X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Flists%2Flistb.ma;h=a95d8a05adb91bcb3eeb7054f200d4759cfe632c;hb=85a42e4a2a4c62818b6a98eff545e58ceb8770a4;hp=f5bf4df9e43589533a511a307ff8281b3f10c1c0;hpb=f050491f27aa5a3d0d151f7268a5ffbfbe7d69df;p=helm.git

diff --git a/matita/matita/lib/basics/lists/listb.ma b/matita/matita/lib/basics/lists/listb.ma
index f5bf4df9e..a95d8a05a 100644
--- a/matita/matita/lib/basics/lists/listb.ma
+++ b/matita/matita/lib/basics/lists/listb.ma
@@ -14,51 +14,65 @@
 
 include "basics/lists/list.ma".
 include "basics/sets.ma".
+include "basics/deqsets.ma".
+
+(********* isnilb *********)
+let rec isnilb A (l: list A) on l ≝
+match l with
+[ nil ⇒ true
+| cons hd tl ⇒ false
+].
 
 (********* search *********)
 
 let rec memb (S:DeqSet) (x:S) (l: list S) on l  ≝
   match l with
   [ nil ⇒ false
-  | cons a tl ⇒ (a == x) ∨ memb S x tl
+  | cons a tl ⇒ (x == a) ∨ memb S x tl
   ].
-  
+
+interpretation "boolean membership" 'mem a l = (memb ? a l).
+
 lemma memb_hd: ∀S,a,l. memb S a (a::l) = true.
 #S #a #l normalize >(proj2 … (eqb_true S …) (refl S a)) //
 qed.
 
 lemma memb_cons: ∀S,a,b,l. 
   memb S a l = true → memb S a (b::l) = true.
-#S #a #b #l normalize cases (b==a) normalize // 
+#S #a #b #l normalize cases (a==b) normalize // 
+qed.
+
+lemma memb_single: ∀S,a,x. memb S a [x] = true → a = x.
+#S #a #x normalize cases (true_or_false … (a==x)) #H
+  [#_ >(\P H) // |>H normalize #abs @False_ind /2/]
 qed.
 
 lemma memb_append: ∀S,a,l1,l2. 
 memb S a (l1@l2) = true →
   memb S a l1= true ∨ memb S a l2 = true.
 #S #a #l1 elim l1 normalize [#l2 #H %2 //] 
-#b #tl #Hind #l2 cases (b==a) normalize /2/ 
+#b #tl #Hind #l2 cases (a==b) normalize /2/ 
 qed. 
 
 lemma memb_append_l1: ∀S,a,l1,l2. 
  memb S a l1= true → memb S a (l1@l2) = true.
 #S #a #l1 elim l1 normalize
   [normalize #le #abs @False_ind /2/
-  |#b #tl #Hind #l2 cases (b==a) normalize /2/ 
+  |#b #tl #Hind #l2 cases (a==b) normalize /2/ 
   ]
 qed. 
 
 lemma memb_append_l2: ∀S,a,l1,l2. 
  memb S a l2= true → memb S a (l1@l2) = true.
 #S #a #l1 elim l1 normalize //
-#b #tl #Hind #l2 cases (b==a) normalize /2/ 
+#b #tl #Hind #l2 cases (a==b) normalize /2/ 
 qed. 
 
 lemma memb_exists: ∀S,a,l.memb S a l = true → 
   ∃l1,l2.l=l1@(a::l2).
 #S #a #l elim l [normalize #abs @False_ind /2/]
 #b #tl #Hind #H cases (orb_true_l … H)
-  [#eqba @(ex_intro … (nil S)) @(ex_intro … tl)
-   >(proj1 … (eqb_true …) eqba) //
+  [#eqba @(ex_intro … (nil S)) @(ex_intro … tl) >(\P eqba) //
   |#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
    @(ex_intro … (b::l1)) @(ex_intro … l2) >eqtl //
   ]
@@ -67,16 +81,15 @@ qed.
 lemma not_memb_to_not_eq: ∀S,a,b,l. 
  memb S a l = false → memb S b l = true → a==b = false.
 #S #a #b #l cases (true_or_false (a==b)) // 
-#eqab >(proj1 … (eqb_true …) eqab) #H >H #abs @False_ind /2/
+#eqab >(\P eqab) #H >H #abs @False_ind /2/
 qed. 
  
