From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Mon, 27 Sep 2010 23:25:57 +0000 (+0000)
Subject: many fixes to setoids for re, 16.1 almost done
X-Git-Tag: make_still_working~2822
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=2c486bbea1d6ffb072d0ff83f9df129b7860f3e1;p=helm.git

many fixes to setoids for re, 16.1 almost done
---

diff --git a/helm/software/matita/nlibrary/depends b/helm/software/matita/nlibrary/depends
index 0f936d55e..96eca527f 100644
--- a/helm/software/matita/nlibrary/depends
+++ b/helm/software/matita/nlibrary/depends
@@ -4,7 +4,7 @@ topology/igft3.ma arithmetics/nat.ma datatypes/bool.ma topology/igft.ma
 basics/functions.ma Plogic/connectives.ma Plogic/equality.ma
 nat/compare.ma datatypes/bool.ma nat/order.ma
 arithmetics/compare.ma arithmetics/nat.ma
-datatypes/list-setoids.ma datatypes/list.ma sets/setoids.ma
+datatypes/list-setoids.ma datatypes/list.ma sets/setoids.ma sets/setoids1.ma
 datatypes/list-theory.ma arithmetics/nat.ma datatypes/list.ma
 logic/pts.ma 
 basics/relations.ma Plogic/connectives.ma
diff --git a/helm/software/matita/nlibrary/depends.dot b/helm/software/matita/nlibrary/depends.dot
index f3f71b61d..0c9dfbc0f 100644
--- a/helm/software/matita/nlibrary/depends.dot
+++ b/helm/software/matita/nlibrary/depends.dot
@@ -19,6 +19,7 @@ digraph g {
   "datatypes/list-setoids.ma" [];
   "datatypes/list-setoids.ma" -> "datatypes/list.ma" [];
   "datatypes/list-setoids.ma" -> "sets/setoids.ma" [];
+  "datatypes/list-setoids.ma" -> "sets/setoids1.ma" [];
   "datatypes/list-theory.ma" [];
   "datatypes/list-theory.ma" -> "arithmetics/nat.ma" [];
   "datatypes/list-theory.ma" -> "datatypes/list.ma" [];
diff --git a/helm/software/matita/nlibrary/depends.png b/helm/software/matita/nlibrary/depends.png
index 63ad7eab1..f1fbaf2f9 100644
Binary files a/helm/software/matita/nlibrary/depends.png and b/helm/software/matita/nlibrary/depends.png differ
diff --git a/helm/software/matita/nlibrary/logic/cprop.ma b/helm/software/matita/nlibrary/logic/cprop.ma
index 1efc042ff..20942ecc8 100644
--- a/helm/software/matita/nlibrary/logic/cprop.ma
+++ b/helm/software/matita/nlibrary/logic/cprop.ma
@@ -48,11 +48,13 @@ ndefinition and_morphism: unary_morphism1 CPROP (unary_morphism1_setoid1 CPROP C
   [ napply (. Ha^-1) | napply (. Hb^-1) | napply (. Ha) | napply (. Hb)] //.
 nqed.
 
-unification hint 0 ≔ A,B ⊢
- mk_unary_morphism1 …
-  (λX.mk_unary_morphism1 … (And X) (prop11 … (and_morphism X)))
-  (prop11 … and_morphism)
-   A B ≡ And A B.
+unification hint 0 ≔ A,B:CProp[0];
+  T ≟ CPROP,
+  MM ≟ mk_unary_morphism1 …
+       (λX.mk_unary_morphism1 … (And X) (prop11 … (and_morphism X)))
+         (prop11 … and_morphism)
+(*-------------------------------------------------------------*) ⊢
+  fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ And A B.
 
 (*
 naxiom daemon: False.
@@ -71,12 +73,14 @@ ndefinition or_morphism: unary_morphism1 CPROP (unary_morphism1_setoid1 CPROP CP
   [ @1; napply (. Ha^-1) | @2; napply (. Hb^-1) | @1; napply (. Ha) | @2; napply (. Hb)] //.
 nqed.
 
