A few integrations (closed an axiom in finset).

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

diff --git a/matita/matita/lib/basics/deqsets.ma b/matita/matita/lib/basics/deqsets.ma
index 2d6434f6578ffc2f4440aa5843388068be8e22ba..c17db5dc0cb4b1e899ae76533a686ad5322fb450 100644 (file)
--- 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).
 
@@ -28,6 +29,10 @@ notation "\P H" non associative with precedence 90
 notation "\b H" non associative with precedence 90 
   for @{(proj2 … (eqb_true ???) $H)}. 
   
+notation < "𝐃" non associative with precedence 90 
+ for @{'bigD}.
+interpretation "DeqSet" 'bigD = (mk_DeqSet ???).
+  
 lemma eqb_false: ∀S:DeqSet.∀a,b:S. 
   (eqb ? a b) = false ↔ a ≠ b.
 #S #a #b % #H 
@@ -57,6 +62,7 @@ qed.
 
 definition DeqBool ≝ mk_DeqSet bool beqb beqb_true.
 
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
 unification hint  0 ≔ ; 
     X ≟ mk_DeqSet bool beqb beqb_true
 (* ---------------------------------------- *) ⊢ 
@@ -71,6 +77,45 @@ example exhint: ∀b:bool. (b == false) = true → b = false.
 #b #H @(\P H).
 qed.
 
+(* option *)
+
+definition eq_option ≝
+  λA:DeqSet.λa1,a2:option A.
+    match a1 with 
+    [ None ⇒ match a2 with [None ⇒ true | _ ⇒ false]
+    | Some a1' ⇒ match a2 with [None ⇒ false | Some a2' ⇒ a1'==a2']].
+
+lemma eq_option_true: ∀A:DeqSet.∀a1,a2:option A.
+  eq_option A a1 a2 = true ↔ a1 = a2.
+#A *
+  [* 
+    [% //
+    |#a1 % normalize #H destruct 
+    ]
+  |#a1 *  
+    [normalize % #H destruct
+    |#a2 normalize %
+      [#Heq >(\P Heq) //
+      |#H destruct @(\b ?) //
+      ]
+  ]
+qed.
+
+definition DeqOption ≝ λA:DeqSet.
+  mk_DeqSet (option A) (eq_option A) (eq_option_true A).
+  
+unification hint  0 ≔ C; 
+    T ≟ carr C,
+    X ≟ DeqOption C
+(* ---------------------------------------- *) ⊢ 
+    option T ≡ carr X.
+
+unification hint  0 ≔ T,a1,a2; 
+    X ≟ DeqOption T
+(* ---------------------------------------- *) ⊢ 
+    eq_option T a1 a2 ≡ eqb X a1 a2.
+
+
 (* pairs *)
 definition eq_pairs ≝
   λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
@@ -144,3 +189,35 @@ 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.
+*)