X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsets.ma;h=113654ad330de847908ae68b3fcf65faaf9c3033;hb=a90c31c1b53222bd6d57360c5ba5c2d0fe7d5207;hp=c8f303a6b407920f101126160a4b6d6333d9acfc;hpb=4377e950998c9c63937582952a79975947aa9a45;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index c8f303a6b..113654ad3 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -34,10 +34,6 @@ 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 }.
@@ -45,51 +41,53 @@ ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈
 ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
 
 nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
-//.nqed.
+ #A; #S; #x; #H; nassumption.
+nqed.
 
 nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
-/3/.nqed.
+ #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
+nqed.
 
 include "properties/relations1.ma".
 
 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
-#A; @(λS,S'. S ⊆ S' ∧ S' ⊆ S); /2/; ##[ #A B; *; /3/]
-#S T U; *; #H1 H2; *; /4/;
+ #A; @
+  [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
+  | #S; @; napply subseteq_refl
+  | #S; #S'; *; #H1; #H2; @; nassumption
+  | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; @; napply subseteq_trans;
+     ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
 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^?).
 
 ndefinition powerclass_setoid: Type[0] → setoid1.
- #A; @(Ω^A);//.
+ #A; @[ napply (Ω^A)| napply seteq ]
 nqed.
 
+include "hints_declaration.ma". 
+
 alias symbol "hint_decl" = "hint_decl_Type2".
-unification hint 0 ≔ A;
-  R ≟ (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A)))
-(*--------------------------------------------------*)⊢ 
-     carr1 R ≡ Ω^A.
+unification hint 0 ≔ A ⊢ carr1 (mk_setoid1 (Ω^A) (eq1 (powerclass_setoid A))) ≡ 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 }.
 
-notation < "Ω term 90 A \atop ≈" non associative with precedence 90 
+notation "Ω term 90 A \atop ≈" non associative with precedence 70 
 for @{ 'ext_powerclass $A }.
 
 interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
@@ -100,11 +98,17 @@ nqed.
 ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
 
 ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
- #A; @ [ napply (λS,S'. S = S') ] /2/.
+ #A; @
+  [ napply (λS,S'. S = S')
+  | #S; napply (refl1 ? (seteq A))
+  | #S; #S'; napply (sym1 ? (seteq A))
+  | #S; #T; #U; napply (trans1 ? (seteq A))]
 nqed.
 
 ndefinition ext_powerclass_setoid: setoid → setoid1.
- #A; @ (ext_seteq A).
+ #A; @
+  [ napply (ext_powerclass A)
+  | napply (ext_seteq A) ]
 nqed.
               
 unification hint 0 ≔ A;
@@ -112,242 +116,106 @@ 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. (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/.
+ ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
+ #A; @
+  [ napply (λx,S. x ∈ S) 
+  | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
+     ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
+     ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
+     ##]
+  ##]
 nqed.
 
-unification hint 0 ≔  AA : setoid, S : 𝛀^AA, x : carr AA;  
-     A ≟ carr AA,
+unification hint 0 ≔  A:setoid, x, S;  
      SS ≟ (ext_carr ? S),
-     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;
+     TT ≟ (mk_binary_morphism1 ??? 
+             (λx:setoid1_of_setoid ?.λS:ext_powerclass_setoid ?. x ∈ S) 
+             (prop21 ??? (mem_ext_powerclass_setoid_is_morph A))),
+     XX ≟ (ext_powerclass_setoid A)
+  (*-------------------------------------*) ⊢ 
+      fun21 (setoid1_of_setoid A) XX CPROP TT x S 
+    ≡ mem A SS x.
+
+nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
+ #A; @
+  [ napply (λS,S'. S ⊆ S')
+  | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
+     [ napply (subseteq_trans … a)
+        [ nassumption | napply (subseteq_trans … b); nassumption ]
+   ##| napply (subseteq_trans … a')
+        [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
 nqed.
 
-(* hints for ∩ *)
+unification hint 0 ≔ A,a,a'
+ (*-----------------------------------------------------------------*) ⊢
+  eq_rel ? (eq A) a a' ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) a a'.
+
 nlemma intersect_is_ext: ∀A. 𝛀^A → 𝛀^A → 𝛀^A.
-#S A B; @ (A ∩ B); #x y Exy; @; *; #H1 H2; @;
-##[##1,2: napply (. Exy^-1╪_1#); nassumption;
-##|##3,4: napply (. Exy‡#); nassumption]
+ #A; #S; #S'; @ (S ∩ S');
+ #a; #a'; #Ha; @; *; #H1; #H2; @ 
+  [##1,2: napply (. Ha^-1‡#); nassumption;
+##|##3,4: napply (. Ha‡#); 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 ? (intersect_is_ext ? B C))) 
+unification hint 0 ≔ 
+  A : setoid, B,C : ext_powerclass A;
+  R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
+  
   (* ------------------------------------------*)  ⊢ 
-    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/.
+    ext_carr A R ≡ intersect ? (ext_carr ? B) (ext_carr ? C).
+
+nlemma intersect_is_morph: 
+ ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
+ #A; @ (λS,S'. S ∩ S');
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
+  [ napply Ha1; nassumption
+  | napply Hb1; nassumption
+  | napply Ha2; nassumption
+  | napply Hb2; nassumption]
 nqed.
 