 lemma memb_map: ∀S1,S2,f,a,l. memb S1 a l= true → 
   memb S2 (f a) (map … f l) = true.
 #S1 #S2 #f #a #l elim l normalize [//]
-#x #tl #memba cases (true_or_false (x==a))
-  [#eqx >eqx >(proj1 … (eqb_true …) eqx) 
-   >(proj2 … (eqb_true …) (refl … (f a))) normalize //
-  |#eqx >eqx cases (f x==f a) normalize /2/
+#x #tl #memba cases (true_or_false (a==x))
+  [#eqx >eqx >(\P eqx) >(\b (refl … (f x))) normalize //
+  |#eqx >eqx cases (f a==f x) normalize /2/
   ]
 qed.
 
@@ -85,11 +98,39 @@ lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
   memb S3 (op a1 a2) (compose S1 S2 S3 op l1 l2) = true.
 #S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
 #x #tl #Hind #l2 #memba1 #memba2 cases (orb_true_l … memba1)
-  [#eqa1 >(proj1 … (eqb_true …) eqa1) @memb_append_l1 @memb_map // 
+  [#eqa1 >(\P eqa1) @memb_append_l1 @memb_map // 
   |#membtl @memb_append_l2 @Hind //
   ]
 qed.
 
+lemma memb_reverse: ∀S:DeqSet.∀a:S.∀l.
+  memb ? a l = true → memb ? a (reverse ? l) = true.
+#S #a #l elim l [normalize //]
+#b #tl #Hind #memba change with ([b]@tl) in match (b::tl);
+>reverse_append cases (orb_true_l … memba) #Hcase
+  [@memb_append_l2 >(\P Hcase) whd in match (reverse ??); @memb_hd
+  |@memb_append_l1 /2/
+  ]
+qed.
+
+lemma mem_to_memb: ∀S:DeqSet.∀a,l. mem S a l → memb S a l = true.
+#S #a #l elim l normalize
+  [@False_ind
+  |#hd #tl #Hind *
+    [#eqa >(\b eqa) %
+    |#Hmem cases (a==hd) // normalize /2/
+    ]
+  ]
+qed.
+
+lemma memb_to_mem: ∀S:DeqSet.∀l,a. memb S a l =true → mem S a l.
+#S #l #a elim l 
+  [normalize #H destruct
+  |#b #tl #Hind #mema cases (orb_true_l … mema) 
+    [#eqab >(\P eqab) %1 % |#memtl %2 @Hind @memtl]
+  ]
+qed.
+
 (**************** unicity test *****************)
 
 let rec uniqueb (S:DeqSet) l on l : bool ≝
@@ -108,9 +149,21 @@ let rec unique_append (S:DeqSet) (l1,l2: list S) on l1 ≝
      if memb S a r then r else a::r
   ].
 
-axiom unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2. 
+lemma memb_unique_append: ∀S,a,l1,l2. 
+memb S a (unique_append S l1 l2) = true →
+  memb S a l1= true ∨ memb S a l2 = true.
+#S #a #l1 elim l1 normalize [#l2 #H %2 //] 
+#b #tl #Hind #l2 cases (true_or_false … (a==b)) #Hab >Hab normalize /2/
+cases (memb S b (unique_append S tl l2)) normalize 
+  [@Hind | >Hab normalize @Hind]   
+qed. 
+
+lemma unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2. 
 (∀x. memb S x l1 = true → P x) → (∀x. memb S x l2 = true → P x) →
 ∀x. memb S x (unique_append S l1 l2) = true → P x. 
+#S #P #l1 #l2 #Hl1 #Hl2 #x #membx cases (memb_unique_append … membx)
+/2/ 
+qed.
 
