lfsx_lfpxs completed!

[helm.git] / matita / matita / lib / basics / finset.ma

diff --git a/matita/matita/lib/basics/finset.ma b/matita/matita/lib/basics/finset.ma
index b8f69811086b42e445549070120a2916ac050c02..df8d0a87c66ae6783735f95a0c1b4791cd3f76f0 100644 (file)
--- a/matita/matita/lib/basics/finset.ma
+++ b/matita/matita/lib/basics/finset.ma
@@ -64,16 +64,12 @@ 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/
+  [whd in ⊢ (??%?→?); #H1 @(absurd  … ltni) @le_to_not_lt 
+   >(eqb_true_to_eq … H1) @le_n
+  |<(notb_notb (memb …)) >Hind normalize /2 by lt_to_le, absurd/
   ]
 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.
@@ -109,6 +105,46 @@ definition initN ≝ λn.
 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.
+
+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).
@@ -169,10 +205,28 @@ 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 [#x #y #Heq @(eq_f … \snd … Heq) | //]
+  |@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) →
@@ -193,3 +247,26 @@ unification hint  0 ≔ C1,C2;
 (* ---------------------------------------- *) ⊢ 
     T1×T2 ≡ FinSetcarr X.
 
+(* graph of a function *)
+
+definition graph_of ≝ λA,B.λf:A→B. 
+  Σp:A×B.f (\fst p) = \snd p.
+
+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.