From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Mon, 29 Apr 2013 09:57:24 +0000 (+0000)
Subject: A few integrations (closed an axiom in finset).
X-Git-Tag: make_still_working~1172
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=ea368a02a071bb99eeb84bf24ab4000acb314d60;p=helm.git

A few integrations (closed an axiom in finset).
---

diff --git a/matita/matita/lib/basics/deqsets.ma b/matita/matita/lib/basics/deqsets.ma
index fb6ad5be4..c17db5dc0 100644
--- a/matita/matita/lib/basics/deqsets.ma
+++ b/matita/matita/lib/basics/deqsets.ma
@@ -14,11 +14,12 @@ include "basics/bool.ma".
 
 (****** DeqSet: a set with a decidbale equality ******)
 
-record DeqSet : Type[1] ≝ { carr :> Type[0];
+record DeqSet : Type[1] ≝ { 
+   carr :> Type[0];
    eqb: carr → carr → bool;
    eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
 }.
-
+    
 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
 interpretation "eqb" 'eqb a b = (eqb ? a b).
 
diff --git a/matita/matita/lib/basics/finset.ma b/matita/matita/lib/basics/finset.ma
index d72388dfb..33b6b664a 100644
--- a/matita/matita/lib/basics/finset.ma
+++ b/matita/matita/lib/basics/finset.ma
@@ -210,9 +210,27 @@ unification hint  0 ≔ C1,C2;
 definition enum_prod ≝ λA,B:DeqSet.λl1.λl2.
   compose ??? (mk_Prod A B) l1 l2.
   
-axiom enum_prod_unique: ∀A,B,l1,l2. 
+lemma enum_prod_unique: ∀A,B,l1,l2. 
   uniqueb A l1 = true → uniqueb B l2 = true →
   uniqueb ? (enum_prod A B l1 l2) = true.
+#A #B #l1 elim l1 //
+  #a #tl #Hind #l2 #H1 #H2 @uniqueb_append 
+  [@unique_map_inj // 
+  |@Hind // @(andb_true_r … H1)
+  |#p #H3 cases (memb_map_to_exists … H3) #b * 
+   #Hmemb #eqp <eqp @(not_to_not ? (memb ? a tl = true))
+    [2: @sym_not_eq @eqnot_to_noteq @sym_eq @(andb_true_l … H1)
+    |elim tl 
+      [normalize //
+      |#a1 #tl1 #Hind2 #H4 cases (memb_append … H4) -H4 #H4
+        [cases (memb_map_to_exists … H4) #b1 * #memb1 #H destruct (H)
+         normalize >(\b (refl ? a)) //
+        |@orb_true_r2 @Hind2 @H4
+        ]
+      ]
+    ]
+  ]
+qed.
 
 lemma enum_prod_complete:∀A,B:DeqSet.∀l1,l2.
   (∀a. memb A a l1 = true) → (∀b.memb B b l2 = true) →
diff --git a/matita/matita/lib/basics/lists/list.ma b/matita/matita/lib/basics/lists/list.ma
index a4ba4cce1..1eaab72ba 100644
--- a/matita/matita/lib/basics/lists/list.ma
+++ b/matita/matita/lib/basics/lists/list.ma
@@ -119,6 +119,7 @@ lemma compare_append : ∀A,l1,l2,l3,l4. l1@l2 = l3@l4 →
     ]
   ]
 qed.
+
 (**************************** iterators ******************************)
 
 let rec map (A,B:Type[0]) (f: A → B) (l:list A) on l: list B ≝
diff --git a/matita/matita/lib/basics/lists/listb.ma b/matita/matita/lib/basics/lists/listb.ma
index a95d8a05a..4413e6ed1 100644
--- a/matita/matita/lib/basics/lists/listb.ma
+++ b/matita/matita/lib/basics/lists/listb.ma
@@ -172,6 +172,32 @@ cases (true_or_false … (memb S a (unique_append S tl l2)))
 #H >H normalize [@Hind //] >H normalize @Hind //
 qed.
 
+lemma uniqueb_append: ∀A,l1,l2. uniqueb A l1 = true → uniqueb A l2 =true → 
+  (∀a. memb A a l1 =true → ¬ memb A a l2 =true) → uniqueb A (l1@l2) = true.
+#A #l1 elim l1 [normalize //] #a #tl #Hind #l2 #Hatl #Hl2 
+#Hmem normalize cut (memb A a (tl@l2)=false)
+  [2:#Hcut >Hcut normalize @Hind //
+    [@(andb_true_r … Hatl) |#x #Hmemx @Hmem @orb_true_r2 //]
+  |@(noteq_to_eqnot ? true) % #Happend cases (memb_append … Happend)
+    [#H1 @(absurd … H1) @sym_not_eq @eqnot_to_noteq 
+     @sym_eq @(andb_true_l … Hatl)
+    |#H @(absurd … H) @Hmem normalize >(\b (refl ? a)) //
+    ]
+  ]
+qed.
+
+lemma memb_map_to_exists: ∀A,B:DeqSet.∀f:A→B.∀l,b. 
+  memb ? b (map ?? f l) = true → ∃a. memb ? a l = true ∧ f a = b.
+#A #B #f #l elim l 
+  [#b normalize #H destruct (H) 
+  |#a #tl #Hind #b #H cases (orb_true_l … H) 
+    [#eqb @(ex_intro … a) <(\P eqb) % // 
+    |#memb cases (Hind … memb) #a * #mema #eqb
+     @(ex_intro … a) /3/
+    ]
+  ]
+qed.
+
 lemma memb_map_inj: ∀A,B:DeqSet.∀f:A→B.∀l,a. injective A B f → 
   memb ? (f a) (map ?? f l) = true → memb ? a l = true.
 #A #B #f #l #a #injf elim l
diff --git a/matita/matita/lib/basics/vectors.ma b/matita/matita/lib/basics/vectors.ma
index a2b7c0186..2ab0a87b1 100644
--- a/matita/matita/lib/basics/vectors.ma
+++ b/matita/matita/lib/basics/vectors.ma
@@ -9,7 +9,7 @@
      \ /   GNU General Public License Version 2   
       V_____________________________________________________________*)
 
-include "basics/finset.ma".
+include "basics/lists/list.ma".
 
 record Vector (A:Type[0]) (n:nat): Type[0] ≝ 
 { vec :> list A;