 lemma unique_append_unique: ∀S,l1,l2. uniqueb S l2 = true →
   uniqueb S (unique_append S l1 l2) = true.
@@ -119,6 +172,30 @@ cases (true_or_false … (memb S a (unique_append S tl l2)))
 #H >H normalize [@Hind //] >H normalize @Hind //
 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
+  [normalize //
+  |#a1 #tl #Hind #Ha cases (orb_true_l … Ha)
+    [#eqf @orb_true_r1 @(\b ?)  >(injf … (\P eqf)) //
+    |#membtl @orb_true_r2 /2/
+    ]
+  ]
+qed.
+
+lemma unique_map_inj: ∀A,B:DeqSet.∀f:A→B.∀l. injective A B f → 
+  uniqueb A l = true → uniqueb B (map ?? f l) = true .
+#A #B #f #l #injf elim l 
+  [normalize //
+  |#a #tl #Hind #Htl @true_to_andb_true
+    [@sym_eq @noteq_to_eqnot @sym_not_eq 
+     @(not_to_not ?? (memb_map_inj … injf …) )
+     <(andb_true_l ?? Htl) /2/
+    |@Hind @(andb_true_r ?? Htl)
+    ]
+  ]
+qed.
+
 (******************* sublist *******************)
 definition sublist ≝ 
   λS,l1,l2.∀a. memb S a l1 = true → memb S a l2 = true.
@@ -134,9 +211,8 @@ applyS le_S_S <length_append @Hind [@(andb_true_r … unique)]
 cases (memb_append … (sub x (orb_true_r2 … membx)))
   [#membxl3 @memb_append_l1 //
   |#membxal4 cases (orb_true_l … membxal4)
-    [#eqax @False_ind lapply (andb_true_l … unique)
-     >(proj1 … (eqb_true …) eqax) >membx normalize /2/
-    |#membxl4 @memb_append_l2 //
+    [#eqxa @False_ind lapply (andb_true_l … unique)
+     <(\P eqxa) >membx normalize /2/ |#membxl4 @memb_append_l2 //
     ]
   ]
 qed.
@@ -145,11 +221,11 @@ lemma sublist_unique_append_l1:
   ∀S,l1,l2. sublist S l1 (unique_append S l1 l2).
 #S #l1 elim l1 normalize [#l2 #S #abs @False_ind /2/]
 #x #tl #Hind #l2 #a 
-normalize cases (true_or_false … (x==a)) #eqxa >eqxa 
-[>(proj1 … (eqb_true …) eqxa) cases (true_or_false (memb S a (unique_append S tl l2)))
-  [#H >H normalize // | #H >H normalize >(proj2 … (eqb_true …) (refl … a)) //]
+normalize cases (true_or_false … (a==x)) #eqax >eqax 
+[<(\P eqax) cases (true_or_false (memb S a (unique_append S tl l2)))
+  [#H >H normalize // | #H >H normalize >(\b (refl … a)) //]
 |cases (memb S x (unique_append S tl l2)) normalize 
-  [/2/ |>eqxa normalize /2/]
+  [/2/ |>eqax normalize /2/]
 ]
 qed.
 
@@ -157,27 +233,56 @@ lemma sublist_unique_append_l2:
   ∀S,l1,l2. sublist S l2 (unique_append S l1 l2).
 #S #l1 elim l1 [normalize //] #x #tl #Hind normalize 
 #l2 #a cases (memb S x (unique_append S tl l2)) normalize
-[@Hind | cases (x==a) normalize // @Hind]
+[@Hind | cases (a==x) normalize // @Hind]
+qed.
+
+lemma decidable_sublist:∀S,l1,l2. 
+  (sublist S l1 l2) ∨ ¬(sublist S l1 l2).
+#S #l1 #l2 elim l1 
+  [%1 #a normalize in ⊢ (%→?); #abs @False_ind /2/
+  |#a #tl * #subtl 
+    [cases (true_or_false (memb S a l2)) #memba
+      [%1 whd #x #membx cases (orb_true_l … membx)
+        [#eqax >(\P eqax) // |@subtl]
+      |%2 @(not_to_not … (eqnot_to_noteq … true memba)) #H1 @H1 @memb_hd
+      ]
+    |%2 @(not_to_not … subtl) #H1 #x #H2 @H1 @memb_cons //
+    ] 
+  ]
 qed.
 
