X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsets.ma;h=c8f303a6b407920f101126160a4b6d6333d9acfc;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=9fd5b7eeb77c78d77de0fb5b220f42e93f84779d;hpb=4efe53bc2098939c255d5b03941212549f89a1bd;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index 9fd5b7eeb..c8f303a6b 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -34,6 +34,10 @@ interpretation "intersect" 'intersects U V = (intersect ? U V).
 ndefinition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
 interpretation "union" 'union U V = (union ? U V).
 
+ndefinition substract ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ ¬ x ∈ V }.
+interpretation "substract" 'minus U V = (substract ? U V).
+
+
 ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
 
 ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
@@ -49,15 +53,15 @@ nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
 include "properties/relations1.ma".
 
 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
- #A; @
-  [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
-  | /2/
-  | #S; #S'; *; /3/
-  | #S; #T; #U; *; #H1; #H2; *; /4/]
+#A; @(λS,S'. S ⊆ S' ∧ S' ⊆ S); /2/; ##[ #A B; *; /3/]
+#S T U; *; #H1 H2; *; /4/;
 nqed.
 
 include "sets/setoids1.ma".
 
+ndefinition singleton ≝ λA:setoid.λa:A.{ x | a = x }.
+interpretation "singl" 'singl a = (singleton ? a).
+
 (* this has to be declared here, so that it is combined with carr *)
 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
 
@@ -65,21 +69,22 @@ ndefinition powerclass_setoid: Type[0] → setoid1.
  #A; @(Ω^A);//.
 nqed.
 
-include "hints_declaration.ma". 
-
 alias symbol "hint_decl" = "hint_decl_Type2".
-unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ Ω^A.
+unification hint 0 ≔ A;
+  R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
+(*--------------------------------------------------*)⊢ 
+     carr1 R ≡ Ω^A.
 
 (************ SETS OVER SETOIDS ********************)
 
 include "logic/cprop.ma".
 
-nrecord ext_powerclass (A: setoid) : Type[1] ≝
- { ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso... 
-                forse lo si vorrebbe dichiarato con un target più lasco 
-                ma la sintassi :> non lo supporta *)
+nrecord ext_powerclass (A: setoid) : Type[1] ≝ { 
+   ext_carr:> Ω^A; (* qui pc viene dichiarato con un target preciso... 
+                      forse lo si vorrebbe dichiarato con un target più lasco 
+                      ma la sintassi :> non lo supporta *)
    ext_prop: ∀x,x':A. x=x' → (x ∈ ext_carr) = (x' ∈ ext_carr) 
- }.
+}.
  
 notation > "𝛀 ^ term 90 A" non associative with precedence 70 
 for @{ 'ext_powerclass $A }.
@@ -107,100 +112,242 @@ unification hint 0 ≔ A;
   (* ----------------------------------------------------- *) ⊢  
                  carr1 R ≡ ext_powerclass A.
 
-(*
-interpretation "prop21 mem" 'prop2 l r = (prop21 (setoid1_of_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
-*)
-    
-(*
-ncoercion ext_carr' : ∀A.∀x:ext_powerclass_setoid A. Ω^A ≝ ext_carr 
-on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?). 
-*)
-
 nlemma mem_ext_powerclass_setoid_is_morph: 
- ∀A. unary_morphism1 (setoid1_of_setoid A) (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
- #A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
- #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
-  [ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
+ ∀A. (setoid1_of_setoid A) ⇒_1 ((𝛀^A) ⇒_1 CPROP).
+#A; napply (mk_binary_morphism1 … (λx:setoid1_of_setoid A.λS: 𝛀^A. x ∈ S));
+#a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H
+[ napply (. (ext_prop … Ha^-1)) | napply (. (ext_prop … Ha)) ] /2/.
 nqed.
 
