X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fsets%2Fsets.ma;h=d4a13507c79bf6790b4d8dce4da16266dc8fb9cc;hb=c8ec2f3e2aaa71efa702f86bacc95e393778a56f;hp=a7e033c15c43119ecec9380f98fa0769fc2c7236;hpb=dc6723959a8e035e2935ca66bd3043de8eb11b42;p=helm.git

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index a7e033c15..d4a13507c 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -28,176 +28,338 @@ interpretation "subseteq" 'subseteq U V = (subseteq ? U V).
 ndefinition overlaps ≝ λA.λU,V.∃x:A.x ∈ U ∧ x ∈ V.
 interpretation "overlaps" 'overlaps U V = (overlaps ? U V).
 
-ndefinition intersect ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∧ x ∈ V }.
+ndefinition intersect ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∧ x ∈ V }.
 interpretation "intersect" 'intersects U V = (intersect ? U V).
 
-ndefinition union ≝ λA.λU,V:Ω \sup A.{ x | x ∈ U ∨ x ∈ V }.
+ndefinition union ≝ λA.λU,V:Ω^A.{ x | x ∈ U ∨ x ∈ V }.
 interpretation "union" 'union U V = (union ? U V).
 
-nlemma subseteq_refl: ∀A.∀S: Ω \sup A. S ⊆ S.
- #A; #S; #x; #H; nassumption.
-nqed.
+ndefinition big_union ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∃i. i ∈ T ∧ x ∈ f i }.
 
-nlemma subseteq_trans: ∀A.∀S,T,U: Ω \sup A. S ⊆ T → T ⊆ U → S ⊆ U.
- #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
-nqed.
+ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈ T → x ∈ f i }.
+
+ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
+
+nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
+//.nqed.
+
+nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
+/3/.nqed.
 
 include "properties/relations1.ma".
 
-ndefinition seteq: ∀A. equivalence_relation1 (Ω \sup A).
- #A; napply mk_equivalence_relation1
+ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
+ #A; @
   [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
-  | #S; napply conj; napply subseteq_refl
-  | #S; #S'; *; #H1; #H2; napply conj; nassumption
-  | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; napply conj; napply subseteq_trans;
-     ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
-nqed. 
+  | /2/
+  | #S; #S'; *; /3/
+  | #S; #T; #U; *; #H1; #H2; *; /4/]
+nqed.
 
 include "sets/setoids1.ma".
 
+(* 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; napply mk_setoid1
-  [ napply (Ω \sup A)
-  | napply seteq ]
-nqed. 
+ #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.
 
 (************ SETS OVER SETOIDS ********************)
 
 include "logic/cprop.ma".
 
-nrecord qpowerclass (A: setoid) : Type[1] ≝
- { pc:> Ω \sup A;
-   mem_ok': ∀x,x':A. x=x' → (x ∈ pc) = (x' ∈ pc) 
+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 70 
+for @{ 'ext_powerclass $A }.
 
-ndefinition qseteq: ∀A. equivalence_relation1 (qpowerclass A).
- #A; napply mk_equivalence_relation1
-  [ napply (λS,S':qpowerclass A. eq_rel1 ? (eq1 (powerclass_setoid A)) S S')
-  | #S; napply (refl1 ? (seteq A))
-  | #S; #S'; napply (sym1 ? (seteq A))
-  | #S; #T; #U; napply (trans1 ? (seteq A))]
+interpretation "extensional powerclass" 'ext_powerclass a = (ext_powerclass a).
+
+ndefinition Full_set: ∀A. 𝛀^A.
+ #A; @[ napply A | #x; #x'; #H; napply refl1]
+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/.
 nqed.
 
-ndefinition qpowerclass_setoid: setoid → setoid1.
- #A; napply mk_setoid1
-  [ napply (qpowerclass A)
-  | napply (qseteq A) ]
+ndefinition ext_powerclass_setoid: setoid → setoid1.
+ #A; @ (ext_seteq A).
 nqed.
+              
+unification hint 0 ≔ A;
+      R ≟ (mk_setoid1 (𝛀^A) (eq1 (ext_powerclass_setoid A)))
+  (* ----------------------------------------------------- *) ⊢  
+                 carr1 R ≡ ext_powerclass A.
 
-unification hint 0 (∀A. (λx,y.True) (carr1 (qpowerclass_setoid A)) (qpowerclass A)).
-ncoercion qpowerclass_hint: ∀A: setoid. ∀S: qpowerclass_setoid A. Ω \sup A ≝ λA.λS.S
- on _S: (carr1 (qpowerclass_setoid ?)) to (Ω \sup ?). 
+(*
+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_ok: ∀A. binary_morphism1 (setoid1_of_setoid A) (qpowerclass_setoid A) CPROP.
- #A; napply mk_binary_morphism1
-  [ napply (λx.λS: qpowerclass_setoid A. x ∈ S) (* CSC: ??? *)
-  | #a; #a'; #b; #b'; #Ha; #Hb; (* CSC: qui *; non funziona *)
-    nwhd; nwhd in ⊢ (? (? % ??) (? % ??)); napply mk_iff; #H
-     [ ncases Hb; #Hb1; #_; napply Hb1; napply (. (mem_ok' …))
-        [ nassumption | napply Ha^-1 | ##skip ]
-   ##| ncases Hb; #_; #Hb2; napply Hb2; napply (. (mem_ok' …))
-        [ nassumption | napply Ha | ##skip ]##]
+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/.
 nqed.
 