-unification hint 0 ≔ A,B ⊢
- mk_unary_morphism1 …
-  (λX.mk_unary_morphism1 … (Or X) (prop11 … (or_morphism X)))
-  (prop11 … or_morphism)
-  A B ≡ Or A B.
-
+unification hint 0 ≔ A,B:CProp[0];
+  T ≟ CPROP,
+  MM ≟ mk_unary_morphism1 …
+       (λX.mk_unary_morphism1 … (Or X) (prop11 … (or_morphism X)))
+         (prop11 … or_morphism)
+(*-------------------------------------------------------------*) ⊢
+  fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ Or A B.
+  
 ndefinition if_morphism: unary_morphism1 CPROP (unary_morphism1_setoid1 CPROP CPROP).
  napply (mk_binary_morphism1 … (λA,B:CProp[0]. A → B));
  #a; #a'; #b; #b'; #Ha; #Hb; @; #H; #x
diff --git a/helm/software/matita/nlibrary/re/re-setoids.ma b/helm/software/matita/nlibrary/re/re-setoids.ma
index dcf1d85fe..3d8682365 100644
--- a/helm/software/matita/nlibrary/re/re-setoids.ma
+++ b/helm/software/matita/nlibrary/re/re-setoids.ma
@@ -26,7 +26,12 @@ naxiom admit : Admit.
 ndefinition if': ∀A,B:CPROP. A = B → A → B.
 #A B; *; /2/. nqed.
 
-ncoercion if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ???? to ∀_:?.?.
+ncoercion if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ? (eq1 CPROP) ?? to ∀_:?.?.
+
+ndefinition ifs': ∀S.∀A,B:Ω^S. A = B → ∀x. x ∈ A → x ∈ B.
+#S A B; *; /2/. nqed.
+
+ncoercion ifs : ∀S.∀A,B:Ω^S. ∀p:A = B.∀x. x ∈ A → x ∈ B ≝ ifs' on _p : eq_rel1 ? (eq1 (powerclass_setoid ?))?? to ∀_:?.?.
 
 (* XXX move to list-setoids-theory.ma *)
 
@@ -205,6 +210,71 @@ ndefinition cat : ∀A:setoid.∀l1,l2:lang A.lang A ≝
   λS.λl1,l2.{ w ∈ list S | ∃w1,w2.w =_0 w1 @ w2 ∧ w1 ∈ l1 ∧ w2 ∈ l2}.
 interpretation "cat lang" 'pc a b = (cat ? a b).
 
+(* hints for cat *)
+nlemma cat_is_morph : ∀A:setoid. (lang A) ⇒_1 (lang A) ⇒_1 (lang A).
+#X; napply (mk_binary_morphism1 … (λA,B:lang X.A · B));
+#A1 A2 B1 B2 EA EB; napply ext_set; #x;
+ncut (∀y,x:list X.(x ∈ B1) =_1 (x ∈ B2)); ##[
+  #_; #y; ncases EA; ncases EB; #h1 h2 h3 h4; @; ##[ napply h1 | napply h2] ##] #YY;
+ncut (∀x,y:list X.(x ∈ A1) =_1 (x ∈ A2)); ##[
+  #y; #y; ncases EA; ncases EB; #h1 h2 h3 h4; @; ##[ napply h3 | napply h4] ##] #XX;
+napply (.=_1 (∑w1, w2. XX w1 w2/ E ; (# ╪_1 E) ╪_1 #));
+napply (.=_1 (∑w1, w2. YY w1 w2/ E ; # ╪_1 E)); //;
+nqed.
+
+nlemma cat_is_ext: ∀A:setoid. (Elang A) → (Elang A) → (Elang A).
+ #S A B; @ (ext_carr … A · ext_carr … B); (* XXX coercion ext_carr che non funge *)
+#x y Exy;
+ncut (∀w1,w2.(x == w1@w2) = (y == w1@w2)); ##[
+  #w1 w2; @; #H; ##[ napply (.= Exy^-1) | napply (.= Exy)] // ] 
+#E; @; #H;
+##[ napply (. (∑w1,w2. (E w1 w2)^-1 / E ; (E ╪_1 #) ╪_1 #)); napply H;
+##| napply (. (∑w1,w2. E w1 w2 / E ; (E ╪_1 #) ╪_1 #)); napply H ]
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, B,C :  Elang A;
+   AA ≟ LIST A,
+   R ≟ mk_ext_powerclass ? 
+         (ext_carr ? B · ext_carr ? C) (ext_prop ? (cat_is_ext ? B C))
+(*--------------------------------------------------------------------*)  ⊢
+    ext_carr AA R ≡ cat A (ext_carr AA B) (ext_carr AA C).
+    
+unification hint 0 ≔ S:setoid, A,B:lang S;
+    T ≟ powerclass_setoid (list S),
+    TT ≟ unary_morphism1_setoid1 T T,
+    MM ≟ mk_unary_morphism1 T TT  
+          (λA:lang S.
+             mk_unary_morphism1 T T 
+               (λB:lang S.A · B) (prop11 T T (cat_is_morph S A)))
+          (prop11 T TT (cat_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T T (fun11 T (unary_morphism1_setoid1 T T) MM A) B ≡ cat S A B.
+   
+nlemma cat_is_ext_morph:∀A:setoid.(Elang A) ⇒_1 (Elang A) ⇒_1 (Elang A).
+#A; napply (mk_binary_morphism1 …  (cat_is_ext …));
+#x1 x2 y1 y2 Ex Ey; napply (prop11 … (cat_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔ AA : setoid, B,C : Elang AA;
+  AAS ≟ LIST AA,
+  T ≟ ext_powerclass_setoid (list AA),
+  TT ≟ unary_morphism1_setoid1 T T,
+  R ≟ mk_unary_morphism1 ??
+          (λS:Elang AA.
+           mk_unary_morphism1 ??
+            (λS':Elang AA.
+              mk_ext_powerclass (list AA) (ext_carr ? S · ext_carr ? S') 
+                (ext_prop (list AA) (cat_is_ext AA S S')))
+            (prop11 ?? (cat_is_ext_morph AA S)))
+          (prop11 ?? (cat_is_ext_morph AA)),
+   BB ≟ ext_carr ? B,
+   CC ≟ ext_carr ? C
+(*------------------------------------------------------*) ⊢
+   ext_carr AAS (fun11 T T (fun11 T TT R B) C) ≡ cat AA BB CC.
+
+(* end hints for cat *)
+
 ndefinition star : ∀A:setoid.∀l:lang A.lang A ≝ 
   λS.λl.{ w ∈ list S | ∃lw.flatten ? lw = w ∧ conjunct ? lw l}. 
 interpretation "star lang" 'pk l = (star ? l).
@@ -221,6 +291,75 @@ match r with
 notation "𝐋 term 70 E" non associative with precedence 75 for @{'L_re $E}.
 interpretation "in_l" 'L_re E = (L_re ? E).
 
