update in groud_2 and models

[helm.git] / helm / software / matita / nlibrary / re / re-setoids.ma

diff --git a/helm/software/matita/nlibrary/re/re-setoids.ma b/helm/software/matita/nlibrary/re/re-setoids.ma
index cfbd7cb31d8c0d5b203654d388d6e64958882ba7..729e8a0ea3a7806d55b8a334d952b613daee4aef 100644 (file)
--- a/helm/software/matita/nlibrary/re/re-setoids.ma
+++ b/helm/software/matita/nlibrary/re/re-setoids.ma
@@ -114,7 +114,7 @@ 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) }.
+notation > "'Elang' S" non associative with precedence 90 for @{ 𝛀^(LIST $S) }.
  
 (* setoid support for re *)
  
@@ -156,47 +156,75 @@ 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.
+
+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; 
-##[##1,2: #a' b b'; ncases a'; nnormalize; /2/ by conj;
-##|#x a' b b'; ncases a'; /2/ by conj;
-##|##4,5: #r1 r2 IH1 IH2 a'; ncases a'; nnormalize; /2/ by conj;
-##|#r IH a'; ncases a'; nnormalize; /2/ by conj; ##]
-nqed.
+#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 S;
-    MM ≟ mk_unary_morphism ??
-          (λA:re S.mk_unary_morphism ?? (λB.A · B) (prop1 ?? (c_is_morph S A)))
-          (prop1 ?? (c_is_morph S)),
+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 ≡ A · B.
+   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; 
-##[##1,2: #a' b b'; ncases a'; nnormalize; /2/ by conj;
-##|#x a' b b'; ncases a'; /2/ by conj;
-##|##4,5: #r1 r2 IH1 IH2 a'; ncases a'; nnormalize; /2/ by conj;
-##|#r IH a'; ncases a'; nnormalize; /2/ by conj; ##]
-nqed.
+#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.
 
-unification hint 0 ≔ S:Alpha, A,B:re S;
-    MM ≟ mk_unary_morphism ??
-          (λA:re S.mk_unary_morphism ?? (λB.A + B) (prop1 ?? (o_is_morph S A)))
-          (prop1 ?? (o_is_morph S)),
+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 (fun1 T (unary_morph_setoid T T) MM A) B ≡ A + B.
+   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.
+
+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.
 
 (* end setoids support for re *)
 
@@ -233,19 +261,23 @@ ncut (∀w1,w2.(x == w1@w2) = (y == w1@w2)); ##[
 nqed.
 
 alias symbol "hint_decl" = "hint_decl_Type1".
-unification hint 0 ≔ A : setoid, B,C :  Elang A;
+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).
+   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 S;
-    T ≟ powerclass_setoid (list S),
+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 S.
+          (λA:lang (carr S).
              mk_unary_morphism1 T T 
-               (λB:lang S.cat S A B) (prop11 T T (cat_is_morph S A)))
+               (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.
@@ -257,15 +289,14 @@ nqed.
 
 unification hint 1 ≔ AA : setoid, B,C : Elang AA;
   AAS ≟ LIST AA,
-  T ≟ ext_powerclass_setoid (list AA),
-  R ≟ mk_unary_morphism1 T (unary_morphism1_setoid1 T T)
-          (λS:Elang AA.
-           mk_unary_morphism1 T T
-            (λS':Elang AA.
-              mk_ext_powerclass (list AA) (cat AA (ext_carr ? S) (ext_carr ? S')) 
-                (ext_prop (list AA) (cat_is_ext AA S S')))
-            (prop11 T T (cat_is_ext_morph AA S)))
-          (prop11 T (unary_morphism1_setoid1 T T) (cat_is_ext_morph 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
 (*------------------------------------------------------*) ⊢
@@ -277,6 +308,50 @@ 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
@@ -302,8 +377,8 @@ ndefinition L_re_is_ext : ∀S:Alpha.∀r:re S.Elang S.
 ##| #e H; @; *; #l; *; #defw1 Pl; @l; @; //; napply (.=_1 defw1); /2/; ##]
 nqed.
 
-unification hint 0 ≔ S : Alpha,e : re S; 
-  SS ≟ LIST S,
+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.
@@ -328,14 +403,13 @@ nlemma L_re_is_morph:∀A:Alpha.(setoid1_of_setoid (re A)) ⇒_1 Ω^(list A).
    @; ##[##1,3: nassumption] /2/; ##]
 nqed.
 
-unification hint 0 ≔ A:Alpha, a:re A;
+unification hint 0 ≔ A:Alpha, a:re (carr (acarr A));
   T ≟ setoid1_of_setoid (RE A),
-  T1 ≟ LIST A,
-  T2 ≟ powerclass_setoid T1,
+  T2 ≟ powerclass_setoid (list (carr (acarr A))),
   MM ≟ mk_unary_morphism1 ?? 
-         (λa:setoid1_of_setoid (RE A).𝐋 a) (prop11 ?? (L_re_is_morph A))
+         (λa:carr1 (setoid1_of_setoid (RE A)).𝐋 a) (prop11 ?? (L_re_is_morph A))
 (*--------------------------------------------------------------------------*) ⊢
-   fun11 T T2 MM a ≡  𝐋 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;
@@ -345,12 +419,13 @@ ncut (𝐋 b =  𝐋 a); ##[ napply (.=_1 (┼_1 E^-1)); // ] #EL;
 ##| napply (. (# ╪_1 ?)); ##[##3: napply H |##2: ##skip ] napply (EL^-1)]
 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))
+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.
 
@@ -413,13 +488,13 @@ 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) 
+    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.    
     
@@ -480,9 +555,9 @@ ndefinition L_pi_ext : ∀S:Alpha.∀r:pitem S.Elang S.
 ##]
 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.
 
@@ -495,6 +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).
@@ -642,10 +762,11 @@ 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;
-    napply (?^-1);
+    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]
@@ -670,15 +791,7 @@ 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.
@@ -704,8 +817,7 @@ nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
     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]
+    napply (.=_1 (((# ╪_1 (┼_1 (erase_bull S e2')) )╪_1 #)╪_1 #));
     //;   
 ##| ncases e2; #e2' b2'; nchange in match (𝐋\p ?) with (?∪?∪?);
     napply (.=_1 (cupA…));
@@ -714,34 +826,22 @@ nlemma odot_dot_aux : ∀S:Alpha.∀e1,e2: pre S.
     //]
 nqed.
 
-STOP
+
 
 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 *)
@@ -749,38 +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: