X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fre%2Fre-setoids.ma;h=729e8a0ea3a7806d55b8a334d952b613daee4aef;hb=7761bd5341f81e4b7d335c0b47bc1e2271f02227;hp=ced520b05abf2276999e04ad0409131b9742cc9b;hpb=27c34e93fc35402111253325b93089a6308dd4bb;p=helm.git

diff --git a/helm/software/matita/nlibrary/re/re-setoids.ma b/helm/software/matita/nlibrary/re/re-setoids.ma
index ced520b05..729e8a0ea 100644
--- a/helm/software/matita/nlibrary/re/re-setoids.ma
+++ b/helm/software/matita/nlibrary/re/re-setoids.ma
@@ -12,76 +12,28 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "datatypes/pairs.ma".
-include "datatypes/bool.ma".
+include "datatypes/pairs-setoids.ma".
+include "datatypes/bool-setoids.ma".
+include "datatypes/list-setoids.ma".
 include "sets/sets.ma".
 
+(*
 ninductive Admit : CProp[0] ≝ .
 naxiom admit : Admit.
+*)
 
-(* single = is for the abstract equality of setoids, == is for concrete 
-   equalities (that may be lifted to the setoid level when needed *)
-notation < "hvbox(a break mpadded width -50% (=)= b)" non associative with precedence 45 for @{ 'eq_low $a $b }.
-notation > "a == b" non associative with precedence 45 for @{ 'eq_low $a $b }.
-
-
-(* XXX move to lists.ma *)
-ninductive list (A:Type[0]) : Type[0] ≝ 
-  | nil: list A
-  | cons: A -> list A -> list A.
-  
-nlet rec eq_list (A : setoid) (l1, l2 : list A) on l1 : CProp[0] ≝ 
-match l1 with
-[ nil ⇒ match l2 return λ_.CProp[0] with [ nil ⇒ True | _ ⇒ False ]
-| cons x xs ⇒ match l2 with [ nil ⇒ False | cons y ys ⇒ x = y ∧ eq_list ? xs ys]].
-   
-interpretation "eq_list" 'eq_low a b = (eq_list ? a b).
-   
-ndefinition LIST : setoid → setoid.
-#S; @(list S); @(eq_list S);
-##[ #l; nelim l; //; #; @; //;
-##| #l1; nelim l1; ##[ #y; ncases y; //] #x xs H y; ncases y; ##[*] #y ys; *; #; @; /2/;
-##| #l1; nelim l1; ##[ #l2 l3; ncases l2; ncases l3; /3/; #z zs y ys; *] 
-    #x xs H l2 l3; ncases l2; ncases l3; /2/; #z zs y yz; *; #H1 H2; *; #H3 H4; @; /3/;##]
-nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ S : setoid;
-  T ≟ carr S,
-  P1 ≟ refl ? (eq0 (LIST S)),
-  P2 ≟ sym ? (eq0 (LIST S)),
-  P3 ≟ trans ? (eq0 (LIST S)),
-  X ≟ mk_setoid (list S) (mk_equivalence_relation ? (eq_list T) P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
-     carr X ≡ list T.
-
-unification hint 0 ≔ S:setoid,a,b:list S;
-   R ≟ eq0 (LIST S),
-   L ≟ (list S)
-(* -------------------------------------------- *) ⊢
-   eq_list S a b ≡ eq_rel L R a b.
-
-notation "hvbox(hd break :: tl)"
-  right associative with precedence 47
-  for @{'cons $hd $tl}.
-
-notation "[ list0 x sep ; ]"
-  non associative with precedence 90
-  for ${fold right @'nil rec acc @{'cons $x $acc}}.
+(* XXX move somewere else *)
+ndefinition if': ∀A,B:CPROP. A = B → A → B.
+#A B; *; /2/. nqed.
 
-notation "hvbox(l1 break @ l2)"
-  right associative with precedence 47
-  for @{'append $l1 $l2 }.
+ncoercion if : ∀A,B:CPROP. ∀p:A = B. A → B ≝ if' on _p : eq_rel1 ? (eq1 CPROP) ?? to ∀_:?.?.
 
-interpretation "nil" 'nil = (nil ?).
-interpretation "cons" 'cons hd tl = (cons ? hd tl).
+ndefinition ifs': ∀S.∀A,B:Ω^S. A = B → ∀x. x ∈ A → x ∈ B.
+#S A B; *; /2/. nqed.
 
-nlet rec append A (l1: list A) l2 on l1 ≝ 
-  match l1 with
-  [ nil ⇒ l2
-  | cons hd tl ⇒ hd :: append A tl l2 ].
+ncoercion ifs : ∀S.∀A,B:Ω^S. ∀p:A = B.∀x. x ∈ A → x ∈ B ≝ ifs' on _p : eq_rel1 ? (eq1 (powerclass_setoid ?))?? to ∀_:?.?.
 
-interpretation "append" 'append l1 l2 = (append ? l1 l2).
+(* XXX move to list-setoids-theory.ma *)
 
 ntheorem append_nil: ∀A:setoid.∀l:list A.l @ [] = l.
 #A;#l;nelim l;//; #a;#l1;#IH;nnormalize;/2/;nqed.
@@ -91,57 +43,37 @@ ndefinition associative ≝ λA:setoid.λf:A → A → A.∀x,y,z.f x (f y z) =
 ntheorem associative_append: ∀A:setoid.associative (list A) (append A).
 #A;#x;#y;#z;nelim x[ napply (refl ???) |#a;#x1;#H;nnormalize;/2/]nqed.
 
-nlet rec flatten S (l : list (list S)) on l : list S ≝ 
-match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
-
 (* end move to list *)
 
+
+(* XXX to undestand what I want inside Alpha 
+   the eqb part should be split away, but when available it should be
+   possible to obtain a leibnitz equality on lemmas proved on setoids
+*)
 interpretation "iff" 'iff a b = (iff a b).  
 
-ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ refl: eq A x x.
+ninductive eq (A:Type[0]) (x:A) : A → CProp[0] ≝ erefl: eq A x x.
 
 nlemma eq_rect_Type0_r':
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (erefl A a) → P x p.
  #A; #a; #x; #p; ncases p; #P; #H; nassumption.
 nqed.
 
 nlemma eq_rect_Type0_r:
- ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (erefl A a) → ∀x.∀p:eq ? x a.P x p.
  #A; #a; #P; #p; #x0; #p0; napply (eq_rect_Type0_r' ??? p0); nassumption.
 nqed.
 
 nlemma eq_rect_CProp0_r':
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (erefl A a) → P x p.
  #A; #a; #x; #p; ncases p; #P; #H; nassumption.
 nqed.
 
 nlemma eq_rect_CProp0_r:
- ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (erefl A a) → ∀x.∀p:eq ? x a.P x p.
  #A; #a; #P; #p; #x0; #p0; napply (eq_rect_CProp0_r' ??? p0); nassumption.
 nqed.
 