-unification hint 0 ≔  A:setoid, x, S;  
+unification hint 0 ≔  AA : setoid, S : 𝛀^AA, x : carr AA;  
+     A ≟ carr AA,
      SS ≟ (ext_carr ? S),
-     TT ≟ (mk_unary_morphism1 … 
-             (λx:setoid1_of_setoid ?.
-               mk_unary_morphism1 …
-                 (λS:ext_powerclass_setoid ?. x ∈ S)
-                 (prop11 … (mem_ext_powerclass_setoid_is_morph A x)))
-             (prop11 … (mem_ext_powerclass_setoid_is_morph A))),
-     XX ≟ (ext_powerclass_setoid A)
-  (*-------------------------------------*) ⊢ 
-      fun11 (setoid1_of_setoid A)
-       (unary_morphism1_setoid1 XX CPROP) TT x S 
-    ≡ mem A SS x.
-
-nlemma subseteq_is_morph: ∀A. unary_morphism1 (ext_powerclass_setoid A)
- (unary_morphism1_setoid1 (ext_powerclass_setoid A) CPROP).
+     TT ≟ (mk_unary_morphism1 ?? 
+             (λx:carr1 (setoid1_of_setoid ?).
+               mk_unary_morphism1 ??
+                 (λS:carr1 (ext_powerclass_setoid ?). x ∈ (ext_carr ? S))
+                 (prop11 ?? (fun11 ?? (mem_ext_powerclass_setoid_is_morph AA) x)))
+             (prop11 ?? (mem_ext_powerclass_setoid_is_morph AA))),
+     T2 ≟ (ext_powerclass_setoid AA)
+(*---------------------------------------------------------------------------*) ⊢ 
+    fun11 T2 CPROP (fun11 (setoid1_of_setoid AA) (unary_morphism1_setoid1 T2 CPROP) TT x) S ≡ mem A SS x.
+
+nlemma set_ext : ∀S.∀A,B:Ω^S.A =_1 B → ∀x:S.(x ∈ A) =_1 (x ∈ B).
+#S A B; *; #H1 H2 x; @; ##[ napply H1 | napply H2] nqed.
+
+nlemma ext_set : ∀S.∀A,B:Ω^S.(∀x:S. (x ∈ A) = (x ∈ B)) → A = B.
+#S A B H; @; #x; ncases (H x); #H1 H2; ##[ napply H1 | napply H2] nqed.
+
+nlemma subseteq_is_morph: ∀A.  𝛀^A ⇒_1 𝛀^A ⇒_1 CPROP.
  #A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
  #a; #a'; #b; #b'; *; #H1; #H2; *; /5/ by mk_iff, sym1, subseteq_trans;
 nqed.
 
-unification hint 0 ≔ A,a,a'
- (*-----------------------------------------------------------------*) ⊢
-  eq_rel ? (eq0 A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
-
+(* hints for ∩ *)
 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
- #A; #S; #S'; @ (S ∩ S');
- #a; #a'; #Ha; @; *; #H1; #H2; @ 
-  [##1,2: napply (. Ha^-1‡#); nassumption;
-##|##3,4: napply (. Ha‡#); nassumption]
+#S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
+##[##1,2: napply (. Exy^-1╪_1#); nassumption;
+##|##3,4: napply (. Exy‡#); nassumption]
 nqed.
 
 alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ 
-  A : setoid, B,C : ext_powerclass A;
-  R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
-  
+unification hint 0 ≔ A : setoid, B,C : 𝛀^A;
+  AA ≟ carr A,
+  BB ≟ ext_carr ? B,
+  CC ≟ ext_carr ? C,
+  R ≟ (mk_ext_powerclass ? 
+        (ext_carr ? B ∩ ext_carr ? C) 
+        (ext_prop ? (intersect_is_ext ? B C))) 
   (* ------------------------------------------*)  ⊢ 
-    ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
-
-nlemma intersect_is_morph:
- ∀A. unary_morphism1 (powerclass_setoid A) (unary_morphism1_setoid1 (powerclass_setoid A) (powerclass_setoid A)).
- #A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
- #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
+    ext_carr A R ≡ intersect AA BB CC.
+    
+nlemma intersect_is_morph: ∀A. Ω^A ⇒_1 Ω^A ⇒_1 Ω^A.
+#A; napply (mk_binary_morphism1 … (λS,S'. S ∩ S'));
+#a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
 nqed.
 
 alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ 
-  A : Type[0], B,C : Ω^A;
-  R ≟ (mk_unary_morphism1 …
-       (λS. mk_unary_morphism1 … (λS'.S ∩ S') (prop11 … (intersect_is_morph A S))) 
-       (prop11 … (intersect_is_morph A)))
-   ⊢ 
-    R B C  ≡ intersect ? B C.
+unification hint 0 ≔ A : Type[0], B,C : Ω^A;
+  T ≟ powerclass_setoid A,
+  R ≟ mk_unary_morphism1 ??
+       (λX. mk_unary_morphism1 ?? 
+         (λY.X ∩ Y) (prop11 ?? (fun11 ?? (intersect_is_morph A) X))) 
+       (prop11 ?? (intersect_is_morph A))
+(*------------------------------------------------------------------------*) ⊢ 
+    fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C  ≡ intersect A B C.
 
 interpretation "prop21 ext" 'prop2 l r =
  (prop11 (ext_powerclass_setoid ?)
   (unary_morphism1_setoid1 (ext_powerclass_setoid ?) ?) ? ?? l ?? r).
 