+(* support for 𝐋 as an extensional set *)
+ndefinition L_re_is_ext : ∀S:Alpha.∀r:re S.Elang S.
+#S r; @(𝐋 r); #w1 w2 E; nelim r; 
+##[ ##1,2: /2/; @; #defw1; napply (.=_0 (defw1 : [ ] = ?)); //; napply (?^-1); //;
+##| #x; @; #defw1; napply (.=_0 (defw1 : [x] = ?)); //; napply (?^-1); //;
+##| #e1 e2 H1 H2; (* not shure I shoud Inline *)
+    @; *; #s1; *; #s2; *; *; #defw1 s1L1 s2L2; 
+    ##[ nlapply (trans … E^-1 defw1); #defw2; 
+    ##| nlapply (trans … E defw1); #defw2; ##] @s1; @s2; /3/;
+##| #e1 e2 H1 H2; napply (H1‡H2); (* good! *)
+##| #e H; @; *; #l; *; #defw1 Pl; @l; @; //; napply (.=_1 defw1); /2/; ##]
+nqed.
+
+unification hint 0 ≔ S : Alpha,e : re S; 
+  SS ≟ LIST S,
+  X ≟ mk_ext_powerclass SS (𝐋 e) (ext_prop SS (L_re_is_ext S e))
+(*-----------------------------------------------------------------*)⊢ 
+  ext_carr SS X ≡ L_re S e.
+
+nlemma L_re_is_morph:∀A:Alpha.(setoid1_of_setoid (re A)) ⇒_1 Ω^(list A).
+#A; @; ##[ napply (λr:re A.𝐋 r); ##] #r1; nelim r1;
+##[##1,2: #r2; ncases r2; //; ##[##1,6: *|##2,7,5,12,10: #a; *|##3,4,8,9: #a1 a2; *]
+##|#x r2; ncases r2; ##[##1,2: *|##4,5: #a1 a2; *|##6: #a; *] #y E; @; #z defz;
+   ncases z in defz; ##[##1,3: *] #zh ztl; ncases ztl; ##[##2,4: #d dl; *; #_; *]
+   *; #defx; #_; @; //; napply (?^-1); napply (.= defx^-1); //; napply (?^-1); //;
+##|#e1 e2 IH1 IH2 r2; ncases r2; ##[##1,2: *|##5: #a1 a2; *|##3,6: #a1; *]
+   #f1 f2; *; #E1 E2; nlapply (IH2 … E2); nlapply (IH1 … E1); #H1 H2;
+   nchange in match (𝐋 (e1 · e2)) with (?·?);
+   napply (.=_1 (H1 ╪_1 H2)); //;
+##|#e1 e2 IH1 IH2 r2; ncases r2; ##[##1,2: *|##4: #a1 a2; *|##3,6: #a1; *]
+   #f1 f2; *; #E1 E2; nlapply (IH2 … E2); nlapply (IH1 … E1); #H1 H2;
+   napply (.=_1 H1╪_1H2); //;
+##|#r IH r2; ncases r2; ##[##1,2: *|##4,5: #a1 a2; *|##3: #a1; *]
+   #e; #defe; nlapply (IH e defe); #H;
+   @; #x; *; #wl; *; #defx Px; @wl; @; //; nelim wl in Px; //; #l ls IH; *; #lr Pr;
+   ##[ nlapply (ifs' … H … lr) | nlapply (ifs' … H^-1 … lr) ] #le; 
+   @; ##[##1,3: nassumption] /2/; ##]
+nqed.
+
+unification hint 0 ≔ A:Alpha, a:re A;
+  T ≟ setoid1_of_setoid (RE A),
+  T1 ≟ LIST A,
+  T2 ≟ powerclass_setoid T1,
+  MM ≟ mk_unary_morphism1 ?? 
+         (λa:setoid1_of_setoid (RE A).𝐋 a) (prop11 ?? (L_re_is_morph A))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T T2 MM a ≡  𝐋 a.
+   
+nlemma L_re_is_ext_morph:∀A:Alpha.(setoid1_of_setoid (re A)) ⇒_1 𝛀^(list A).
+#A; @; ##[ #a; napply (L_re_is_ext ? a); ##] #a b E; @; #x H;
+##[ nchange with (x ∈ 𝐋 b); napply (. #╪_1?); 
+    ##[ nchange with (𝐋 b = ?); napply (.= ┼_1 E^-1); napply #| ##skip]
+    nassumption;
+##| nchange with (x ∈ 𝐋 a); napply (. #╪_1?); 
+    ##[ nchange with (𝐋 a = ?); napply (.= ┼_1 E); napply #| ##skip]
+    nassumption; ##]
+nqed.
+            
+unification hint 1 ≔  AA : Alpha, a: re AA;
+  T ≟ RE AA, T1 ≟ LIST AA, TT ≟ ext_powerclass_setoid T1,
+  R ≟ mk_unary_morphism1 ??
+       (λa:setoid1_of_setoid T.
+         mk_ext_powerclass ? (𝐋 a) (ext_prop ? (L_re_is_ext AA a)))
+            (prop11 ?? (L_re_is_ext_morph AA))
+(*------------------------------------------------------*) ⊢
+   ext_carr T1 (fun11 (setoid1_of_setoid T) TT R a) ≡ L_re AA a.
+
+(* end support for 𝐋 as an extensional set *)
+
 ninductive pitem (S: Type[0]) : Type[0] ≝
    pz: pitem S
  | pe: pitem S
@@ -278,13 +417,13 @@ unification hint 0 ≔ SS:Alpha;
     P1 ≟ refl ? (eq0 (PITEM SS)),
     P2 ≟ sym ? (eq0 (PITEM SS)),
     P3 ≟ trans ? (eq0 (PITEM SS)),
-    R ≟ mk_setoid (pitem S) (mk_equivalence_relation (pitem A) (eq_pitem SS) P1 P2 P3)
-(*---------------------------*)⊢
+    R ≟ mk_setoid (pitem S) 
+         (mk_equivalence_relation (pitem A) (eq_pitem SS) P1 P2 P3)
+(*-----------------------------------------------------------------*)⊢
     carr R ≡ pitem A.
     