-unification hint 0 (∀A,x,S. (λx,y.True) (fun21 ??? (mem_ok A) x S) (mem A S x)).
+unification hint 0 ≔  A:setoid, x, S;  
+     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).
+ #A; napply (mk_binary_morphism1 … (λS,S':𝛀^A. S ⊆ S'));
+ #a; #a'; #b; #b'; *; #H1; #H2; *; /5/.
+nqed.
+
+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.
+ #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 : ext_powerclass A;
+  R ≟ (mk_ext_powerclass ? (B ∩ C) (ext_prop ? (intersect_is_ext ? B C)))
   
-nlemma subseteq_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) CPROP.
- #A; napply mk_binary_morphism1
-  [ napply (λS,S': qpowerclass_setoid ?. S ⊆ S')
-  | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; napply mk_iff; #H
-     [ napply (subseteq_trans … a' a) (* anche qui, perche' serve a'? *)
-        [ nassumption | napply (subseteq_trans … a b); nassumption ]
-   ##| napply (subseteq_trans … a a') (* anche qui, perche' serve a'? *)
-        [ nassumption | napply (subseteq_trans … a' b'); nassumption ] ##]
-nqed.
-
-nlemma intersect_ok: ∀A. binary_morphism1 (qpowerclass_setoid A) (qpowerclass_setoid A) (qpowerclass_setoid A).
- #A; napply mk_binary_morphism1
-  [ #S; #S'; napply mk_qpowerclass
+  (* ------------------------------------------*)  ⊢ 
+    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/.
+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.
+
+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)).
+ #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))) ,
+       BB ≟ (ext_carr ? B),
+       CC ≟ (ext_carr ? C)
+   (* ------------------------------------------------------*) ⊢ 
+            ext_carr A (R B C) ≡ intersect (carr A) BB CC.
+
+(*
+alias symbol "hint_decl" = "hint_decl_Type2".
+unification hint 0 ≔
+  A : setoid, B,C : 𝛀^A ;
+  CC ≟ (ext_carr ? C),
+  BB ≟ (ext_carr ? B),
+  C1 ≟ (carr1 (powerclass_setoid (carr A))),
+  C2 ≟ (carr1 (ext_powerclass_setoid A))
+  ⊢ 
+     eq_rel1 C1 (eq1 (powerclass_setoid (carr A))) BB CC ≡ 
+          eq_rel1 C2 (eq1 (ext_powerclass_setoid A)) B C.
+          
+unification hint 0 ≔
+  A, B : CPROP ⊢ iff A B ≡ eq_rel1 ? (eq1 CPROP) A B.    
+    
+nlemma test: ∀U.∀A,B:𝛀^U. A ∩ B = A →
+ ∀x,y. x=y → x ∈ A → y ∈ A ∩ B.
+ #U; #A; #B; #H; #x; #y; #K; #K2;
+  alias symbol "prop2" = "prop21 mem".
+  alias symbol "invert" = "setoid1 symmetry".
+  napply (. K^-1‡H);
+  nassumption;
+nqed. 
+
+
+nlemma intersect_ok: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) (ext_powerclass_setoid A).
+ #A; @
+  [ #S; #S'; @
      [ napply (S ∩ S')
-     | #a; #a'; #Ha; nwhd in ⊢ (? ? ? % %); napply mk_iff; *; #H1; #H2; napply conj
-        [##1,2: napply (. (mem_ok' …)^-1) [##3,6: nassumption |##1,4: nassumption |##*: ##skip]
-      ##|##3,4: napply (. (mem_ok' …)) [##2,5: nassumption |##1,4: nassumption |##*: ##skip]##]##]
- ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; napply conj; #x; nwhd in ⊢ (% → %); #H
-      [ napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
+     | #a; #a'; #Ha;
+        nwhd in ⊢ (? ? ? % %); @; *; #H1; #H2; @
+        [##1,2: napply (. Ha^-1‡#); nassumption;
+      ##|##3,4: napply (. Ha‡#); nassumption]##]
+ ##| #a; #a'; #b; #b'; #Ha; #Hb; nwhd; @; #x; nwhd in ⊢ (% → %); #H
+      [ alias symbol "invert" = "setoid1 symmetry".
+        alias symbol "refl" = "refl".
+alias symbol "prop2" = "prop21".
+napply (. ((#‡Ha^-1)‡(#‡Hb^-1))); nassumption
       | napply (. ((#‡Ha)‡(#‡Hb))); nassumption ]##]
 nqed.
 