-nlemma intersect_is_ext_morph: 
- ∀A. unary_morphism1 (ext_powerclass_setoid A)
-  (unary_morphism1_setoid1 (ext_powerclass_setoid A) (ext_powerclass_setoid A)).
+nlemma intersect_is_ext_morph: ∀A. 𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
  #A; napply (mk_binary_morphism1 … (intersect_is_ext …));
  #a; #a'; #b; #b'; #Ha; #Hb; napply (prop11 … (intersect_is_morph A)); nassumption.
 nqed.
 
 unification hint 1 ≔ 
-      A:setoid, B,C : 𝛀^A;
-      R ≟ (mk_unary_morphism1 …
-              (λS:ext_powerclass_setoid A.
-               mk_unary_morphism1 ??
-                (λS':ext_powerclass_setoid A.
-                  mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S')))
-                (prop11 … (intersect_is_ext_morph A S))) 
-              (prop11 … (intersect_is_ext_morph A))) ,
+      AA : setoid, B,C : 𝛀^AA;
+      A ≟ carr AA,
+      T ≟ ext_powerclass_setoid AA,
+      R ≟ (mk_unary_morphism1 ?? (λX:𝛀^AA.
+               mk_unary_morphism1 ?? (λY:𝛀^AA.
+                  mk_ext_powerclass AA 
+                    (ext_carr ? X ∩ ext_carr ? Y) 
+                    (ext_prop AA (intersect_is_ext ? X Y)))
+                (prop11 ?? (fun11 ?? (intersect_is_ext_morph AA) X))) 
+              (prop11 ?? (intersect_is_ext_morph AA))) ,
        BB ≟ (ext_carr ? B),
        CC ≟ (ext_carr ? C)
-   (* ------------------------------------------------------*) ⊢ 
-            ext_carr A (R B C) ≡ intersect (carr A) BB CC.
+   (* ---------------------------------------------------------------------------------------*) ⊢ 
+      ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ intersect A BB CC.
+
+
+(* hints for ∪ *)
+nlemma union_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
+#X; napply (mk_binary_morphism1 … (λA,B.A ∪ B));
+#A1 A2 B1 B2 EA EB; napply ext_set; #x;
+nchange in match (x ∈ (A1 ∪ B1)) with (?∨?);
+napply (.= (set_ext ??? EA x)‡#);
+napply (.= #‡(set_ext ??? EB x)); //;
+nqed.
+
+nlemma union_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+ #S A B; @ (A ∪ B); #x y Exy; @; *; #H1; 
+##[##1,3: @; ##|##*: @2 ]
+##[##1,3: napply (. (Exy^-1)╪_1#) 
+##|##2,4: napply (. Exy╪_1#)]
+nassumption;
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, B,C :  𝛀^A;
+   AA ≟ carr A,
+   BB ≟ ext_carr ? B,
+   CC ≟ ext_carr ? C,
+   R ≟ mk_ext_powerclass ? 
+         (ext_carr ? B ∪ ext_carr ? C) (ext_prop ? (union_is_ext ? B C))
+(*-------------------------------------------------------------------------*)  ⊢
+    ext_carr A R ≡ union AA BB CC.
+
+unification hint 0 ≔ S:Type[0], A,B:Ω^S;
+  T ≟ powerclass_setoid S,
+  MM ≟ mk_unary_morphism1 ??
+        (λA.mk_unary_morphism1 ?? 
+          (λB.A ∪ B) (prop11 ?? (fun11 ?? (union_is_morph S) A)))
+        (prop11 ?? (union_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A ∪ B.
+   
+nlemma union_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
+#A; napply (mk_binary_morphism1 …  (union_is_ext …));
+#x1 x2 y1 y2 Ex Ey; napply (prop11 … (union_is_morph A)); nassumption.
+nqed.
+            
+unification hint 1 ≔
+  AA : setoid, B,C : 𝛀^AA;
+  A ≟ carr AA,
+  T ≟ ext_powerclass_setoid AA,  
+  R ≟ mk_unary_morphism1 ?? (λX:𝛀^AA.
+           mk_unary_morphism1 ?? (λY:𝛀^AA.
+              mk_ext_powerclass AA 
+               (ext_carr ? X ∪ ext_carr ? Y) (ext_prop AA (union_is_ext ? X Y)))
+            (prop11 ?? (fun11 ?? (union_is_ext_morph AA) X)))
+          (prop11 ?? (union_is_ext_morph AA)),
+   BB ≟ (ext_carr ? B),
+   CC ≟ (ext_carr ? C)
+(*------------------------------------------------------*) ⊢
