X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fbasics%2Ffinset.ma;h=d72388dfb19953c525c6f43256f0796e44c8eec4;hb=723514507b1e2e58c0a8c7bcc39c5d1301b910bc;hp=1b972ec06c7cdb8db859838572c93bef7813fcd6;hpb=65a3b93b01f2d00960c56df3563b879f36f3cbfd;p=helm.git

diff --git a/matita/matita/lib/basics/finset.ma b/matita/matita/lib/basics/finset.ma
index 1b972ec06..d72388dfb 100644
--- a/matita/matita/lib/basics/finset.ma
+++ b/matita/matita/lib/basics/finset.ma
@@ -11,60 +11,248 @@
 
 include "basics/lists/listb.ma".
 
-(****** DeqSet: a set with a decidbale equality ******)
+(****** DeqSet: a set with a decidable equality ******)
 
 record FinSet : Type[1] ≝ 
-{ carr:> DeqSet;
-  enum: list carr;
-  enum_complete: ∀x.memb carr x enum = true;
-  enum_unique: uniqueb carr enum = true
+{ FinSetcarr:> DeqSet;
+  enum: list FinSetcarr;
+  enum_unique: uniqueb FinSetcarr enum = true;
+  enum_complete : ∀x:FinSetcarr. memb ? x enum = true
 }.
 
-(*
-definition DeqBool ≝ mk_DeqSet bool beqb beqb_true.
+notation < "𝐅" non associative with precedence 90 
+ for @{'bigF}.
+interpretation "FinSet" 'bigF = (mk_FinSet ???).
+
+(* bool *)
+lemma bool_enum_unique: uniqueb ? [true;false] = true.
+// qed.
+
+lemma bool_enum_complete: ∀x:bool. memb ? x [true;false] = true.
+* // qed.
+
+definition FinBool ≝ 
+  mk_FinSet DeqBool [true;false] bool_enum_unique bool_enum_complete.
 
 unification hint  0 ≔ ; 
-    X ≟ mk_DeqSet bool beqb beqb_true
+    X ≟ FinBool
 (* ---------------------------------------- *) ⊢ 
-    bool ≡ carr X.
-    
-unification hint  0 ≔ b1,b2:bool; 
-    X ≟ mk_DeqSet bool beqb beqb_true
-(* ---------------------------------------- *) ⊢ 
-    beqb b1 b2 ≡ eqb X b1 b2.
+    bool ≡ FinSetcarr X.
+
+(* nat segment *)
+
+lemma eqbnat_true : ∀n,m. eqb n m = true ↔ n = m.
+#n #m % [@eqb_true_to_eq | @eq_to_eqb_true]
+qed.
+
+definition DeqNat ≝ mk_DeqSet nat eqb eqbnat_true.
+
+lemma lt_to_le : ∀n,m. n < m → n ≤ m.
+/2/ qed-.
+ 
+let rec enumnaux n m ≝ 
+  match n return (λn.n ≤ m → list (Σx.x < m)) with 
+    [ O ⇒ λh.[ ] | S p ⇒ λh:p < m.(mk_Sig ?? p h)::enumnaux p m (lt_to_le p m h)].
     
-example exhint: ∀b:bool. (b == false) = true → b = false. 
-#b #H @(\P H).
+definition enumn ≝ λn.enumnaux n n (le_n n).
+
+definition Nat_to ≝ λn. DeqSig DeqNat (λi.i<n).
+
+(* lemma prova : ∀n. carr (Nat_to n) = (Σx.x<n). // *)
+
+lemma memb_enumn: ∀m,n,i:DeqNat. ∀h:n ≤ m. ∀k: i < m. n ≤ i →
+  (¬ (memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h))) = true.
+#m #n elim n -n // #n #Hind #i #ltm #k #ltni @sym_eq @noteq_to_eqnot @sym_not_eq
+% #H cases (orb_true_l … H)
+  [#H1 @(absurd … (\P H1)) % #Hfalse
+   cut (∀A,P,a,a1,h,h1.mk_Sig A P a h = mk_Sig A P a1 h1 → a = a1)
+   [#A #P #a #a1 #h #h1 #H destruct (H) %] #Hcut 
+    lapply (Hcut nat (λi.i<m) i n ? ? Hfalse) #Hfalse @(absurd … ltni)
+    @le_to_not_lt >Hfalse @le_n
+  |<(notb_notb (memb …)) >Hind normalize /2/
+  ]
+qed. 
+
+
+lemma enumn_unique_aux: ∀n,m. ∀h:n ≤ m. uniqueb (Nat_to m) (enumnaux n m h) = true.
+#n elim n -n // #n #Hind #m #h @true_to_andb_true // @memb_enumn //
+qed.
+ 
+lemma enumn_unique: ∀n.uniqueb (Nat_to n) (enumn n) = true.
+#n @enumn_unique_aux
+qed.
+
+(* definition ltb ≝ λn,m.leb (S n) m. *)
+lemma enumn_complete_aux: ∀n,m,i.∀h:n ≤m.∀k:i<m.i<n → 
+  memb (Nat_to m) (mk_Sig ?? i k) (enumnaux n m h) = true.
+#n elim n -n
+  [normalize #n #i #_ #_ #Hfalse @False_ind /2/ 
+  |#n #Hind #m #i #h #k #lein whd in ⊢ (??%?);
+   cases (le_to_or_lt_eq … (le_S_S_to_le … lein))
+    [#ltin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = false)
+      [normalize @not_eq_to_eqb_false @lt_to_not_eq @ltin] 
+       #Hcut >Hcut @Hind //
+    |#eqin cut (eqb (Nat_to m) (mk_Sig ?? i k) (mk_Sig ?? n h) = true)
+     [normalize @eq_to_eqb_true //
+     |#Hcut >Hcut %
+    ]
+  ]
 qed.
 