-(* XXX move to bool *)
-interpretation "bool eq" 'eq_low a b = (eq bool a b). 
-
-ndefinition BOOL : setoid.
-@bool; @(eq bool); ncases admit.nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-alias id "refl" = "cic:/matita/ng/properties/relations/refl.fix(0,1,3)".
-unification hint 0 ≔ ;
-  P1 ≟ refl ? (eq0 BOOL),
-  P2 ≟ sym ? (eq0 BOOL),
-  P3 ≟ trans ? (eq0 BOOL),
-  X ≟ mk_setoid bool (mk_equivalence_relation ? (eq bool) P1 P2 P3)
-(*-----------------------------------------------------------------------*) ⊢
-     carr X ≡ bool.
-
-unification hint 0 ≔ a,b;
-   R ≟ eq0 BOOL,
-   L ≟ bool
-(* -------------------------------------------- *) ⊢
-   eq bool a b ≡ eq_rel L R a b.
-
 nrecord Alpha : Type[1] ≝ { 
    acarr :> setoid;
    eqb: acarr → acarr → bool; 
@@ -149,7 +81,9 @@ nrecord Alpha : Type[1] ≝ {
 }.
  
 interpretation "eqb" 'eq_low a b = (eqb ? a b).
+(* end alpha *)
 
+(* re *)
 ninductive re (S: Type[0]) : Type[0] ≝
    z: re S
  | e: re S
@@ -158,6 +92,32 @@ ninductive re (S: Type[0]) : Type[0] ≝
  | o: re S → re S → re S
  | k: re S → re S.
  
+notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
+notation > "a ^ *" non associative with precedence 75 for @{ 'pk $a}.
+interpretation "star" 'pk a = (k ? a).
+interpretation "or" 'plus a b = (o ? a b).
+           
+notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
+interpretation "cat" 'pc a b = (c ? a b).
+
+(* to get rid of \middot *)
+ncoercion c  : ∀S.∀p:re S.  re S →  re S   ≝ c  on _p : re ?  to ∀_:?.?.
+
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
+
+notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
+interpretation "epsilon" 'epsilon = (e ?).
+
+notation "0" non associative with precedence 90 for @{ 'empty_r }.
+interpretation "empty" 'empty_r = (z ?).
+
+notation > "'lang' S" non associative with precedence 90 for @{ Ω^(list $S) }.
+notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(LIST $S) }.
+ 
+(* setoid support for re *)
+ 
 nlet rec eq_re (S:Alpha) (a,b : re S) on a : CProp[0] ≝ 
   match a with
   [ z ⇒ match b with [ z ⇒ True | _ ⇒ False]
@@ -186,8 +146,6 @@ ndefinition RE : Alpha → setoid.
         #r2 r3; /3/; ##]##]
 nqed.
 
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-alias id "carr" = "cic:/matita/ng/sets/setoids/carr.fix(0,0,1)".
 unification hint 0 ≔ A : Alpha;
   S ≟ acarr A,
   T ≟ carr S,
@@ -198,44 +156,81 @@ unification hint 0 ≔ A : Alpha;
 (*-----------------------------------------------------------------------*) ⊢
      carr X ≡ re T.
 
-unification hint 0 ≔ A:Alpha,a,b:re A;
+unification hint 0 ≔ A:Alpha, a,b:re (carr (acarr A));
    R ≟ eq0 (RE A),
-   L ≟ re A
+   L ≟ re (carr (acarr A))
 (* -------------------------------------------- *) ⊢
    eq_re A a b ≡ eq_rel L R a b.
+   
+(* XXX This seems to be a pattern for equations in setoid(0) *)
+unification hint 0 ≔ AA;
+   A  ≟  carr (acarr AA),
+   R  ≟ setoid1_of_setoid (RE AA)
+(*-----------------------------------------------*) ⊢
+   re A ≡ carr1 R.
 
-notation < "a \sup ⋇" non associative with precedence 90 for @{ 'pk $a}.
-notation > "a ^ *" non associative with precedence 75 for @{ 'pk $a}.
-interpretation "star" 'pk a = (k ? a).
-interpretation "or" 'plus a b = (o ? a b).
-           
-notation "a · b" non associative with precedence 60 for @{ 'pc $a $b}.
-interpretation "cat" 'pc a b = (c ? a b).
-
-(* to get rid of \middot *)
-ncoercion c  : ∀S.∀p:re S.  re S →  re S   ≝ c  on _p : re ?  to ∀_:?.?.
-
-notation < "a" non associative with precedence 90 for @{ 'ps $a}.
-notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
-interpretation "atom" 'ps a = (s ? a).
-
-notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
-interpretation "epsilon" 'epsilon = (e ?).
+alias symbol "hint_decl" (instance 1) = "hint_decl_CProp2".
+unification hint 0 ≔ S : Alpha, x,y: re (carr (acarr S));
+  SS ≟ RE S,
+  TT ≟ setoid1_of_setoid SS,
+  T ≟ carr1 TT
+(*-----------------------------------------*) ⊢ 
+  eq_re S x y ≡ eq_rel1 T (eq1 TT) x y.    
+
+(* contructors are morphisms *)
+nlemma c_is_morph : ∀A:Alpha.(re A) ⇒_0 (re A) ⇒_0 (re A).
+#A; napply (mk_binary_morphism … (λs1,s2:re A. s1 · s2)); #a; nelim a; /2/ by conj; nqed.
+
+(* XXX This is the good format for hints about morphisms, fix the others *)
+alias symbol "hint_decl" (instance 1) = "hint_decl_Type0".
+unification hint 0 ≔ S:Alpha, A,B:re (carr (acarr S));
+    SS ≟ carr (acarr S),
+    MM ≟ mk_unary_morphism ?? (λA.
+           mk_unary_morphism ?? 
+             (λB.A · B) (prop1 ?? (fun1 ?? (c_is_morph S) A)))
+           (prop1 ?? (c_is_morph S)),
+    T ≟ RE S
+(*--------------------------------------------------------------------------*) ⊢
+   fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ c SS A B.
+
+nlemma o_is_morph : ∀A:Alpha.(re A) ⇒_0 (re A) ⇒_0 (re A).
+#A; napply (mk_binary_morphism … (λs1,s2:re A. s1 + s2)); #a; nelim a;  /2/ by conj; nqed.
+
+unification hint 0 ≔ S:Alpha, A,B:re (carr (acarr S));
+    SS ≟ carr (acarr S),
+    MM ≟ mk_unary_morphism ?? (λA.
+           mk_unary_morphism ?? 
+             (λB.A + B) (prop1 ?? (fun1 ?? (o_is_morph S) A)))
+           (prop1 ?? (o_is_morph S)),
+    T ≟ RE S
+(*--------------------------------------------------------------------------*) ⊢
+   fun1 T T (fun1 T (unary_morph_setoid T T) MM A) B ≡ o SS A B.
+
+nlemma k_is_morph : ∀A:Alpha.(re A) ⇒_0 (re A).
+#A; @(λs1:re A. s1^* ); #a; nelim a; /2/ by conj; nqed.
+
+unification hint 0 ≔ S:Alpha, A:re (carr (acarr S));
+    SS ≟ carr (acarr S),
+    MM ≟ mk_unary_morphism ?? (λB.B^* ) (prop1 ?? (k_is_morph S)),
+    T ≟ RE S
+(*--------------------------------------------------------------------------*) ⊢
+   fun1 T T MM A ≡ k SS A.
+   
+nlemma s_is_morph : ∀A:Alpha.A ⇒_0 (re A).
+#A; @(λs1:A. s ? s1 ); #x y E; //; nqed.
 