+   ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ union A BB CC.
+
+
+(* hints for - *)
+nlemma substract_is_morph : ∀A. Ω^A ⇒_1 (Ω^A ⇒_1 Ω^A).
+#X; napply (mk_binary_morphism1 … (λA,B.A - B));
+#A1 A2 B1 B2 EA EB; napply ext_set; #x;
+nchange in match (x ∈ (A1 - B1)) with (?∧?);
+napply (.= (set_ext ??? EA x)‡#); @; *; #H H1; @; //; #H2; napply H1;
+##[ napply (. (set_ext ??? EB x)); ##| napply (. (set_ext ??? EB^-1 x)); ##] //;
+nqed.
+
+nlemma substract_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
+ #S A B; @ (A - B); #x y Exy; @; *; #H1 H2; @; ##[##2,4: #H3; napply H2]
+##[##1,4: napply (. Exy╪_1#); // ##|##2,3: napply (. Exy^-1╪_1#); //]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, B,C :  𝛀^A;
+   AA ≟ carr A,
+   BB ≟ ext_carr ? B,
+   CC ≟ ext_carr ? C,
+   R ≟ mk_ext_powerclass ? 
+         (ext_carr ? B - ext_carr ? C) 
+         (ext_prop ? (substract_is_ext ? B C))
+(*---------------------------------------------------*)  ⊢
+    ext_carr A R ≡ substract AA BB CC.
+
+unification hint 0 ≔ S:Type[0], A,B:Ω^S;
+  T ≟ powerclass_setoid S,  
+  MM ≟ mk_unary_morphism1 ??
+        (λA.mk_unary_morphism1 ?? 
+          (λB.A - B) (prop11 ?? (fun11 ?? (substract_is_morph S) A)))
+        (prop11 ?? (substract_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ A - B.
+   
+nlemma substract_is_ext_morph:∀A.𝛀^A ⇒_1 𝛀^A ⇒_1 𝛀^A.
+#A; napply (mk_binary_morphism1 …  (substract_is_ext …));
+#x1 x2 y1 y2 Ex Ey; napply (prop11 … (substract_is_morph A)); nassumption.
+nqed.
+            
+unification hint 1 ≔
+  AA : setoid, B,C : 𝛀^AA;
+  A ≟ carr AA,
+  T ≟ ext_powerclass_setoid AA,    
+  R ≟ mk_unary_morphism1 ?? (λX:𝛀^AA.
+           mk_unary_morphism1 ?? (λY:𝛀^AA.
+              mk_ext_powerclass AA 
+                (ext_carr ? X - ext_carr ? Y) 
+                (ext_prop AA (substract_is_ext ? X Y)))
+            (prop11 ?? (fun11 ?? (substract_is_ext_morph AA) X)))
+          (prop11 ?? (substract_is_ext_morph AA)),
+   BB ≟ (ext_carr ? B),
+   CC ≟ (ext_carr ? C)
+(*------------------------------------------------------*) ⊢
+   ext_carr AA (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) R B) C) ≡ substract A BB CC.
+
+(* hints for {x} *)
+nlemma single_is_morph : ∀A:setoid. (setoid1_of_setoid A) ⇒_1 Ω^A.
+#X; @; ##[ napply (λx.{(x)}); ##] 
+#a b E; napply ext_set; #x; @; #H; /3/; nqed.
+
+nlemma single_is_ext: ∀A:setoid. A → 𝛀^A.
+#X a; @; ##[ napply ({(a)}); ##] #x y E; @; #H; /3/; nqed. 
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, a : carr A;
+   R ≟ (mk_ext_powerclass ? {(a)} (ext_prop ? (single_is_ext ? a)))
+(*-------------------------------------------------------------------------*)  ⊢
+    ext_carr A R ≡ singleton A a.
+
+unification hint 0 ≔ A:setoid, a : carr A;
+  T ≟ setoid1_of_setoid A,
+  AA ≟ carr A,
+  MM ≟ mk_unary_morphism1 ?? 
+         (λa:carr1 (setoid1_of_setoid A).{(a)}) (prop11 ?? (single_is_morph A))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T (powerclass_setoid AA) MM a ≡ {(a)}.
+   
+nlemma single_is_ext_morph:∀A:setoid.(setoid1_of_setoid A) ⇒_1 𝛀^A.
+#A; @; ##[ #a; napply (single_is_ext ? a); ##] #a b E; @; #x; /3/; nqed.
+            
+unification hint 1 ≔ AA : setoid, a: carr AA;
+  T ≟ ext_powerclass_setoid AA,
+  R ≟ mk_unary_morphism1 ??
+       (λa:carr1 (setoid1_of_setoid AA).
+         mk_ext_powerclass AA {(a)} (ext_prop ? (single_is_ext AA a)))
+            (prop11 ?? (single_is_ext_morph AA))
+(*------------------------------------------------------*) ⊢
+   ext_carr AA (fun11 (setoid1_of_setoid AA) T R a) ≡ singleton AA a.
+
 