-unification hint 0 (∀A.∀U,V.(λx,y.True) (fun21 ??? (intersect_ok A) U V) (intersect A U V)).
+(* unfold if intersect, exposing fun21 *)
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ 
+  A : setoid, B,C : ext_powerclass A ⊢ 
+    pc A (fun21 …
+            (mk_binary_morphism1 …
+              (λS,S':qpowerclass_setoid A.mk_qpowerclass ? (S ∩ S') (mem_ok' ? (intersect_ok ? S S'))) 
+              (prop21 … (intersect_ok A))) 
+            B
+            C) 
+    ≡ intersect ? (pc ? B) (pc ? C).
 
 nlemma test: ∀A:setoid. ∀U,V:qpowerclass A. ∀x,x':setoid1_of_setoid A. x=x' → x ∈ U ∩ V → x' ∈ U ∩ V.
- #A; #U; #V; #x; #x'; #H; #p;
-  (* CSC: senza la change non funziona! *)
-  nchange with (x' ∈ (fun21 ??? (intersect_ok A) U V));
-  napply (. (H^-1‡#)); nassumption.
+ #A; #U; #V; #x; #x'; #H; #p; napply (. (H^-1‡#)); nassumption.
 nqed.
+*)
 
-(*
-(* qui non funziona una cippa *)
-ndefinition image: ∀A,B. (carr A → carr B) → Ω \sup A → Ω \sup B ≝
- λA,B:setoid.λf:carr A → carr B.λSa:Ω \sup A.
-  {y | ∃x. x ∈ Sa ∧ eq_rel (carr B) (eq B) ? ?(*(f x) y*)}.
-  ##[##2: napply (f x); ##|##3: napply y]
- #a; #a'; #H; nwhd; nnormalize; (* per togliere setoid1_of_setoid *) napply (mk_iff ????);
- *; #x; #Hx; napply (ex_intro … x)
-  [ napply (. (#‡(#‡#))); 
-
-ndefinition counter_image: ∀A,B. (A → B) → Ω \sup B → Ω \sup A ≝
+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) (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}.
-*)
 
 (******************* compatible equivalence relations **********************)
 
 nrecord compatible_equivalence_relation (A: setoid) : Type[1] ≝
  { rel:> equivalence_relation A;
-   compatibility: ∀x,x':A. x=x' → eq_rel ? rel x x' (* coercion qui non va *)
+   compatibility: ∀x,x':A. x=x' → rel x x'
  }.
 
 ndefinition quotient: ∀A. compatible_equivalence_relation A → setoid.
- #A; #R; napply mk_setoid
-  [ napply A
-  | napply R]
+ #A; #R; @ A R; 
 nqed.
 
 (******************* first omomorphism theorem for sets **********************)
 
 ndefinition eqrel_of_morphism:
  ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
- #A; #B; #f; napply mk_compatible_equivalence_relation
-  [ napply mk_equivalence_relation
-     [ napply (λx,y. f x = f y)
-     | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
-##| #x; #x'; #H; nwhd; napply (.= (†H)); napply refl ]
+ #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).
- #A; #R; napply mk_unary_morphism
-  [ napply (λx.x) | #a; #a'; #H; napply (compatibility ? R … H) ]
+ #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; napply mk_unary_morphism
-  [ napply f | #a; #a'; #H; nassumption]
+  unary_morphism (quotient … (eqrel_of_morphism … f)) B.
+ #A; #B; #f; @ [ napply f ] //.
 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).
- #A; #B; #f; #x; napply refl;
-nqed.
+  ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
+//. nqed.
 
-ndefinition surjective ≝ λA,B.λf:unary_morphism A B. ∀y.∃x. f x = y.
+alias symbol "eq" = "setoid eq".
+ndefinition surjective ≝
+ 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.λf:unary_morphism A B. ∀x,x'. f x = f x' → x = x'.
+ndefinition injective ≝
+ 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: unary_morphism A B. surjective ?? (canonical_proj ? (eqrel_of_morphism ?? f)).
- #A; #B; #f; nwhd; #y; napply (ex_intro … y); napply refl.
-nqed.
+ ∀A,B.∀f: unary_morphism A 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. injective ?? (quotiented_mor ?? f).
- #A; #B; #f; nwhd; #x; #x'; #H; nassumption.
-nqed.
\ No newline at end of file
+ ∀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:> 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; *; /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] ≝
+ { iso_f:> unary_morphism A B;
+   f_closed: ∀x. x ∈ pc A S → fun1 ?? iso_f x ∈ pc B T}.
+   
+   
+ncheck (λA:?.
+   λB:?.
+    λS:?.
+     λT:?.
+      λxxx:isomorphism A B S T.
+       match xxx
+       return λxxx:isomorphism A B S T.
+               ∀x: carr A.
+                ∀x_72: mem (carr A) (pc A S) x.
+                 mem (carr B) (pc B T) (fun1 A B ((λ_.?) A B S T xxx) x)
+        with [ mk_isomorphism _ yyy ⇒ yyy ] ).   
+   
+   ;
+ }.
+*)
\ No newline at end of file