-(* pairs *)
-definition eq_pairs ≝
-  λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
+lemma enumn_complete: ∀n.∀i:Nat_to n. memb ? i (enumn n) = true.
+#n whd in ⊢ (%→?); * #i #ltin @enumn_complete_aux //
+qed.
+
+definition initN ≝ λn.
+  mk_FinSet (Nat_to n) (enumn n) (enumn_unique n) (enumn_complete n).
 
-lemma eq_pairs_true: ∀A,B:DeqSet.∀p1,p2:A×B.
-  eq_pairs A B p1 p2 = true ↔ p1 = p2.
-#A #B * #a1 #b1 * #a2 #b2 %
-  [#H cases (andb_true …H) #eqa #eqb >(\P eqa) >(\P eqb) //
-  |#H destruct normalize >(\b (refl … a2)) >(\b (refl … b2)) //
+example tipa: ∀n.∃x: initN (S n). pi1 … x = n.
+#n @ex_intro [whd @mk_Sig [@n | @le_n] | //] qed.
+
+(* option *)
+definition enum_option ≝ λA:DeqSet.λl.
+  None A::(map ?? (Some A) l).
+  
+lemma enum_option_def : ∀A:FinSet.∀l. 
+  enum_option A l = None A :: (map ?? (Some A) l).
+// qed.
+
+lemma enum_option_unique: ∀A:DeqSet.∀l. 
+  uniqueb A l = true → 
+    uniqueb ? (enum_option A l) = true.
+#A #l #uA @true_to_andb_true
+  [generalize in match uA; -uA #_ elim l [%]
+   #a #tl #Hind @sym_eq @noteq_to_eqnot % #H 
+   cases (orb_true_l … (sym_eq … H))
+    [#H1 @(absurd (None A = Some A a)) [@(\P H1) | % #H2 destruct]
+    |-H #H >H in Hind; normalize /2/
+    ]
+  |@unique_map_inj // #a #a1 #H destruct %
   ]
 qed.
 
-definition DeqProd ≝ λA,B:DeqSet.
-  mk_DeqSet (A×B) (eq_pairs A B) (eq_pairs_true A B).
+lemma enum_option_complete: ∀A:DeqSet.∀l. 
+  (∀x:A. memb A x l = true) →
+    ∀x:DeqOption A. memb ? x (enum_option A l) = true.
+#A #l #Hl * // #a @memb_cons @memb_map @Hl
+qed.
+    
+definition FinOption ≝ λA:FinSet.
+  mk_FinSet (DeqOption A) 
+   (enum_option A (enum A)) 
+   (enum_option_unique … (enum_unique A))
+   (enum_option_complete … (enum_complete A)).
+
+unification hint  0 ≔ C; 
+    T ≟ FinSetcarr C,
+    X ≟ FinOption C
+(* ---------------------------------------- *) ⊢ 
+    option T ≡ FinSetcarr X.
+
+(* sum *)
+definition enum_sum ≝ λA,B:DeqSet.λl1.λl2.
+  (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
   