 unification hint 0 ≔ S:Alpha,a,b:pitem S;
-   R ≟ PITEM S,
-   L ≟ (pitem S)
+   R ≟ PITEM S,  L ≟ (pitem S)
 (* -------------------------------------------- *) ⊢
    eq_pitem S a b ≡ eq_rel L (eq0 R) a b.    
     
@@ -332,18 +471,15 @@ ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
 ##| #x; @; *;
 ##| #x; @; #H; nchange in H with ([?] =_0 ?); ##[ napply ((.=_0 H) E); ##]
     napply ((.=_0 H) E^-1);
-##| #e1 e2 H1 H2; (*
-    nchange in match (w1 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?);
-    nchange in match (w2 ∈ 𝐋\p (?·?)) with ((∃_.?)∨?); good! *)
+##| #e1 e2 H1 H2;
     napply (.= (#‡H2));
     ncut (∀x1,x2. (w1 = (x1@x2)) = (w2 = (x1@x2)));##[
       #x1 x2; @; #X; ##[ napply ((.= E^-1) X) | napply ((.= E) X) ] ##] #X;
     napply ((∑w1,w2. X w1 w2 / H ; (H╪_1#)╪_1#) ╪_1 #); 
-##| #e1 e2 H1 H2; napply (H1‡H2); (* good! *)
+##| #e1 e2 H1 H2; napply (H1‡H2); 
 ##| #e H; 
     ncut (∀x1,x2.(w1 = (x1@x2)) = (w2 = (x1@x2)));##[
       #x1 x2; @; #X; ##[ napply ((.= E^-1) X) | napply ((.= E) X) ] ##] #X;
-    (* nnormalize in ⊢ (???%%); good! (but a bit too hard) *)
     napply (∑w1,w2. X w1 w2 / H ; (H╪_1#)╪_1#); 
 ##]
 nqed.
@@ -454,10 +590,7 @@ nqed.
 
 (* theorem 16: 2 *)
 nlemma oplus_cup : ∀S:Alpha.∀e1,e2:pre S.𝐋\p (e1 ⊕ e2) = 𝐋\p e1 ∪ 𝐋\p e2.
-#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2; (* oh my!
-nwhd in ⊢ (???(??%)?);
-nchange in ⊢(???%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
-nchange in ⊢(???(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2)); *)
+#S r1; ncases r1; #e1 b1 r2; ncases r2; #e2 b2;
 napply (.=_1 #╪_1 (epsilon_or ???));
 napply (.=_1 (cupA…)^-1);
 napply (.=_1 (cupA…)╪_1#);
@@ -541,6 +674,16 @@ napply (. ((defw1 : [ ] = ?)^-1 ╪_0 #)╪_1#);
 napply Hw2; 
 nqed.
 
+(* XXX This seems to be a pattern for equations *)
+alias symbol "hint_decl" (instance 1) = "hint_decl_CProp2".
+unification hint 0 ≔ S : Alpha, x,y: re S;
+  SS ≟ RE S,
+  TT ≟ setoid1_of_setoid SS,
+  T ≟ carr1 TT
+(*-----------------------------------------*) ⊢ 
+  eq_re S x y ≡ eq_rel1 T (eq1 TT) x y.    
+(* XXX the previous hint does not work *)
+
 (* theorem 16: 1 → 3 *)
 nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
       𝐋\p (•(\fst e2)) =  𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| → 
@@ -555,16 +698,19 @@ nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
     napply (.=_1 (# ╪_1 (cupC …))); napply (.=_1 (cupA …)); 
     napply (.=_1 (# ╪_1 (cupA …)^-1)); (* XXX slow, but not because of disamb! *)
     ncut (𝐋\p e2'' ∪ ϵ b2'' =  𝐋\p e2' ∪ 𝐋  |e2'|); ##[
-      nchange with (𝐋\p 〈e2'',b2''〉 =  𝐋\p e2' ∪ 𝐋 |e2'|);
       napply (?^-1); napply (.=_1 th1^-1); //;##] #E;
     napply (.=_1 (# ╪_1 (E ╪_1 #)));
-    STOP  
-      
-    nrewrite > (eta_lp ? e2); 
-    nchange in match (𝐋\p 〈\fst e2,?〉) with (𝐋\p e2'∪ ϵ b2');
-    nrewrite > (cup_dotD …); nrewrite > (epsilon_dot…);       
-    nrewrite > (cupC ? (𝐋\p e2')…); nrewrite > (cupA…);nrewrite > (cupA…);
-    nrewrite < (erase_bull S e2') in ⊢ (???(??%?)); //;
+    napply (?^-1);
+    napply (.=_1 (cup_dotD …) ╪_1 #);
+    napply (.=_1 (# ╪_1 (epsilon_dot …)) ╪_1 #); 
+    napply (?^-1);
+    napply (.=_1 # ╪_1 ((cupC …) ╪_1 #));
+    napply (.=_1 (cupA …)^-1);
+    napply (.=_1 (cupA …)^-1 ╪_1 #);
+    napply (.=_1 (cupA …));
+    nlapply (erase_bull S e2'); #XX;
+    napply (.=_1 (((# ╪_1 (┼_1 ?) )╪_1 #)╪_1 #)); ##[##2: napply XX; ##| ##skip]
+    //;   
 ##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
     nchange in match (𝐋\p ?) with (?∪?);
     nchange in match (𝐋\p (e1'·?)) with (?∪?);
diff --git a/helm/software/matita/nlibrary/sets/setoids1.ma b/helm/software/matita/nlibrary/sets/setoids1.ma
index 48b7d3fcc..068334183 100644
--- a/helm/software/matita/nlibrary/sets/setoids1.ma
+++ b/helm/software/matita/nlibrary/sets/setoids1.ma
@@ -35,12 +35,19 @@ ndefinition setoid1_of_setoid: setoid → setoid1.
  #s; @ (carr s); @ (eq0…) (refl…) (sym…) (trans…);
 nqed.
 
