From 30172d86a0cd979e44ae3343655f6ce55043dd52 Mon Sep 17 00:00:00 2001
From: Andrea Asperti <andrea.asperti@unibo.it>
Date: Fri, 25 May 2012 08:58:36 +0000
Subject: [PATCH] New definition of finset.

---
 matita/matita/lib/basics/deqsets.ma |  31 +++++++++
 matita/matita/lib/basics/finset.ma  | 101 ++++++++++++++++++++++------
 matita/matita/lib/basics/types.ma   |   3 +
 3 files changed, 115 insertions(+), 20 deletions(-)

diff --git a/matita/matita/lib/basics/deqsets.ma b/matita/matita/lib/basics/deqsets.ma
index 45afabe59..38c9a1e05 100644
--- a/matita/matita/lib/basics/deqsets.ma
+++ b/matita/matita/lib/basics/deqsets.ma
@@ -149,3 +149,34 @@ unification hint  0 ≔ T1,T2,p1,p2;
 (* ---------------------------------------- *) ⊢ 
     eq_sum T1 T2 p1 p2 ≡ eqb X p1 p2.
 
+(* sigma *)
+definition eq_sigma ≝ 
+  λA:DeqSet.λP:A→Prop.λp1,p2:Σx:A.P x.
+  match p1 with 
+  [mk_Sig a1 h1 ⇒ 
+    match p2 with 
+    [mk_Sig a2 h2 ⇒ a1==a2]].
+ 
+(* uses proof irrelevance *)
+lemma eq_sigma_true: ∀A:DeqSet.∀P.∀p1,p2:Σx.P x.
+  eq_sigma A P p1 p2 = true ↔ p1 = p2.
+#A #P * #a1 #Ha1 * #a2 #Ha2 %
+  [normalize #eqa generalize in match Ha1; >(\P eqa) // 
+  |#H >H @(\b ?) //
+  ]
+qed.
+
+definition DeqSig ≝ λA:DeqSet.λP:A→Prop.
+  mk_DeqSet (Σx:A.P x) (eq_sigma A P) (eq_sigma_true A P).
+  
+unification hint  0 ≔ C,P; 
+    T ≟ carr C,
+    X ≟ DeqSig C P
+(* ---------------------------------------- *) ⊢ 
+    Σx:T.P x ≡ carr X.
+
+unification hint  0 ≔ T,P,p1,p2; 
+    X ≟ DeqSig T P
+(* ---------------------------------------- *) ⊢ 
+    eq_sigma T P p1 p2 ≡ eqb X p1 p2.
+
diff --git a/matita/matita/lib/basics/finset.ma b/matita/matita/lib/basics/finset.ma
index 85f4ef3ba..b8f698110 100644
--- a/matita/matita/lib/basics/finset.ma
+++ b/matita/matita/lib/basics/finset.ma
@@ -16,7 +16,8 @@ include "basics/lists/listb.ma".
 record FinSet : Type[1] ≝ 
 { FinSetcarr:> DeqSet;
   enum: list FinSetcarr;
-  enum_unique: uniqueb FinSetcarr enum = true
+  enum_unique: uniqueb FinSetcarr enum = true;
+  enum_complete : ∀x:FinSetcarr. memb ? x enum = true
 }.
 
 notation < "𝐅" non associative with precedence 90 
@@ -27,7 +28,11 @@ interpretation "FinSet" 'bigF = (mk_FinSet ???).
 lemma bool_enum_unique: uniqueb ? [true;false] = true.
 // qed.
 
-definition FinBool ≝ mk_FinSet DeqBool [true;false] bool_enum_unique.
+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 ≟ FinBool
@@ -42,39 +47,77 @@ qed.
 
 definition DeqNat ≝ mk_DeqSet nat eqb eqbnat_true.
 
-let rec enumn n ≝ 
-  match n with [ O ⇒ [ ] | S p ⇒ p::enumn p ].
+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)].
+    
+definition enumn ≝ λn.enumnaux n n (le_n n).
+
+definition Nat_to ≝ λn. DeqSig DeqNat (λi.i<n).
 
-lemma memb_enumn: ∀m,n. n ≤ m →  (¬ (memb DeqNat m (enumn n))) = true.
-#m #n elim n // #n1 #Hind #ltm  @sym_eq @noteq_to_eqnot @sym_not_eq
+(* 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)) @sym_not_eq /2/ 
+  [#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 DeqNat (enumn n) = true.
-#n elim n // #m #Hind @true_to_andb_true /2/ 
+ 
+lemma enumn_unique: ∀n.uniqueb (Nat_to n) (enumn n) = true.
+#n @enumn_unique_aux
 qed.
 
-definition initN ≝ λn.mk_FinSet DeqNat (enumn n) (enumn_unique n).
-
-example tipa: ∀n.∃x: initN (S n). x = n.
-#n @(ex_intro … n) // 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.
 
-example inject : ∃f: initN 2 → initN 4. injective ?? f.
-@(ex_intro … S) // 
+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).
+
+example tipa: ∀n.∃x: initN (S n). pi1 … x = n.
+#n @ex_intro [whd @mk_Sig [@n | @le_n] | //] qed.
+
 (* sum *)
 definition enum_sum ≝ λA,B:DeqSet.λl1.λl2.
   (map ?? (inl A B) l1) @ (map ?? (inr A B) l2).
   
-lemma enumAB_def : ∀A,B:FinSet.∀l1,l2. enum_sum A B l1 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 enumAB_unique: ∀A,B:DeqSet.∀l1,l2. 
+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 
@@ -97,10 +140,21 @@ lemma enumAB_unique: ∀A,B:DeqSet.∀l1,l2.
   ]
 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)) 
-   (enumAB_unique … (enum_unique A) (enum_unique B)).
+   (enum_sum_unique … (enum_unique A) (enum_unique B))
+   (enum_sum_complete … (enum_complete A) (enum_complete B)).
 
 include alias "basics/types.ma".
 
@@ -120,10 +174,17 @@ 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 A) (enum B) (enum_unique A) (enum_unique 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,
diff --git a/matita/matita/lib/basics/types.ma b/matita/matita/lib/basics/types.ma
index b7a022ad2..94777e071 100644
--- a/matita/matita/lib/basics/types.ma
+++ b/matita/matita/lib/basics/types.ma
@@ -79,6 +79,9 @@ lemma sub_pi2 : ∀A.∀P,P':A → Prop. (∀x.P x → P' x) → ∀x:Σx:A.P x.
 #A #P #P' #H1 * #x #H2 @H1 @H2
 qed.
 
+lemma inj_mk_Sig: ∀A,P.∀x. x = mk_Sig A P (pi1 A P x) (pi2 A P x).
+#A #P #x cases x //
+qed-. 
 (* Prod *)
 
 record Prod (A,B:Type[0]) : Type[0] ≝ {
-- 
2.39.2