-notation "0" non associative with precedence 90 for @{ 'empty_r }.
-interpretation "empty" 'empty_r = (z ?).
+unification hint 0 ≔ S:Alpha, a: carr (acarr S);
+    SS ≟ carr (acarr S),
+    MM ≟ mk_unary_morphism ?? (λb.s ? b ) (prop1 ?? (s_is_morph S)),
+    T ≟ RE S, T1 ≟ acarr S
+(*--------------------------------------------------------------------------*) ⊢
+   fun1 T1 T MM a ≡ s SS a.
 
-notation > "'lang' S" non associative with precedence 90 for @{ Ω^(list $S) }.
-notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(list $S) }.
+(* end setoids support for re *)
 
 nlet rec conjunct S (l : list (list S)) (L : lang S) on l: CProp[0] ≝
 match l with [ nil ⇒ True | cons w tl ⇒ w ∈ L ∧ conjunct ? tl L ].
 
-(*
-ndefinition sing_lang : ∀A:setoid.∀x:A.Ω^A ≝ λS.λx.{ w | x = w }.
-interpretation "sing lang" 'singl x = (sing_lang ? x).
-*)
-
 interpretation "subset construction with type" 'comprehension t \eta.x = 
   (mk_powerclass t x).
 
@@ -243,10 +238,120 @@ 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,
+   BB ≟ ext_carr AA B,
+   CC ≟ ext_carr AA C,
+   R ≟ mk_ext_powerclass AA
+         (cat A (ext_carr AA B) (ext_carr AA C)) 
+         (ext_prop AA (cat_is_ext A B C))
+(*----------------------------------------------------------*)  ⊢
+    ext_carr AA R ≡ cat A BB CC.
+    
+unification hint 0 ≔ S:setoid, A,B:lang (carr S);
+    T ≟ powerclass_setoid (list (carr S)),
+    MM ≟ mk_unary_morphism1 T (unary_morphism1_setoid1 T T)
+          (λA:lang (carr S).
+             mk_unary_morphism1 T T 
+               (λB:lang (carr S).cat S A B) 
+               (prop11 T T (fun11 ?? (cat_is_morph S) A)))
+          (prop11 T (unary_morphism1_setoid1 T T) (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 AAS,
+  R ≟ mk_unary_morphism1 T (unary_morphism1_setoid1 T T) (λX:Elang AA.
+           mk_unary_morphism1 T T (λY:Elang AA.
+             mk_ext_powerclass AAS 
+               (cat AA (ext_carr ? X) (ext_carr ? Y)) 
+               (ext_prop AAS (cat_is_ext AA X Y)))
+             (prop11 T T (fun11 ?? (cat_is_ext_morph AA) X)))
+           (prop11 T (unary_morphism1_setoid1 T T) (cat_is_ext_morph AA)),
+   BB ≟ ext_carr ? B,
+   CC ≟ ext_carr ? C
+(*------------------------------------------------------*) ⊢
+   ext_carr AAS (fun11 T T (fun11 T (unary_morphism1_setoid1 T T) 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).
 
+(* hints for star *)
+nlemma star_is_morph : ∀A:setoid. (lang A) ⇒_1 (lang A).
+#X; @(λA:lang X.A^* ); #a1 a2 E; @; #x; *; #wl; *; #defx Px; @wl; @; //;
+nelim wl in Px; //; #s tl IH; *; #a1s a1tl; /4/; nqed.
+
+nlemma star_is_ext: ∀A:setoid. (Elang A) → (Elang A).
+ #S A; @ ((ext_carr … A) ^* ); #w1 w2 E; @; *; #wl; *; #defw1 Pwl;
+ @wl; @; //; napply (.=_0 defw1); /2/; nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : setoid, B :  Elang A;
+   AA ≟ LIST A,
+   BB ≟ ext_carr AA B,
+   R ≟ mk_ext_powerclass ? 
+         ((ext_carr ? B)^* ) (ext_prop ? (star_is_ext ? B))
+(*--------------------------------------------------------------------*)  ⊢
+    ext_carr AA R ≡ star A BB.
+    
+unification hint 0 ≔ S:setoid, A:lang (carr S);
+    T ≟ powerclass_setoid (list (carr S)),
+    MM ≟ mk_unary_morphism1 T T 
+               (λB:lang (carr S).star S B) (prop11 T T (star_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T T MM A ≡ star S A.
+   
+nlemma star_is_ext_morph:∀A:setoid.(Elang A) ⇒_1 (Elang A).
+#A; @(star_is_ext …);
+#x1 x2 Ex; napply (prop11 … (star_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔ AA : setoid, B : Elang AA;
+  AAS ≟ LIST AA,
+  T ≟ ext_powerclass_setoid AAS,
+  R ≟ mk_unary_morphism1 T T
+            (λS:Elang AA.
+              mk_ext_powerclass AAS (star AA (ext_carr ? S)) 
+                (ext_prop AAS (star_is_ext AA S)))
+            (prop11 T T (star_is_ext_morph AA)),
+   BB ≟ ext_carr ? B
+(*------------------------------------------------------*) ⊢
+   ext_carr AAS (fun11 T T R B) ≡ star AA BB.
+
+(* end hints for star *)
+
 notation > "𝐋 term 70 E" non associative with precedence 75 for @{L_re ? $E}.
 nlet rec L_re (S : Alpha) (r : re S) on r : lang S ≝ 
 match r with
@@ -259,9 +364,72 @@ 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).
 
-notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
-ndefinition orb ≝ λa,b:bool. match a with [ true ⇒ true | _ ⇒ b ].
-interpretation "orb" 'orb a b = (orb a b).
+(* 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 (carr (acarr S)); 
+  SS ≟ LIST (acarr 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 (carr (acarr A));
+  T ≟ setoid1_of_setoid (RE A),
+  T2 ≟ powerclass_setoid (list (carr (acarr A))),
+  MM ≟ mk_unary_morphism1 ?? 
+         (λa:carr1 (setoid1_of_setoid (RE A)).𝐋 a) (prop11 ?? (L_re_is_morph A))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 T T2 MM a ≡  L_re A 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;
+ncut (𝐋 b =  𝐋 a); ##[ napply (.=_1 (┼_1 E^-1)); // ] #EL;
+@; #x H; nchange in H ⊢ % with (x ∈ 𝐋 ?); 
+##[ napply (. (# ╪_1 ?)); ##[##3: napply H |##2: ##skip ] napply EL;
+##| napply (. (# ╪_1 ?)); ##[##3: napply H |##2: ##skip ] napply (EL^-1)]
+nqed.
+            
+unification hint 1 ≔  AA : Alpha, a: re (carr (acarr AA));
+  T ≟ RE AA, T1 ≟ LIST (acarr AA), T2 ≟ setoid1_of_setoid T, 
+  TT ≟ ext_powerclass_setoid T1,
+  R ≟ mk_unary_morphism1 T2 TT
+       (λa:carr1 (setoid1_of_setoid T).
+         mk_ext_powerclass T1 (𝐋 a) (ext_prop T1 (L_re_is_ext AA a)))
+            (prop11 T2 TT (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
@@ -272,6 +440,19 @@ ninductive pitem (S: Type[0]) : Type[0] ≝
  | po: pitem S → pitem S → pitem S
  | pk: pitem S → pitem S.
  
+interpretation "pstar" 'pk a = (pk ? a).
+interpretation "por" 'plus a b = (po ? a b).
+interpretation "pcat" 'pc a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "ppatom" 'pp a = (pp ? a).
+(* to get rid of \middot *)
+ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S  ≝ pc on _p : pitem ? to ∀_:?.?.
+interpretation "patom" 'ps a = (ps ? a).
+interpretation "pepsilon" 'epsilon = (pe ?).
+interpretation "pempty" 'empty_r = (pz ?). 
+ 
+(* setoids for pitem *)
 nlet rec eq_pitem  (S : Alpha) (p1, p2 : pitem S) on p1 : CProp[0] ≝ 
  match p1 with
  [ pz ⇒ match p2 with [ pz ⇒ True | _ ⇒ False]
@@ -307,50 +488,18 @@ 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 (carr 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)
+unification hint 0 ≔ S:Alpha,a,b:pitem (carr (acarr S));
+   R ≟ PITEM S,  L ≟ pitem (carr (acarr S))
 (* -------------------------------------------- *) ⊢
    eq_pitem S a b ≡ eq_rel L (eq0 R) a b.    
     
-(* XXX move to pair.ma *)   
-nlet rec eq_pair (A, B : setoid) (a : A × B) (b : A × B) on a : CProp[0] ≝ 
-  match a with [ mk_pair a1 a2 ⇒ 
-  match b with [ mk_pair b1 b2 ⇒ a1 = b1 ∧ a2 = b2 ]].
-
-interpretation "eq_pair" 'eq_low a b = (eq_pair ?? a b). 
-
-nlemma PAIR : ∀A,B:setoid. setoid.
-#A B; @(A × B); @(eq_pair …);
-##[ #ab; ncases ab; #a b; @; napply #;
-##| #ab cd; ncases ab; ncases cd; #a1 a2 b1 b2; *; #E1 E2;
-    @; napply (?^-1); //;
-##| #a b c; ncases a; ncases b; ncases c; #c1 c2 b1 b2 a1 a2;
-    *; #E1 E2; *; #E3 E4; @; ##[ napply (.= E1); //] napply (.= E2); //.##]
-nqed.
-
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ AA, BB;
-    A ≟ carr AA, B ≟ carr BB,
-    P1 ≟ refl ? (eq0 (PAIR AA BB)),
-    P2 ≟ sym ? (eq0 (PAIR AA BB)),
-    P3 ≟ trans ? (eq0 (PAIR AA BB)),
-    R ≟ mk_setoid (A × B) (mk_equivalence_relation ? (eq_pair …) P1 P2 P3)
-(*---------------------------------------------------------------------------*)⊢
-    carr R ≡ A × B.
- 
-unification hint 0 ≔ S1,S2,a,b;
-   R ≟ PAIR S1 S2,
-   L ≟ (pair S1 S2)
-(* -------------------------------------------- *) ⊢
-   eq_pair S1 S2 a b ≡ eq_rel L (eq0 R) a b.    
+(* end setoids for pitem *)
 
-(* end move to pair *) 
- 
 ndefinition pre ≝ λS.pitem S × bool.
 
 notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
@@ -358,18 +507,6 @@ interpretation "fst" 'fst x = (fst ? ? x).
 notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
 interpretation "snd" 'snd x = (snd ? ? x).
 
-interpretation "pstar" 'pk a = (pk ? a).
-interpretation "por" 'plus a b = (po ? a b).
-interpretation "pcat" 'pc a b = (pc ? a b).
-notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
-notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
-interpretation "ppatom" 'pp a = (pp ? a).
-(* to get rid of \middot *)
-ncoercion pc : ∀S.∀p:pitem S. pitem S → pitem S  ≝ pc on _p : pitem ? to ∀_:?.?.
-interpretation "patom" 'ps a = (ps ? a).
-interpretation "pepsilon" 'epsilon = (pe ?).
-interpretation "pempty" 'empty_r = (pz ?).
-
 notation > "|term 19 e|" non associative with precedence 70 for @{forget ? $e}.
 nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
  match l with
@@ -380,6 +517,7 @@ nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
   | pc E1 E2 ⇒ (|E1| · |E2|)
   | po E1 E2 ⇒ (|E1| + |E2|)
   | pk E ⇒ |E|^* ].
+  
 notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
 interpretation "forget" 'forget a = (forget ? a).
 
@@ -397,92 +535,33 @@ notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'L_pi $E}
 notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'L_pi $E}.
 interpretation "in_pl" 'L_pi E = (L_pi ? E).
 
-(* The caml, as some patches for it *)
-ncoercion setoid1_of_setoid : ∀s:setoid. setoid1 ≝ setoid1_of_setoid on _s: setoid to setoid1.   
-   
-alias symbol "hint_decl" (instance 1) = "hint_decl_CProp2".
-unification hint 0 ≔ S : setoid, x,y;
-  SS ≟ LIST S,
-  TT ≟ setoid1_of_setoid SS
-(*-----------------------------------------*) ⊢ 
-  eq_list S x y ≡ eq_rel1 ? (eq1 TT) x y.
-  
-unification hint 0 ≔ SS : setoid;
-  S ≟ carr SS,
-  TT ≟ setoid1_of_setoid (LIST SS)
-(*-----------------------------------------------------------------*) ⊢ 
-  list S ≡ carr1 TT.
-
-(* XXX Ex setoid support *)
-nlemma Sig: ∀S,T:setoid.∀P: S → (T → CPROP).
-  ∀y,z:T.y = z → (∀x.y=z → P x y = P x z)  → (Ex S (λx.P x y)) =_1 (Ex S (λx.P x z)).
-#S T P y z Q E; @; *; #x Px; @x; nlapply (E x Q); *; /2/; nqed.
-
-notation "∑" non associative with precedence 90 for @{Sig ?????}.
-
-nlemma test : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y → (Ex S (λw.ee x w ∧ True)) =_1 (Ex S (λw.ee y w ∧ True)).
-#S m x y E;
-napply (.=_1 (∑ E (λw,H.(H ╪_1 #)╪_1 #))).
-napply #.
-nqed.
-
-nlemma test2 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y → 
-   (True ∧ (Ex S (λw.ee x w ∧ True))) =_1 (True ∧ (Ex S (λw.ee y w ∧ True))).
-#S m x y E;
-napply (.=_1 #╪_1(∑ E (λw,H.(H ╪_1 #) ╪_1 #))).
-napply #.
-nqed.
-
-nlemma ex_setoid : ∀S:setoid.(S ⇒_1 CPROP) → setoid.
-#T P; @ (Ex T (λx:T.P x)); @;
-##[ #H1 H2; napply True |##*: //; ##]
-nqed.
-
-unification hint 0 ≔ T,P ; S ≟ (ex_setoid T P) ⊢
- Ex T (λx:T.P x) ≡ carr S.
-
-nlemma test3 : ∀S:setoid. ∀ee: S ⇒_1 S ⇒_1 CPROP.
- ∀x,y:setoid1_of_setoid S.x =_1 y → 
-   ((Ex S (λw.ee x w ∧ True) ∨ True)) =_1 ((Ex S (λw.ee y w ∧ True) ∨ True)).
-#S m x y E;
-napply (.=_1 (∑ E (λw,H.(H ╪_1 #) ╪_1 #)) ╪_1 #).
-napply #.
-nqed.
-(* Ex setoid support end *)
-
+(* set support for 𝐋\p *)
 ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
 #S r; @(𝐋\p r); #w1 w2 E; nelim r; 
 ##[ ##1,2: /2/;
 ##| #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 ((∃_.?)∨?);
-    napply (.= (#‡H2));
-    napply (.=_1 (∑ E (λx1,H1.∑ E (λx2,H2.?)))╪_1 #); ##[
-      ncut ((w1 = (x1@x2)) = (w2 = (x1@x2)));##[
-        @; #X; ##[ napply ((.= H1^-1) X) | napply ((.= H2) X) ] ##] #X;
-      napply ( (X‡#)‡#); ##]
-    napply #;
 ##| #e1 e2 H1 H2;
-    nnormalize in ⊢ (???%%);
-    napply (H1‡H2);
-##| #e H; nnormalize in ⊢ (???%%);
-    napply (.=_1 (∑ E (λx1,H1.∑ E (λx2,H2.?)))); ##[
-      ncut ((w1 = (x1@x2)) = (w2 = (x1@x2)));##[
-        @; #X; ##[ napply ((.= H1^-1) X) | napply ((.= H2) X) ] ##] #X;
-      napply ((X‡#)‡#); ##]
-    napply #;##] 
+    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); 
+##| #e H; 
+    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#); 
+##]
 nqed.
 
-unification hint 0 ≔ S : Alpha,e : pitem S; 
-  SS ≟ (list S),
-  X ≟ (mk_ext_powerclass SS (𝐋\p e) (ext_prop SS (L_pi_ext S e)))
+unification hint 0 ≔ S : Alpha,e : pitem (carr (acarr S)); 
+  SS ≟ LIST (acarr S),
+  X ≟ mk_ext_powerclass SS (𝐋\p e) (ext_prop SS (L_pi_ext S e))
 (*-----------------------------------------------------------------*)⊢ 
   ext_carr SS X ≡ 𝐋\p e.
+
+(* end set support for 𝐋\p *)  
   
 ndefinition epsilon ≝ 
   λS:Alpha.λb.match b return λ_.lang S with [ true ⇒ { [ ] } | _ ⇒ ∅ ].
@@ -491,6 +570,51 @@ interpretation "epsilon" 'epsilon = (epsilon ?).
 notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
 interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
 
+(* hints for epsilon *)
+nlemma epsilon_is_morph : ∀A:Alpha. (setoid1_of_setoid bool) ⇒_1 (lang A).
+#X; @; ##[#b; napply(ϵ b)] #a1 a2; ncases a1; ncases a2; //; *; nqed.
+
+nlemma epsilon_is_ext: ∀A:Alpha. (setoid1_of_setoid bool) → (Elang A).
+ #S b; @(ϵ b); #w1 w2 E; ncases b; @; ##[##3,4:*] 
+nchange in match (w1 ∈ ϵ true) with ([] =_0 w1);
+nchange in match (w2 ∈ ϵ true) with ([] =_0 w2); #H; napply (.= H); /2/;
+nqed.
+
+alias symbol "hint_decl" = "hint_decl_Type1".
+unification hint 0 ≔ A : Alpha, B :  bool;
+   AA ≟ LIST (acarr A),
+   R ≟ mk_ext_powerclass ? 
+         (ϵ B) (ext_prop ? (epsilon_is_ext ? B))
+(*--------------------------------------------------------------------*)  ⊢
+    ext_carr AA R ≡ epsilon A B.
+    
+unification hint 0 ≔ S:Alpha, A:bool;
+    B ≟ setoid1_of_setoid BOOL,
+    T ≟ powerclass_setoid (list (carr (acarr S))),
+    MM ≟ mk_unary_morphism1 B T 
+               (λB.ϵ B) (prop11 B T (epsilon_is_morph S))
+(*--------------------------------------------------------------------------*) ⊢
+   fun11 B T MM A ≡ epsilon S A.
+   
+nlemma epsilon_is_ext_morph:∀A:Alpha. (setoid1_of_setoid bool) ⇒_1 (Elang A).
+#A; @(epsilon_is_ext …);
+#x1 x2 Ex; napply (prop11 … (epsilon_is_morph A)); nassumption.
+nqed.
+
+unification hint 1 ≔ AA : Alpha, B : bool;
+  AAS ≟ LIST (acarr AA), 
+  BB ≟ setoid1_of_setoid BOOL,
+  T ≟ ext_powerclass_setoid AAS,
+  R ≟ mk_unary_morphism1 BB T
+            (λS.
+              mk_ext_powerclass AAS (epsilon AA S) 
+                (ext_prop AAS (epsilon_is_ext AA S)))
+            (prop11 BB T (epsilon_is_ext_morph AA))
+(*------------------------------------------------------*) ⊢
+   ext_carr AAS (fun11 BB T R B) ≡ epsilon AA B.
+
+(* end hints for epsilon *)
+
 ndefinition L_pr ≝ λS : Alpha.λp:pre S.  𝐋\p\ (\fst p) ∪ ϵ (\snd p).
   
 interpretation "L_pr" 'L_pi E = (L_pr ? E).
@@ -498,10 +622,10 @@ interpretation "L_pr" 'L_pi E = (L_pr ? E).
 nlemma append_eq_nil : ∀S:setoid.∀w1,w2:list S. [ ] = w1 @ w2 → w1 = [ ].
 #S w1; ncases w1; //. nqed.
   
-(* lemma 12 *)
+(* lemma 12 *) (* XXX: a case where Leibnitz equality could be exploited for H *)
 nlemma epsilon_in_true : ∀S:Alpha.∀e:pre S. [ ] ∈ 𝐋\p e = (\snd e = true).
-#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//; 
-*; ##[##2:*] nelim e;
+#S r; ncases r; #e b; @; ##[##2: #H; ncases b in H; ##[##2:*] #; @2; /2/; ##] 
+ncases b; //; *; ##[##2:*] nelim e;
 ##[ ##1,2: *; ##| #c; *; ##| #c; *| ##7: #p H;
 ##| #r1 r2 H G; *; ##[##2: nassumption; ##]
 ##| #r1 r2 H1 H2; *; /2/ by {}]
@@ -580,18 +704,9 @@ nchange in match (ϵ false) with ∅;
 ##| napply (.= (cup0 ? {[]})^-1); napply cupC; ##]
 nqed.
 
-(* XXX move somewere else *)
-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 ∀_:?.?.
-
 (* 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;
-nwhd in ⊢ (???(??%)?);
-nchange in ⊢(???%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
-nchange in ⊢(???(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
 napply (.=_1 #╪_1 (epsilon_or ???));
 napply (.=_1 (cupA…)^-1);
 napply (.=_1 (cupA…)╪_1#);
@@ -605,10 +720,12 @@ nqed.
 (* XXX problem: auto does not find # (refl) when it has a concrete == *)
 nlemma odotEt : ∀S:Alpha.∀e1,e2:pitem S.∀b2:bool.
   〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
-#S e1 e2 b2; ncases b2; nnormalize; @; //; @; napply refl; nqed.
+#S e1 e2 b2; ncases b2; @; /3/ by refl, conj, I; nqed.
 
+(*
 nlemma LcatE : ∀S:Alpha.∀e1,e2:pitem S.
   𝐋\p (e1 · e2) =  𝐋\p e1 · 𝐋  |e2| ∪ 𝐋\p e2. //; nqed.
+*)
 
 nlemma cup_dotD : ∀S:Alpha.∀p,q,r:lang S.(p ∪ q) · r = (p · r) ∪ (q · r). 
 #S p q r; napply ext_set; #w; nnormalize; @; 
@@ -629,9 +746,6 @@ nlemma erase_star : ∀S:Alpha.∀e1:pitem S.𝐋 |e1|^* = 𝐋 |e1^*|. //; nqed
 nlemma mem_single : ∀S:setoid.∀a,b:S. a ∈ {(b)} → a = b.
 #S a b; nnormalize; /2/; nqed.
 
-notation < "[\setoid\emsp\of\emsp term 19 x]" non associative with precedence 90 for @{'mk_setoid $x}.
-interpretation "mk_setoid" 'mk_setoid x = (mk_setoid x ?).
-
 nlemma cup_sub: ∀S.∀A,B:𝛀^S.∀x. ¬ (x ∈ A) → A ∪ (B - {(x)}) = (A ∪ B) - {(x)}.
 #S A B x H; napply ext_set; #w; @; 
 ##[ *; ##[ #wa; @; ##[@;//] #H2; napply H; napply (. (mem_single ??? H2)^-1╪_1#); //]
@@ -648,87 +762,86 @@ nlemma subK : ∀S.∀a:Ω^S. a - a = ∅.
 nlemma subW : ∀S.∀a,b:Ω^S.∀w.w ∈ (a - b) → w ∈ a.
 #S a b w; nnormalize; *; //; nqed.
 