-
+alias symbol "hint_decl" = "hint_decl_CProp2".
 alias symbol "hint_decl" (instance 1) = "hint_decl_Type2".
-unification hint 0 ≔ A,x,y
+unification hint 0 ≔ A,x,y;
+   T  ≟ carr A, 
+   R  ≟ setoid1_of_setoid A,
+   T1 ≟ carr1 R
+(*-----------------------------------------------*) ⊢
+   eq_rel T (eq0 A) x y ≡ eq_rel1 T1 (eq1 R) x y.
+
+unification hint 0 ≔ A;
+   R  ≟ setoid1_of_setoid A
 (*-----------------------------------------------*) ⊢
-   eq_rel ? (eq0 A) x y ≡ eq_rel1 ? (eq1 (setoid1_of_setoid A)) x y.
-(* XXX capire come mai questa hint non funziona se porto su (setoid1_of_setoid A) *)
+   carr A ≡ carr1 R.
 
 interpretation "setoid1 eq" 'eq t x y = (eq_rel1 ? (eq1 t) x y).
 interpretation "setoid eq" 'eq t x y = (eq_rel ? (eq0 t) x y).
diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index aae969ed2..d547fbbbf 100644
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -148,7 +148,7 @@ nqed.
 (* hints for ∩ *)
 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‡#); nassumption;
+##[##1,2: napply (. Exy^-1╪_1#); nassumption;
 ##|##3,4: napply (. Exy‡#); nassumption]
 nqed.
 
@@ -166,11 +166,12 @@ 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 ??
        (λS. mk_unary_morphism1 ?? (λS'.S ∩ S') (prop11 ?? (intersect_is_morph A S))) 
        (prop11 ?? (intersect_is_morph A))
 (*------------------------------------------------------------------------*) ⊢ 
-    fun11 ?? (fun11 ?? R B) C  ≡ intersect A B C.
+    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 ?)
@@ -222,11 +223,12 @@ unification hint 0 ≔
     ext_carr A R ≡ union ? (ext_carr ? B) (ext_carr ? C).
 
 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 ?? (union_is_morph S A)))
         (prop11 ?? (union_is_morph S))
 (*--------------------------------------------------------------------------*) ⊢
-   fun11 ?? (fun11 ?? MM A) B ≡ A ∪ B.
+   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 …));
@@ -271,11 +273,12 @@ unification hint 0 ≔
     ext_carr A R ≡ substract ? (ext_carr ? B) (ext_carr ? C).
 
 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 ?? (substract_is_morph S A)))
         (prop11 ?? (substract_is_morph S))
 (*--------------------------------------------------------------------------*) ⊢
-   fun11 ?? (fun11 ?? MM A) B ≡ A - B.
+   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 …));
@@ -312,10 +315,11 @@ unification hint 0 ≔ A : setoid, a:A;
     ext_carr A R ≡ singleton A a.
 
 unification hint 0 ≔ A:setoid, a:A;
+  T ≟ setoid1_of_setoid A,
   MM ≟ mk_unary_morphism1 ?? 
          (λa:setoid1_of_setoid A.{(a)}) (prop11 ?? (single_is_morph A))
 (*--------------------------------------------------------------------------*) ⊢
-   fun11 ?? MM a ≡ {(a)}.
+   fun11 T (powerclass_setoid A) 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.