 alias symbol "hint_decl" = "hint_decl_Type1".
-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. 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.
+unification hint 0 ≔ 
+  A : Type[0], B,C : Ω^A;
+  R ≟ (mk_binary_morphism1 …
+       (λS,S'.S ∩ S') 
+       (prop21 … (intersect_is_morph A)))
+   ⊢ 
+    fun21 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A) R B C 
+  ≡ intersect ? B C.
+
+interpretation "prop21 ext" 'prop2 l r = (prop21 (ext_powerclass_setoid ?) (ext_powerclass_setoid ?) ? ? ???? l r).
+
+nlemma intersect_is_ext_morph: 
+ ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
+ #A; @ (intersect_is_ext …); nlapply (prop21 … (intersect_is_morph A));
+#H; #a; #a'; #b; #b'; #H1; #H2; napply H; 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 (intersect_is_ext ? X Y)))
-                (prop11 ?? (fun11 ?? (intersect_is_ext_morph AA) X))) 
-              (prop11 ?? (intersect_is_ext_morph AA))) ,
+      A:setoid, B,C : 𝛀^A;
+      R ≟ (mk_binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A)
+              (λS,S':carr1 (ext_powerclass_setoid A).
+                mk_ext_powerclass A (S∩S') (ext_prop A (intersect_is_ext ? S S'))) 
+              (prop21 … (intersect_is_ext_morph A))) ,
        BB ≟ (ext_carr ? B),
        CC ≟ (ext_carr ? C)
-   (* ---------------------------------------------------------------------------------------*) ⊢ 
-      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.
-
+   (* ------------------------------------------------------*) ⊢ 
+            ext_carr A
+             (fun21 
+              (ext_powerclass_setoid A) 
+              (ext_powerclass_setoid A) 
+              (ext_powerclass_setoid A) R B C) ≡ 
+            intersect (carr A) BB CC.
 
 (*
 alias symbol "hint_decl" = "hint_decl_Type2".
@@ -409,7 +277,7 @@ nqed.
 
 ndefinition image: ∀A,B. (carr A → carr B) → Ω^A → Ω^B ≝
  λA,B:setoid.λf:carr A → carr B.λSa:Ω^A.
-  {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq0 B) (f x) y}.
+  {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) (f x) y}.
 
 ndefinition counter_image: ∀A,B. (carr A → carr B) → Ω^B → Ω^A ≝
  λA,B,f,Sb. {x | ∃y. y ∈ Sb ∧ f x = y}.
@@ -428,55 +296,73 @@ nqed.
 (******************* first omomorphism theorem for sets **********************)
 
 ndefinition eqrel_of_morphism:
- ∀A,B. A ⇒_0 B → compatible_equivalence_relation A.
+ ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
  #A; #B; #f; @
-  [ @ [ napply (λx,y. f x = f y) ] /2/;
+  [ @
+     [ napply (λx,y. f x = f y)
+     | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
-napply (.= (†H)); // ]
+napply (.= (†H)); napply refl ]
 nqed.
 
-ndefinition canonical_proj: ∀A,R. A ⇒_0 (quotient A R).
+ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
  #A; #R; @
-  [ napply (λx.x) |  #a; #a'; #H; napply (compatibility … R … H) ]
+  [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
 nqed.
 
 ndefinition quotiented_mor:
- ∀A,B.∀f:A ⇒_0 B.(quotient … (eqrel_of_morphism … f)) ⇒_0 B.
- #A; #B; #f; @ [ napply f ] //.
+ ∀A,B.∀f:unary_morphism A B.
+  unary_morphism (quotient … (eqrel_of_morphism … f)) B.
+ #A; #B; #f; @
+  [ napply f | #a; #a'; #H; nassumption]
 nqed.
 
 nlemma first_omomorphism_theorem_functions1:
  ∀A,B.∀f: unary_morphism A B.
   ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
-//. nqed.
+ #A; #B; #f; #x; napply refl;
+nqed.
 
 alias symbol "eq" = "setoid eq".
 ndefinition surjective ≝
- λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:A ⇒_0 B.
+ λA,B.λS: ext_powerclass A.λT: ext_powerclass B.λf:unary_morphism A B.
   ∀y. y ∈ T → ∃x. x ∈ S ∧ f x = y.
 
 ndefinition injective ≝
- λA,B.λS: ext_powerclass A.λf:A ⇒_0 B.
+ λA,B.λS: ext_powerclass A.λf:unary_morphism A B.
   ∀x,x'. x ∈ S → x' ∈ S → f x = f x' → x = x'.
 
 nlemma first_omomorphism_theorem_functions2:
- ∀A,B.∀f:A ⇒_0 B. 
+ ∀A,B.∀f: unary_morphism A B. 
    surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
-/3/. nqed.
+ #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl; 
+ (* bug, prova @ I refl *)
+nqed.
 
 nlemma first_omomorphism_theorem_functions3:
- ∀A,B.∀f:A ⇒_0 B. 
+ ∀A,B.∀f: unary_morphism A 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:> A ⇒_0 B;
+ { iso_f:> unary_morphism A 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; *; #xU; #xV; napply H; nassumption;
+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; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
+nqed. 
+
+nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
+#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
+nqed. 
 
 (*
 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝
@@ -499,35 +385,3 @@ ncheck (λA:?.
    ;
  }.
 *)
-
-(* 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.
-