+lemma enum_sum_def : ∀A,B:FinSet.∀l1,l2. enum_sum A B l1 l2 = 
+  (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
+// qed.
+
+lemma enum_sum_unique: ∀A,B:DeqSet.∀l1,l2. 
+  uniqueb A l1 = true → uniqueb B l2 = true → 
+    uniqueb ? (enum_sum A B l1 l2) = true.
+#A #B #l1 #l2 elim l1 
+  [#_ #ul2 @unique_map_inj // #b1 #b2 #Hinr destruct //
+  |#a #tl #Hind #uA #uB @true_to_andb_true 
+    [@sym_eq @noteq_to_eqnot % #H 
+     cases (memb_append … (sym_eq … H))
+      [#H1 @(absurd (memb ? a tl = true)) 
+        [@(memb_map_inj …H1) #a1 #a2 #Hinl destruct //
+        |<(andb_true_l … uA) @eqnot_to_noteq //
+        ]
+      |elim l2
+        [normalize #H destruct 
+        |#b #tlB #Hind #membH cases (orb_true_l … membH)
+          [#H lapply (\P H) #H1 destruct |@Hind]
+        ]
+      ] 
+    |@Hind // @(andb_true_r … uA)
+    ]
+  ]
+qed.
+
+lemma enum_sum_complete: ∀A,B:DeqSet.∀l1,l2. 
+  (∀x:A. memb A x l1 = true) →
+  (∀x:B. memb B x l2 = true) →
+    ∀x:DeqSum A B. memb ? x (enum_sum A B l1 l2) = true.
+#A #B #l1 #l2 #Hl1 #Hl2 *
+  [#a @memb_append_l1 @memb_map @Hl1
+  |#b @memb_append_l2 @memb_map @Hl2
+  ]
+qed.
+    
+definition FinSum ≝ λA,B:FinSet.
+  mk_FinSet (DeqSum A B) 
+   (enum_sum A B (enum A) (enum B)) 
+   (enum_sum_unique … (enum_unique A) (enum_unique B))
+   (enum_sum_complete … (enum_complete A) (enum_complete B)).
+
+include alias "basics/types.ma".
+
 unification hint  0 ≔ C1,C2; 
-    T1 ≟ carr C1,
-    T2 ≟ carr C2,
-    X ≟ DeqProd C1 C2
+    T1 ≟ FinSetcarr C1,
+    T2 ≟ FinSetcarr C2,
+    X ≟ FinSum C1 C2
 (* ---------------------------------------- *) ⊢ 
-    T1×T2 ≡ carr X.
+    T1+T2 ≡ FinSetcarr X.
+
+(* prod *)
 
-unification hint  0 ≔ T1,T2,p1,p2; 
-    X ≟ DeqProd T1 T2
+definition enum_prod ≝ λA,B:DeqSet.λl1.λl2.
+  compose ??? (mk_Prod A B) l1 l2.
+  
+axiom enum_prod_unique: ∀A,B,l1,l2. 
+  uniqueb A l1 = true → uniqueb B l2 = true →
+  uniqueb ? (enum_prod A B l1 l2) = true.
+
+lemma enum_prod_complete:∀A,B:DeqSet.∀l1,l2.
+  (∀a. memb A a l1 = true) → (∀b.memb B b l2 = true) →
+    ∀p. memb ? p (enum_prod A B l1 l2) = true.
+#A #B #l1 #l2 #Hl1 #Hl2 * #a #b @memb_compose // 
+qed.
+
+definition FinProd ≝ 
+λA,B:FinSet.mk_FinSet (DeqProd A B)
+  (enum_prod A B (enum A) (enum B)) 
+  (enum_prod_unique A B … (enum_unique A) (enum_unique B)) 
+  (enum_prod_complete A B … (enum_complete A) (enum_complete B)).
+
+unification hint  0 ≔ C1,C2; 
+    T1 ≟ FinSetcarr C1,
+    T2 ≟ FinSetcarr C2,
+    X ≟ FinProd C1 C2
 (* ---------------------------------------- *) ⊢ 
-    eq_pairs T1 T2 p1 p2 ≡ eqb X p1 p2.
+    T1×T2 ≡ FinSetcarr X.
+
+(* graph of a function *)
+
+definition graph_of ≝ λA,B.λf:A→B. 
+  Σp:A×B.f (\fst p) = \snd p.
 
-example hint2: ∀b1,b2. 
-  〈b1,true〉==〈false,b2〉=true → 〈b1,true〉=〈false,b2〉.
-#b1 #b2 #H @(\P H).
-*)
\ No newline at end of file
+definition graph_enum ≝ λA,B:FinSet.λf:A→B. 
+  filter ? (λp.f (\fst p) == \snd p) (enum (FinProd A B)).
+  
+lemma graph_enum_unique : ∀A,B,f.
+  uniqueb ? (graph_enum A B f) = true.
+#A #B #f @uniqueb_filter @(enum_unique (FinProd A B))
+qed.
+
+lemma graph_enum_correct: ∀A,B:FinSet.∀f:A→B. ∀a,b.
+memb ? 〈a,b〉 (graph_enum A B f) = true → f a = b.
+#A #B #f #a #b #membp @(\P ?) @(filter_true … membp)
+qed.
+
+lemma graph_enum_complete: ∀A,B:FinSet.∀f:A→B. ∀a,b.
+f a = b → memb ? 〈a,b〉 (graph_enum A B f) = true.
+#A #B #f #a #b #eqf @memb_filter_l [@(\b eqf)]
+@enum_prod_complete //
+qed.