
@@ -74,22 +75,24 @@ Module Properties (M: S).
   
   Lemma subset_refl : s[<=]s. 
   Proof.
-  unfold Subset; intuition.
+  apply Subset_refl.
   Qed.
 
-  Lemma subset_antisym : s[<=]s' -> s'[<=]s -> s[=]s'.
+  Lemma subset_trans : s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.
   Proof.
-  unfold Subset, Equal; intuition.
+  apply Subset_trans.
   Qed.
 
-  Lemma subset_trans : s1[<=]s2 -> s2[<=]s3 -> s1[<=]s3.
+  Lemma subset_antisym : s[<=]s' -> s'[<=]s -> s[=]s'.
   Proof.
-  unfold Subset; intuition.
+  unfold Subset, Equal; intuition.
   Qed.
 
+  Hint Immediate subset_refl : set.
+
   Lemma subset_equal : s[=]s' -> s[<=]s'.
   Proof.
-  unfold Subset, Equal; firstorder.
+  unfold Subset, Equal; intros; rewrite <- H; auto.
   Qed.
 
   Lemma subset_empty : empty[<=]s.
@@ -120,7 +123,7 @@ Module Properties (M: S).
 
   Lemma in_subset : In x s1 -> s1[<=]s2 -> In x s2.
   Proof.
-  unfold Subset; intuition.
+  intros; rewrite <- H0; auto.
   Qed.
  
   Lemma double_inclusion : s1[=]s2 <-> s1[<=]s2 /\ s2[<=]s1.
@@ -224,6 +227,21 @@ Module Properties (M: S).
   unfold Equal; intros; set_iff; tauto.
   Qed.
 
+  Lemma union_remove_add_1 : 
+   union (remove x s) (add x s') [=] union (add x s) (remove x s').
+  Proof.
+  unfold Equal; intros; set_iff.
+  destruct (eq_dec x a); intuition.
+  Qed.
+
+  Lemma union_remove_add_2 : In x s -> 
+   union (remove x s) (add x s') [=] union s s'.
+  Proof.
+  unfold Equal; intros; set_iff.
+  destruct (eq_dec x a); intuition.
+  left; eauto with set.
+  Qed.
+
   Lemma union_subset_1 : s [<=] union s s'.
   Proof.
   unfold Subset; intuition.
-- 
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