 (********************* filtering *****************)
 
+lemma memb_filter_memb: ∀S,f,a,l. 
+  memb S a (filter S f l) = true → memb S a l = true.
+#S #f #a #l elim l [normalize //]
+#b #tl #Hind normalize (cases (f b)) normalize 
+cases (a==b) normalize // @Hind
+qed.
+  
+lemma uniqueb_filter : ∀S,l,f.
+ uniqueb S l = true → uniqueb S (filter S f l) = true.
+#S #l #f elim l //
+#a #tl #Hind #Hunique cases (andb_true … Hunique)
+#memba #uniquetl cases (true_or_false … (f a)) #Hfa
+  [>filter_true // @true_to_andb_true
+    [@sym_eq @noteq_to_eqnot @(not_to_not … (eqnot_to_noteq … (sym_eq … memba)))
+     #H @sym_eq @(memb_filter_memb … f) <H // 
+    |/2/
+    ]
+  |>filter_false /2/
+  ]
+qed.
+  
 lemma filter_true: ∀S,f,a,l. 
   memb S a (filter S f l) = true → f a = true.
 #S #f #a #l elim l [normalize #H @False_ind /2/]
 #b #tl #Hind cases (true_or_false (f b)) #H
 normalize >H normalize [2:@Hind]
-cases (true_or_false (b==a)) #eqab
-  [#_ <(proj1 … (eqb_true …) eqab) // | >eqab normalize @Hind]
+cases (true_or_false (a==b)) #eqab
+  [#_ >(\P eqab) // | >eqab normalize @Hind]
 qed. 
   
-lemma memb_filter_memb: ∀S,f,a,l. 
-  memb S a (filter S f l) = true → memb S a l = true.
-#S #f #a #l elim l [normalize //]
-#b #tl #Hind normalize (cases (f b)) normalize 
-cases (b==a) normalize // @Hind
-qed.
-  
 lemma memb_filter: ∀S,f,l,x. memb S x (filter ? f l) = true → 
 memb S x l = true ∧ (f x = true).
 /3/ qed.
@@ -185,12 +290,22 @@ memb S x l = true ∧ (f x = true).
 lemma memb_filter_l: ∀S,f,x,l. (f x = true) → memb S x l = true →
 memb S x (filter ? f l) = true.
 #S #f #x #l #fx elim l normalize //
-#b #tl #Hind cases (true_or_false (b==x)) #eqbx
-  [>(proj1 … (eqb_true … ) eqbx) >(proj2 … (eqb_true …) (refl … x))
-   >fx normalize >(proj2 … (eqb_true …) (refl … x)) normalize //
-  |>eqbx cases (f b) normalize [>eqbx normalize @Hind| @Hind]
+#b #tl #Hind cases (true_or_false (x==b)) #eqxb
+  [<(\P eqxb) >(\b (refl … x)) >fx normalize >(\b (refl … x)) normalize //
+  |>eqxb cases (f b) normalize [>eqxb normalize @Hind| @Hind]
   ]
 qed. 
 
+(********************* exists *****************)
 
+let rec exists (A:Type[0]) (p:A → bool) (l:list A) on l : bool ≝
+match l with
+[ nil ⇒ false
+| cons h t ⇒ orb (p h) (exists A p t)
+].
 
+lemma Exists_exists : ∀A,P,l.
+  Exists A P l →
+  ∃x. P x.
+#A #P #l elim l [ * | #hd #tl #IH * [ #H %{hd} @H | @IH ]
+qed.