+alias symbol "eclose" (instance 3) = "eclose".
 nlemma erase_bull : ∀S:Alpha.∀a:pitem S. |\fst (•a)| = |a|.
 #S a; nelim a; // by {};
-##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| · |e2|);
-
-finqui: manca · morfismo, oppure un lemma che dice che == è come Leibnitz.
-
-    nrewrite < IH1; nrewrite < IH2;  
-    nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
-    ncases (•e1); #e3 b; ncases b; nnormalize;
-    ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
-##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| + .|e2|);
-    nrewrite < IH2; nrewrite < IH1;
-    nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
+##[ #e1 e2 IH1 IH2;
+    napply (?^-1); 
+    napply (.=_0 (IH1^-1)╪_0 (IH2^-1));
+    nchange in match (•(e1 · ?)) with (?⊙?);
+    ncases (•e1); #e3 b; ncases b; ##[ nnormalize; ncases (•e2); /3/ by refl, conj]
+    napply (.=_0 #╪_0 (IH2)); //;
+##| #e1 e2 IH1 IH2; napply (?^-1);
+    napply (.=_0 (IH1^-1)╪_0(IH2^-1));
+    nchange in match (•(e1+?)) with (?⊕?);
     ncases (•e1); ncases (•e2); //]
-##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
-    nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
 nqed.
 
-nlemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
+(*
+nlemma eta_lp : ∀S:Alpha.∀p:pre S. 𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
 #S p; ncases p; //; nqed.
+*)
+
+(* XXX coercion ext_carr non applica *)
+nlemma epsilon_dot: ∀S:Alpha.∀p:Elang S. {[]} · (ext_carr ? p) = p. 
+#S e; napply ext_set; #w; @; ##[##2: #Hw; @[]; @w; @; //; @; //; napply #; (* XXX auto *) ##]
+*; #w1; *; #w2; *; *; #defw defw1 Hw2; 
+napply (. defw╪_1#); 
+napply (. ((defw1 : [ ] = ?)^-1 ╪_0 #)╪_1#);
+napply Hw2; 
+nqed.
+
 
-nlemma epsilon_dot: ∀S.∀p:word S → Prop. {[]} · p = p. 
-#S e; napply extP; #w; nnormalize; @; ##[##2: #Hw; @[]; @w; /3/; ##]
-*; #w1; *; #w2; *; *; #defw defw1 Hw2; nrewrite < defw; nrewrite < defw1;
-napply Hw2; nqed.
 
 (* theorem 16: 1 → 3 *)
-nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
-      𝐋\p (•(\fst e2)) =  𝐋\p (\fst e2) ∪ 𝐋 .|\fst e2| → 
-         𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
+nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
+      𝐋\p (•(\fst e2)) =  𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| → 
+         𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
 #S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
-##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2); 
+##[ nchange in match (〈?,true〉⊙?) with 〈?,?〉; 
+    nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2); 
     nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
-    nchange in ⊢ (??%?) with (?∪?); 
-    nchange in ⊢ (??(??%?)?) with (?∪?);
-    nchange in match (𝐋\p 〈?,?〉) with (?∪?);
-    nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
-    nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
-    nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 .|e2'|); ##[##2:
-      nchange with (𝐋\p 〈e2'',b2''〉 =  𝐋\p e2' ∪ 𝐋 .|e2'|); 
-      ngeneralize in match th1;
-      nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
-    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 ⊢ (???(??%?)); //;
-##| ncases e2; #e2' b2'; nchange in match (〈e1',false〉⊙?) with 〈?,?〉;
-    nchange in match (𝐋\p ?) with (?∪?);
-    nchange in match (𝐋\p (e1'·?)) with (?∪?);
-    nchange in match (𝐋\p 〈e1',?〉) with (?∪?);
-    nrewrite > (cup0…); 
-    nrewrite > (cupA…); //;##]
+    napply (.=_1 (# ╪_1 (epsilon_or …))); (* XXX … is too slow if combined with .= *)
+    nchange in match b2'' with b2''; (* XXX some unfoldings happened *)
+    nchange in match b2' with b2';
+    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'|); ##[
+      napply (?^-1); napply (.=_1 th1^-1); //;##] #E;
+    napply (.=_1 (# ╪_1 (E ╪_1 #)));
+    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 …));
+    napply (.=_1 (((# ╪_1 (┼_1 (erase_bull S e2')) )╪_1 #)╪_1 #));
+    //;   
+##| ncases e2; #e2' b2'; nchange in match (𝐋\p ?) with (?∪?∪?);
+    napply (.=_1 (cupA…));
+    napply (?^-1); nchange in match (𝐋\p 〈?,false〉) with (?∪?);
+    napply (.=_1 ((cup0…)╪_1#)╪_1#);
+    //]
 nqed.
 
+
+
 nlemma sub_dot_star : 
-  ∀S.∀X:word S → Prop.∀b. (X - ϵ b) · X^* ∪ {[]} = X^*.
-#S X b; napply extP; #w; @;
-##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+  ∀S:Alpha.∀X:Elang S.∀b. (X - ϵ b) · (ext_carr … X)^* ∪ {[]} = (ext_carr … X)^*.
+#S X b; napply ext_set; #w; @;
+##[ *; ##[##2: #defw; @[]; @; //]
     *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
-    @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+    @ (w1 :: lw); @; ##[ napply (.=_0 # ╪_0 flx); napply (?^-1); //]
     @; //; napply (subW … sube);
-##| *; #wl; *; #defw Pwl; nrewrite < defw; nelim wl in Pwl; ##[ #_; @2; //]
-    #w' wl' IH; *; #Pw' IHp; nlapply (IH IHp); *;
-    ##[ *; #w1; *; #w2; *; *; #defwl' H1 H2;
-        @; ncases b in H1; #H1; 
-        ##[##2: nrewrite > (sub0…); @w'; @(w1@w2);
-                nrewrite > (associative_append ? w' w1 w2);
-                nrewrite > defwl'; @; ##[@;//] @(wl'); @; //;
-           ##| ncases w' in Pw';
-               ##[ #ne; @w1; @w2; nrewrite > defwl'; @; //; @; //;
-               ##| #x xs Px; @(x::xs); @(w1@w2); 
-                   nrewrite > (defwl'); @; ##[@; //; @; //; @; nnormalize; #; ndestruct]
-                   @wl'; @; //; ##] ##]
-        ##| #wlnil; nchange in match (flatten ? (w'::wl')) with (w' @ flatten ? wl');
-            nrewrite < (wlnil); nrewrite > (append_nil…); ncases b;
-            ##[ ncases w' in Pw'; /2/; #x xs Pxs; @; @(x::xs); @([]);
-                nrewrite > (append_nil…); @; ##[ @; //;@; //; nnormalize; @; #; ndestruct]
-                @[]; @; //;
-            ##| @; @w'; @[]; nrewrite > (append_nil…); @; ##[##2: @[]; @; //] 
-                @; //; @; //; @; *;##]##]##] 
+##| *; #wl; *; #defw Pwl; napply (. (defw^-1 ╪_1 #));
+    nelim wl in Pwl; /2/;
+    #s tl IH; *; #Xs Ptl; ncases s in Xs; ##[ #; napply IH; //] #x xs Xxxs;
+    @; @(x :: xs); @(flatten ? tl); @; 
+      ##[ @; ##[ napply #] @; ##[nassumption] ncases b; *; ##]
+    nelim tl in Ptl; ##[ #; @[]; /2/] #w ws IH; *; #Xw Pws; @(w :: ws); @; ##[ napply #]
+    @; //;##]
 nqed.
 