 (*
 alias symbol "hint_decl" = "hint_decl_Type2".
@@ -281,21 +428,20 @@ nqed.
 (******************* first omomorphism theorem for sets **********************)
 
 ndefinition eqrel_of_morphism:
- ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
+ ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
  #A; #B; #f; @
   [ @ [ napply (λx,y. f x = f y) ] /2/;
 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
 napply (.= (†H)); // ]
 nqed.
 
-ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
+ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
  #A; #R; @
   [ napply (λx.x) |  #a; #a'; #H; napply (compatibility … R … H) ]
 nqed.
 
 ndefinition quotiented_mor:
- ∀A,B.∀f:unary_morphism A B.
-  unary_morphism (quotient … (eqrel_of_morphism … f)) B.
+ ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
  #A; #B; #f; @ [ napply f ] //.
 nqed.
 
@@ -306,41 +452,31 @@ nlemma first_omomorphism_theorem_functions1:
 
 alias symbol "eq" = "setoid eq".
 ndefinition surjective ≝
- λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
+ λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
   ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
 
 ndefinition injective ≝
- λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
+ λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
   ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
 
 nlemma first_omomorphism_theorem_functions2:
- ∀A,B.∀f: unary_morphism A B. 
+ ∀A,B.∀f:A ⇒_0 B. 
    surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
 /3/. nqed.
 
 nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f: unary_morphism A B. 
+ ∀A,B.∀f:A ⇒_0 B. 
    injective … (Full_set ?) (quotiented_mor … f).
  #A; #B; #f; nwhd; #x; #x'; #Hx; #Hx'; #K; nassumption.
 nqed.
 
 nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B) : Type[0] ≝
- { iso_f:> unary_morphism A B;
+ { iso_f:> A ⇒_0 B;
    f_closed: ∀x. x ∈ S → iso_f x ∈ T;
    f_sur: surjective … S T iso_f;
    f_inj: injective … S iso_f
  }.
 
-nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
-#A; #U; #V; #W; *; #H; #x; *; /2/.
-nqed.
-
-nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
-#A; #U; #V; #W; #H; #H1; #x; *; /2/.
-nqed.
-
-nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
-/3/. nqed.
 
 (*
 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
@@ -362,4 +498,36 @@ ncheck (λA:?.
    
    ;
  }.
-*)
\ No newline at end of file
+*)
+
+(* Set theory *)
+
+nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
+#A; #U; #V; #W; *; #H; #x; *; /2/.
+nqed.
+
+nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
+#A; #U; #V; #W; #H; #H1; #x; *; /2/.
+nqed.
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+/3/. nqed.
+
+nlemma cupC : ∀S. ∀a,b:Ω^S.a ∪ b = b ∪ a.
+#S a b; @; #w; *; nnormalize; /2/; nqed.
+
+nlemma cupID : ∀S. ∀a:Ω^S.a ∪ a = a.
+#S a; @; #w; ##[*; //] /2/; nqed.
+
+(* XXX Bug notazione \cup, niente parentesi *)
+nlemma cupA : ∀S.∀a,b,c:Ω^S.a ∪ b ∪ c = a ∪ (b ∪ c).
+#S a b c; @; #w; *; /3/; *; /3/; nqed.
+
+ndefinition Empty_set : ∀A.Ω^A ≝ λA.{ x | False }.
+
+notation "∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty set" 'empty = (Empty_set ?).
+
+nlemma cup0 :∀S.∀A:Ω^S.A ∪ ∅ = A.
+#S p; @; #w; ##[*; //| #; @1; //] *; nqed.
+