 (* theorem 16: 1 *)
@@ -736,38 +849,54 @@ alias symbol "pc" (instance 13) = "cat lang".
 alias symbol "in_pl" (instance 23) = "in_pl".
 alias symbol "in_pl" (instance 5) = "in_pl".
 alias symbol "eclose" (instance 21) = "eclose".
-ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 𝐋 .|e|.
+ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 𝐋 |e|.
 #S e; nelim e; //;
-  ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
-  ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
+  ##[ #a; napply ext_set; #w; @; *; /3/ by or_introl, or_intror;
+  ##| #a; napply ext_set; #w; @; *; /3/ by or_introl; *;
   ##| #e1 e2 IH1 IH2;  
-      nchange in ⊢ (??(??(%))?) with (•e1 ⊙ 〈e2,false〉);
-      nrewrite > (odot_dot_aux S (•e1) 〈e2,false〉 IH2);
-      nrewrite > (IH1 …); nrewrite > (cup_dotD …);
-      nrewrite > (cupA …); nrewrite > (cupC ?? (𝐋\p ?) …);
-      nchange in match (𝐋\p 〈?,?〉) with (𝐋\p e2 ∪ {}); nrewrite > (cup0 …);
-      nrewrite < (erase_dot …); nrewrite < (cupA …); //;
+      nchange in match (•(e1·e2)) with (•e1 ⊙ 〈e2,false〉);
+      napply (.=_1 (odot_dot_aux ?? 〈e2,false〉 IH2));
+      napply (.=_1 (IH1 ╪_1 #) ╪_1 #);
+      napply (.=_1 (cup_dotD …) ╪_1 #);
+      napply (.=_1 (cupA …));
+      napply (.=_1 # ╪_1 ((erase_dot ???)^-1 ╪_1 (cup0 ??)));
+      napply (.=_1 # ╪_1 (cupC…));
+      napply (.=_1 (cupA …)^-1); //;
   ##| #e1 e2 IH1 IH2;
-      nchange in match (•(?+?)) with (•e1 ⊕ •e2); nrewrite > (oplus_cup …);
-      nrewrite > (IH1 …); nrewrite > (IH2 …); nrewrite > (cupA …);
-      nrewrite > (cupC ? (𝐋\p e2)…);nrewrite < (cupA ??? (𝐋\p e2)…);
-      nrewrite > (cupC ?? (𝐋\p e2)…); nrewrite < (cupA …); 
-      nrewrite < (erase_plus …); //.
+      nchange in match (•(?+?)) with (•e1 ⊕ •e2);
+      napply (.=_1 (oplus_cup …));
+      napply (.=_1 IH1 ╪_1 IH2);
+      napply (.=_1 (cupA …));
+      napply (.=_1 # ╪_1 (# ╪_1 (cupC…)));
+      napply (.=_1 # ╪_1 (cupA ????)^-1);
+      napply (.=_1 # ╪_1 (cupC…));
+      napply (.=_1 (cupA ????)^-1);
+      napply (.=_1 # ╪_1 (erase_plus ???)^-1); //;
   ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
-      nchange in match (𝐋\p ?) with  (𝐋\p 〈e'^*,true〉);
+      nchange in match (𝐋\p ?) with (𝐋\p 〈e'^*,true〉);
       nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
-      nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
-      nrewrite > (erase_bull…e);
-      nrewrite > (erase_star …);
-      nrewrite > (?: 𝐋\p e' =  𝐋\p e ∪ (𝐋 .|e| - ϵ b')); ##[##2:
-        nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH; 
-        ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH; 
-            nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
-        ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
-      nrewrite > (cup_dotD…); nrewrite > (cupA…); 
-      nrewrite > (?: ((?·?)∪{[]} = 𝐋 .|e^*|)); //;
-      nchange in match (𝐋 .|e^*|) with ((𝐋. |e|)^* ); napply sub_dot_star;##]
- nqed.
+      (* nwhd in match (𝐋\p e'^* ); (* XXX bug uncertain *) *)
+      nchange in ⊢ (???(??%?)?) with (𝐋\p e' · ?);
+      napply (.=_1 (# ╪_1 (┼_1 (┼_0 (erase_bull S e)))) ╪_1 #);
+      napply (.=_1 (# ╪_1 (erase_star …)) ╪_1 #);
+      ncut ( 𝐋\p e' =  𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[
+        nchange in IH : (???%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH; 
+        ##[ #IH; napply (?^-1); napply (.=_1 (cup_sub … (not_epsilon_lp…)));
+            napply (.=_1 (IH^-1 ╪_1 #)); 
+            alias symbol "invert" = "setoid1 symmetry". 
+            (* XXX too slow if ambiguous, since it tries with a ? (takes 12s) then
+               tries with sym0 and fails immediately, then with sym1 that is OK *)
+            napply (.=_1 (cup_sub …(not_epsilon_lp …))^-1);
+            napply (.=_1 # ╪_1 (subK…)); napply (.=_1 (cup0…)); //;
+        ##| #IH;  napply (?^-1); napply (.=_1 # ╪_1 (sub0 …));
+            napply (.=_1 IH^-1); napply (.=_1 (cup0 …)); //; ##]##] #EE;
+      napply (.=_1 (EE ╪_1 #) ╪_1 #);
+      napply (.=_1 (cup_dotD…) ╪_1 #);
+      napply (.=_1 (cupA…));
+      napply (.=_1 # ╪_1 (sub_dot_star…)); //; ##]
+nqed.
+
+STOP
 
 (* theorem 16: 3 *)      
 nlemma odot_dot: