From 737f7dca7da2379a725182776e8c01c3ceae1cc0 Mon Sep 17 00:00:00 2001
From: Enrico Tassi <enrico.tassi@inria.fr>
Date: Thu, 15 Jul 2010 16:16:50 +0000
Subject: [PATCH] re 16.4 almost done

---
 helm/software/matita/nlibrary/re/re.ma | 581 +++++++++++++++----------
 1 file changed, 344 insertions(+), 237 deletions(-)

diff --git a/helm/software/matita/nlibrary/re/re.ma b/helm/software/matita/nlibrary/re/re.ma
index 6b6a98524..7b6580c87 100644
--- a/helm/software/matita/nlibrary/re/re.ma
+++ b/helm/software/matita/nlibrary/re/re.ma
@@ -12,21 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
-(*include "logic/connectives.ma".*)
-(*include "logic/equality.ma".*)
 include "datatypes/list.ma".
 include "datatypes/pairs.ma".
 include "arithmetics/nat.ma".
 
-(*include "Plogic/equality.ma".*)
-
 interpretation "iff" 'iff a b = (iff a b).  
 
-nrecord Alpha : Type[1] ≝
- { carr :> Type[0];
+nrecord Alpha : Type[1] ≝ { carr :> Type[0];
    eqb: carr → carr → bool;
    eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
- }.
+}.
  
 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
 interpretation "eqb" 'eqb a b = (eqb ? a b).
@@ -62,186 +57,50 @@ interpretation "epsilon" 'epsilon = (e ?).
 notation "∅" non associative with precedence 90 for @{ 'empty }.
 interpretation "empty" 'empty = (z ?).
 
-notation > "w ∈ E" non associative with precedence 45 for @{in_l ? $w $E}.
-ninductive in_l (S : Alpha) : word S → re S → Prop ≝
- | in_e: [ ] ∈ ϵ
- | in_s: ∀x:S. [x] ∈ `x
- | in_c: ∀w1,w2,e1,e2. w1 ∈ e1 → w2 ∈ e2 → w1@w2 ∈ e1 · e2
- | in_o1: ∀w,e1,e2. w ∈ e1 → w ∈ e1 + e2
- | in_o2: ∀w,e1,e2. w ∈ e2 → w ∈ e1 + e2
- | in_ke: ∀e. [ ] ∈ e^*
- | in_ki: ∀w1,w2,e. w1 ∈ e → w2 ∈ e^* → w1@w2 ∈ e^*.
-interpretation "in_l" 'mem w l = (in_l ? w l).
-
-notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
-interpretation "orb" 'orb a b = (orb a b).
-
-notation "a && b" left associative with precedence 40 for @{'andb $a $b}.
-interpretation "andb" 'andb a b = (andb a b).
-
-notation "~~ a" non associative with precedence 40 for @{'notb $a}.
-interpretation "notb" 'notb a = (notb a).
-
-nlet rec weq (S : Alpha) (l1, l2 : word S) on l1 : bool ≝ 
-match l1 with 
-[ nil ⇒ match l2 with [ nil ⇒ true | cons _ _ ⇒ false ]
-| cons x xs ⇒ match l2 with [ nil ⇒ false | cons y ys ⇒ (x == y) && weq S xs ys]].
-
-ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
-notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
-interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
-
-interpretation "qew" 'eqb a b = (weq ? a b).
-
-ndefinition is_epsilon ≝ λA.λw:word A. w == [ ].
-ndefinition is_empty ≝ λA.λw:word A.false.
-ndefinition is_char ≝ λA,x.λw:word A. w == [ x ].
-
-nlemma andP : ∀a,b.(a && b) = true ↔ (a = true ∧ b = true).
-#a b; ncases a; ncases b; nnormalize; @; ##[##2,4,6,8: *] /2/;
-nqed.
-
-nlemma orP : ∀a,b.(a || b) = true ↔ (a = true ∨ b = true).
-#a b; ncases a; ncases b; nnormalize; @; ##[##2,4,6,8: *] /2/;
-nqed.
-
-nlemma iff_l2r : ∀a,p.a = true ↔ p → a = true → p.
-#a p; *; /2/;
-nqed.
+nlet rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝ 
+match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
 
-nlemma iff_r2l : ∀a,p.a = true ↔ p → p → a = true.
-#a p; *; /2/;
-nqed.
+nlet rec conjunct (S : Alpha) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
+match l with [ nil ⇒ ? | cons w tl ⇒ r w ∧ conjunct ? tl r ]. napply True. nqed.
 
-ncoercion xx : ∀a,p.∀H:a = true ↔ p. a = true → p ≝ iff_l2r
-on _H : (? = true) ↔ ? to ∀_:?. ?.
+ndefinition empty_lang ≝ λS.λw:word S.False.
+notation "{}" non associative with precedence 90 for @{'empty_lang}.
+interpretation "empty lang" 'empty_lang = (empty_lang ?).
 
-ncoercion yy : ∀a,p.∀H:a = true ↔ p. p → a = true ≝ iff_r2l
-on _H : (? = true) ↔ ? to ∀_:?. ?.
+ndefinition sing_lang ≝ λS.λx,w:word S.x=w.
+notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.
+interpretation "sing lang" 'sing_lang x = (sing_lang ? x).
 
-ndefinition wAlpha : Alpha → Alpha. #A; @ (word A) (weq A).
-#x; nelim x; ##[ #y; ncases y; /2/; #x xs; @; nnormalize; #; ndestruct; ##]
-#x xs; #IH y; nelim y; ##[ @; nnormalize; #; ndestruct; ##]
-#y ys; *; #H1 H2; @; #H3;
-##[ ##2: ncases (IH ys); #_; #H; ndestruct; nrewrite > (iff_r2l ?? (eqb_true ???) ?); //; napply H; //]
-nrewrite > (iff_l2r ?? (eqb_true ? x y) ?); nnormalize in H3; ncases (x == y) in H3; nnormalize; /2/;
-##[ #H; ncases (IH ys); #E; #_; nrewrite > (E H); //] #; ndestruct;
-nqed.
+ndefinition union : ∀S,l1,l2,w.Prop ≝ λS.λl1,l2.λw:word S.l1 w ∨ l2 w.
+interpretation "union lang" 'union a b = (union ? a b).
 
-alias symbol "hint_decl" (instance 1) = "hint_decl_Type1".
-unification hint 0 ≔ T; Y ≟ T, X ≟  (wAlpha T) ⊢ carr X ≡ word Y.
-unification hint 0 ≔ T; Y ≟ T, X ≟  (wAlpha T) ⊢ carr X ≡ list Y.
-unification hint 0 ≔ T,x,y; Y ≟ T, X ≟ (wAlpha T) ⊢ eqb X x y ≡ weq Y x y.
-
-nlet rec ex_split (A : Alpha) (p1,p2 : word A → bool) (w : word A) on w : bool ≝
-  match w with
-  [ nil ⇒ p1 [ ] && p2 [ ]
-  | cons x xs ⇒ p1 [ ] && p2 (x::xs) || ex_split … (λw.p1 (x :: w)) p2 xs].
-
-nlemma ex_splitP : 
-  ∀A,w,p1,p2. ex_split A p1 p2 w = true ↔ 
-              ∃w1,w2. w = w1 @ w2 ∧ p1 w1 = true ∧ p2 w2 = true.
-#A w; nelim w;
-##[ #p1 p2; @;
-   ##[ #H; @[ ]; @[ ]; ncases (iff_l2r ?? (andP ??) H); (* bug coercions *) 
-       #E1 E2; nrewrite > E1; nrewrite > E2; /3/;
-   ##| *; #w1; *;#w2; *; *; ncases w1; ncases w2; nnormalize; #abs H1 H2; #;
-       ndestruct; nrewrite > H1 ; nrewrite > H2; //]
-##| #x xs IH p1 p2; @;
-   ##[ #H; ncases (iff_l2r ?? (orP ??) H);
-      ##[ #H1; ncases (iff_l2r ?? (andP ??) H1); #p1T p2T;
-          @[ ]; @(x::xs); nnormalize; /3/;
-      ##| #E; ncases (iff_l2r ?? (IH ??) E); #l1; *; #l2; *; *; #defxs p1T p2T;
-          @(x :: l1); @l2; ndestruct; /3/; ##]
-    ##| *; #w1; *; #w2; *; *; ncases w1;
-       ##[ nnormalize in ⊢ (% → ?); ncases w2; ##[ #; ndestruct] #y ys defw2 p1T p2T;
-           nchange with ((p1 [ ] && p2 (x::xs) || ex_split ? (λw0.p1 (x::w0)) p2 xs) = true);
-           napply (iff_r2l ?? (orP ??)); @1; napply (iff_r2l ?? (andP ??));
-           ndestruct; /2/;
-       ##| #y ys; nnormalize in ⊢ (% → ?); #E p1T p2T;
-           nchange with ((p1 [ ] && p2 (x::xs) || ex_split ? (λw0.p1 (x::w0)) p2 xs) = true);
-           napply (iff_r2l ?? (orP ??)); @2; napply (iff_r2l ?? (IH ??));
-           @ys; @w2; ndestruct; /3/; ##]##]##]
-nqed.
+ndefinition cat : ∀S,l1,l2,w.Prop ≝ 
+  λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2.
+interpretation "cat lang" 'pc a b = (cat ? a b).
 
-nlet rec allb (A : Alpha) (p,fresh_p : word A → bool) (w : word A) on w : bool ≝
-  match w with
-  [ nil ⇒ p [ ]
-  | cons x xs ⇒ p [x] && (xs == [ ] || allb … fresh_p fresh_p xs) 
-                || allb … (λw.p (x :: w)) fresh_p xs].
-
-nlemma allbP :
-  ∀A,w,p.allb A p p w = true ↔ 
-    ∃w1,w2.w = w1 @ w2 ∧ p w1 = true ∧ (w2 = [ ] ∨ allb ? p p w2 = true).
-#A w; nelim w;
-##[ #p; @; 
-   ##[ #H; @[ ]; @[ ]; nnormalize in H; /4/ by conj, or_introl;
-   ##| *; #w1; *; #w2; ncases w1;
-       ##[ *; *; nnormalize in ⊢ (% → ?); #defw2 pnil; *; ##[ #; ndestruct] //;
-       ##| #y ys; *; *; nnormalize in ⊢ (% → ?); #; ndestruct; ##]##]
-##| #y ys IH p; @;
-   ##[ #E; ncases (iff_l2r ?? (orP ??) E);
-       ##[ #H; ncases (iff_l2r ?? (andP ??) H); #px allys;
-           nlapply (iff_l2r ?? (orP ??) allys); *; 
-           ##[ #defys; @[y]; @[ ]; nrewrite > (iff_l2r ?? (eqb_true ? ys ?) defys);
-               /4/ by conj, or_introl;
-           ##| #IHa; ncases (iff_l2r ?? (IH ?) IHa); #z; *; #zs; *; *;
-               #defys pz; *; 
-               ##[ #; ndestruct; @[y]; @z;
-                   nrewrite > (append_nil ? z) in IHa; /4/ by or_intror, conj;
-               ##| #allzs; @[y]; @(z@zs); nrewrite > defys; /3/ by or_intror, conj;##]##]
-       ##| #allbp;
-               ; 
-                   
-           
+ndefinition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l. 
+interpretation "star lang" 'pk l = (star ? l).
 
-nlet rec in_lb (A : Alpha) (e : re A) on e : word A → bool ≝ 
-  match e with
-  [ e ⇒ is_epsilon …
-  | z ⇒ is_empty …
-  | s x ⇒ is_char … x
-  | o e1 e2 ⇒ λw.in_lb … e1 w || in_lb … e2 w
-  | c e1 e2 ⇒ ex_split … (in_lb A e1) (in_lb A e2)
-  | k e ⇒ allb … (in_lb A e) (in_lb A e)].
-  
-  
-nlemma equiv_l_lb : ∀A,e,w. w ∈ e ↔ in_lb A e w = true.
-#A e; nelim e; nnormalize;
-##[ #w; @; ##[##2: #; ndestruct] #H; ninversion H; #; ndestruct;
-##| #w; @; ##[##2: #H; nrewrite > (l2r ??? H); //; ##]
-    #H; ninversion H; #; ndestruct; //;
-##| #x w; @; ##[ #H; ninversion H; #; ndestruct; nrewrite > (r2l ????); //; ##]
-    #H; nrewrite > (l2r ??? H); @2;
-##| #e1 e2 IH1 IH2 w; @; #E; 
-    ##[ ninversion E; ##[##1,2,4,5,6,7: #; ndestruct]
-        #w1 w2 e1 e2 w1r1 w2r2 H1 H2 defw defr1r2; ndestruct;
-        nlapply (IH1 w1); *; #IH1; #_; nlapply (IH1 w1r1);
-        nlapply (IH2 w2); *; #IH2; #_; nlapply (IH2 w2r2);
-        nelim w1;nnormalize; ncases w2; //; nnormalize;
-
-    ##[ nelim w; ##[ nnormalize; //] #x xs IH E; nnormalize;
-    nlapply (IH1 [x]); nlapply (IH2 xs);
-    ncases (in_lb A e1 [x]); ncases (in_lb A e2 xs); nnormalize; *; #E1 E2; *; #E3 E4; /2/;
-    ##[ ncases xs in IH E3 E4; nnormalize; //; #xx xs H; #_;
-    
-     *; nnormalize;
-
-
-nlemma in_l_inv_e:
- ∀S.∀w:word S. w ∈ ∅ → w = [].
- #S; #w; #H; ninversion H; #; ndestruct;
-nqed.
+notation > "𝐋 term 90 E" non associative with precedence 90 for @{in_l ? $E}.
+nlet rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝ 
+match r with
+[ z ⇒ {}
+| e ⇒ { [ ] }
+| s x ⇒ { [x] }
+| c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
+| o r1 r2 ⇒  𝐋 r1 ∪ 𝐋 r2
+| k r1 ⇒ (𝐋 r1) ^*].
+notation "𝐋 term 90 E" non associative with precedence 90 for @{'in_l $E}.
+interpretation "in_l" 'in_l E = (in_l ? E).
+interpretation "in_l mem" 'mem w l = (in_l ? l w).
 
-nlemma in_l_inv_s: ∀S.∀w:word S.∀x. w ∈ `x → w = [x].
-#S w x H; ninversion H; #; ndestruct; //.
-nqed.
+notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
+interpretation "orb" 'orb a b = (orb a b).
 
-nlemma in_l_inv_c:
- ∀S.∀w:word S.∀E1,E2. w ∈ E1 · E2 → ∃w1.∃w2. w = w1@w2 ∧ w1 ∈ E1 ∧ w2 ∈ E2.
-#S w e1 e2 H; ninversion  H; ##[##1,2,4,5,6,7: #; ndestruct; ##]
-#w1 w2 r1 r2 w1r1 w2r2; #_; #_; #defw defe; @w1; @w2; ndestruct; /3/.
-nqed.
+ndefinition if_then_else ≝ λT:Type[0].λe,t,f.match e return λ_.T with [ true ⇒ t | false ⇒ f].
+notation > "'if' term 19 e 'then' term 19 t 'else' term 19 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 90 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
+interpretation "if_then_else" 'if_then_else e t f = (if_then_else ? e t f).
 
 ninductive pitem (S: Alpha) : Type[0] ≝
    pz: pitem S
@@ -273,8 +132,8 @@ nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
   | pe ⇒ ϵ
   | ps x ⇒ `x
   | pp x ⇒ `x
-  | pc E1 E2 ⇒ .|E1| .|E2|
-  | po E1 E2 ⇒ .|E1| + .|E2|
+  | pc E1 E2 ⇒ (.|E1| · .|E2|)
+  | po E1 E2 ⇒ (.|E1| + .|E2|)
   | pk E ⇒ .|E|^* ].
 notation < ".|term 19 e|" non associative with precedence 90 for @{'forget $e}.
 interpretation "forget" 'forget a = (forget ? a).
@@ -284,53 +143,53 @@ 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).
 
-notation > "w .∈ E" non associative with precedence 40 for @{in_pl ? $w $E}.
-ninductive in_pl (S: Alpha): word S → pitem S → Prop ≝
- | in_pp:  ∀x:S.[x] .∈ `.x
- | in_pc1: ∀w1,w2:word S.∀e1,e2:pitem S. 
-             w1 .∈ e1 → w2 ∈ .|e2| → (w1@w2) .∈ e1 · e2
- | in_pc2: ∀w,e1,e2. w .∈ e2 → w .∈ e1 · e2
- | in_po1: ∀w,e1,e2. w .∈ e1 → w .∈ e1 + e2
- | in_po2: ∀w,e1,e2. w .∈ e2 → w .∈ e1 + e2
- | in_pki: ∀w1,w2,e. w1 .∈ e → w2 ∈ .|e|^* → (w1@w2) .∈ e^*.
- 
-interpretation "in_pl" 'in_pl w l = (in_pl ? w l).
- 
-ndefinition in_prl ≝ λS : Alpha.λw:word S.λp:pre S.
-  (w = [ ] ∧ \snd p = true) ∨ w .∈ (\fst p).
+notation > "𝐋\p\ term 90 E" non associative with precedence 90 for @{in_pl ? $E}.
+nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ 
+match r with
+[ pz ⇒ {}
+| pe ⇒ {}
+| ps _ ⇒ {}
+| pp x ⇒ { [x] }
+| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 .|r2| ∪ 𝐋\p\ r2
+| po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
+| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (.|r1|^* ) ].
+notation > "𝐋\p term 90 E" non associative with precedence 90 for @{'in_pl $E}.
+notation "𝐋\sub(\p) term 90 E" non associative with precedence 90 for @{'in_pl $E}.
+interpretation "in_pl" 'in_pl E = (in_pl ? E).
+interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
+
+ndefinition epsilon ≝ λS,b.if b then { ([ ] : word S) } else {}.
+
+interpretation "epsilon" 'epsilon = (epsilon ?).
+notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
+interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
+
+ndefinition in_prl ≝ λS : Alpha.λp:pre S.  𝐋\p\ (\fst p) ∪ ϵ (\snd p).
   
-notation > "w .∈ E" non associative with precedence 40 for @{'in_pl $w $E}.  
-notation < "w\shy .∈\shy E" non associative with precedence 40 for @{'in_pl $w $E}.   
-interpretation "in_prl" 'in_pl w l = (in_prl ? w l).
-
-
-
-nlemma append_eq_nil : 
-  ∀S.∀w1,w2:word S. [ ] = w1 @ w2 → w1 = [ ].
-#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct;
-nqed.
-
-nlemma append_eq_nil_r : 
-  ∀S.∀w1,w2:word S. [ ] = w1 @ w2 → w2 = [ ].
-#S w1; nelim w1; ##[ #w2 H; nrewrite > H; // ]
-#x tl IH w2; nnormalize; #abs; ndestruct;
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_pl E = (in_prl ? E).
+
+nlemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
+#S w1; nelim w1; //. #x tl IH w2; nnormalize; #abs; ndestruct; nqed.
+
+(* lemma 12 *)
+nlemma epsilon_in_true : ∀S.∀e:pre S. [ ] ∈ e ↔ \snd e = true.
+#S r; ncases r; #e b; @; ##[##2: #H; nrewrite > H; @2; /2/; ##] ncases b;//; 
+nnormalize; *; ##[##2:*] nelim e;
+##[ ##1,2: *; ##| #c; *; ##| #c; nnormalize; #; ndestruct; ##| ##7: #p H;
+##| #r1 r2 H G; *; ##[##2: /3/ by or_intror]
+##| #r1 r2 H1 H2; *; /3/ by or_intror, or_introl; ##]
+*; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//;
 nqed.
 
-nlemma lemma16 : 
-  ∀S.∀e:pre S. [ ] .∈ e ↔ \snd e = true.
-#S p; ncases p; #e b; @; ##[##2: #H; nrewrite > H; @; @; //. ##]
-ncases b; //; *; ##[*; //] nelim e; 
-##[##1,2: #abs; ninversion abs; #; ndestruct;
-##|##3,4: #x abs; ninversion abs; #; ndestruct;
-##|#p1 p2 H1 H2 H; ninversion H; ##[##1,3,4,5,6: #; ndestruct; /2/. ##]
-   #w1 w2 r1 r2 w1r1 w2fr2 H3 H4 Ep1p2; ndestruct;
-   nrewrite > (append_eq_nil … H4) in w1r1; /2/ by {};
-##|#r1 r2 H1 H2 H; ninversion H; #; ndestruct; /2/ by {};
-##|#r H1 H2; ninversion H2; ##[##1,2,3,4,5: #; ndestruct; ##]
-   #w1 w2 r1 w1r1 w1er1 H11 H21 H31;
-   nrewrite > (append_eq_nil … H21) in w1r1 H1;
-   nrewrite > (?: r = r1); /2/ by {};
-   ndestruct; //. ##]
+nlemma not_epsilon_lp : ∀S.∀e:pitem S. ¬ (𝐋\p e [ ]).
+#S e; nelim e; nnormalize; /2/ by nmk;
+##[ #; @; #; ndestruct;
+##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
+    nrewrite > (append_eq_nil …H…); /2/;
+##| #r1 r2 n1 n2; @; *; /2/;
+##| #r n; @; *; #w1; *; #w2; *; *; #H;     
+    nrewrite > (append_eq_nil …H…); /2/;##]
 nqed.
 
 ndefinition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
@@ -339,10 +198,9 @@ interpretation "oplus" 'oplus a b = (lo ? a b).
 
 ndefinition lc ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa,b:pre S.
    match a with [ mk_pair e1 b1 ⇒
-   match b with [ mk_pair e2 b2 ⇒
    match b1 with 
-   [ false ⇒ 〈e1 · e2, b2〉 
-   | true ⇒ match bcast ? e2 with [ mk_pair e2' b2' ⇒ 〈e1 · e2', b2 || b2'〉 ]]]].
+   [ false ⇒ 〈e1 · \fst b, \snd b〉 
+   | true ⇒ 〈e1 · \fst (bcast ? (\fst b)),\snd b || \snd (bcast ? (\fst b))〉]].
    
 notation < "a ⊙ b" left associative with precedence 60 for @{'lc $op $a $b}.
 interpretation "lc" 'lc op a b = (lc ? op a b).
@@ -352,9 +210,9 @@ ndefinition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λa:pre S.
    match a with [ mk_pair e1 b1 ⇒
    match b1 with 
    [ false ⇒ 〈e1^*, false〉 
-   | true ⇒ match bcast ? e1 with [ mk_pair e1' b1' ⇒ 〈e1'^*, true〉 ]]].
+   | true ⇒ 〈(\fst (bcast ? e1))^*, true〉]].
 
-notation < "a \sup ⊛" non associative with precedence 90 for @{'lk ? $a}.
+notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.
 interpretation "lk" 'lk op a = (lk ? op a).
 notation > "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
 
@@ -367,7 +225,7 @@ nlet rec eclose (S: Alpha) (E: pitem S) on E : pre S ≝
   | pp x ⇒ 〈 `.x, false 〉
   | po E1 E2 ⇒ •E1 ⊕ •E2
   | pc E1 E2 ⇒ •E1 ⊙ 〈 E2, false 〉 
-  | pk E ⇒ 〈E,true〉^⊛].
+  | pk E ⇒ 〈(\fst (•E))^*,true〉].
 notation < "• x" non associative with precedence 60 for @{'eclose $x}.
 interpretation "eclose" 'eclose x = (eclose ? x).
 notation > "• x" non associative with precedence 60 for @{'eclose $x}.
@@ -375,15 +233,264 @@ notation > "• x" non associative with precedence 60 for @{'eclose $x}.
 ndefinition reclose ≝ λS:Alpha.λp:pre S.let p' ≝ •\fst p in 〈\fst p',\snd p || \snd p'〉.
 interpretation "reclose" 'eclose x = (reclose ? x).
 
-nlemma lemma19_2 :
- ∀S:Alpha.∀e1,e2:pre S.∀w. w .∈ e1 ⊕ e2 → w .∈ e1 ∨ w .∈ e2.
-#S e1 e2 w H; nnormalize in H; ncases H;
-##[ *; #defw; ncases e1; #p b; ncases b; nnormalize;
-    ##[ #_; @1; @1; /2/ by conj;
-    ##| #H1; @2; @1; /2/ by conj; ##]
-##| #H1; ninversion H1; #; ndestruct; /4/ by or_introl, or_intror; ##]
+ndefinition eq_f1 ≝ λS.λa,b:word S → Prop.∀w.a w ↔ b w.
+notation > "A =1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+notation "A =\sub 1 B" non associative with precedence 45 for @{'eq_f1 $A $B}.
+interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+
+naxiom extP : ∀S.∀p,q:word S → Prop.(p =1 q) → p = q.
+
+nlemma epsilon_or : ∀S:Alpha.∀b1,b2. ϵ(b1 || b2) = ϵ b1 ∪ ϵ b2. ##[##2: napply S]
+#S b1 b2; ncases b1; ncases b2; napply extP; #w; nnormalize; @; /2/; *; //; *;
+nqed.
+
+nlemma cupA : ∀S.∀a,b,c:word S → Prop.a ∪ b ∪ c = a ∪ (b ∪ c).
+#S a b c; napply extP; #w; nnormalize; @; *; /3/; *; /3/; nqed.
+
+nlemma cupC : ∀S. ∀a,b:word S → Prop.a ∪ b = b ∪ a.
+#S a b; napply extP; #w; @; *; nnormalize; /2/; 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;
+nwhd in ⊢ (??(??%)?);
+nchange in ⊢(??%?) with (𝐋\p (e1 + e2) ∪ ϵ (b1 || b2));
+nchange in ⊢(??(??%?)?) with (𝐋\p (e1) ∪ 𝐋\p (e2));
+nrewrite > (epsilon_or S …); nrewrite > (cupA S (𝐋\p e1) …);
+nrewrite > (cupC ? (ϵ b1) …); nrewrite < (cupA S (𝐋\p e2) …);
+nrewrite > (cupC ? ? (ϵ b1) …); nrewrite < (cupA …); //;
+nqed.
+
+nlemma odotEt : 
+  ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
+#S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
+
+nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) =  𝐋\p e1 · 𝐋 .|e2| ∪ 𝐋\p e2. //; nqed.
+
+nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r). 
+#S p q r; napply extP; #w; nnormalize; @; 
+##[ *; #x; *; #y; *; *; #defw; *; /7/ by or_introl, or_intror, ex_intro, conj;
+##| *; *; #x; *; #y; *; *; /7/ by or_introl, or_intror, ex_intro, conj; ##]
+nqed.
+
+nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
+#S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
+
+nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 .|e1 · e2| =  𝐋 .|e1| · 𝐋 .|e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
+nqed.
+
+nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 .|e1 + e2| =  𝐋 .|e1| ∪ 𝐋 .|e2|.
+#S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
+
+nlemma erase_star : ∀S.∀e1:pitem S.𝐋 .|e1|^* = 𝐋 .|e1^*|. //; nqed.
+
+ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
+interpretation "substract" 'minus a b = (substract ? a b).
+
+nlemma cup_sub: ∀S.∀a,b:word S → Prop. ¬ (a []) → a ∪ (b - {[]}) = (a ∪ b) - {[]}.
+#S a b c; napply extP; #w; nnormalize; @; *; /4/; *; /4/; nqed.
+
+nlemma sub0 : ∀S.∀a:word S → Prop. a - {} = a.
+#S a; napply extP; #w; nnormalize; @; /3/; *; //; nqed.
+
+nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
+#S a; napply extP; #w; nnormalize; @; *; /2/; nqed.
+
+nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
+#S a b w; nnormalize; *; //; nqed.
+
+nlemma erase_bull : ∀S.∀a:pitem S. .|\fst (•a)| = .|a|.
+#S a; nelim a; // by {};
+##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| · .|e2|);
+    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));
+    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〉.
+#S p; ncases p; //; 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.
+#S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
+##[ nwhd in ⊢ (??(??%)?); 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…); //;##]
+nqed.
+
+(* theorem 16: 1 *)
+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|.
+#S e; nelim e; //;
+  ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
+  ##| #a; napply extP; #w; nnormalize; @; *; /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 …); //;
+  ##| #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 …); //.
+  ##| #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'^* ) ∪ {[ ]});
+      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 ⊢ (???%) with ((𝐋. |e|)^* ); napply extP; #w; @;
+      ##[ *; ##[##2: nnormalize; #defw; nrewrite < defw; @[]; @; //]
+          *; #w1; *; #w2; *; *; #defw sube; *; #lw; *; #flx cj;
+          @ (w1 :: lw); nrewrite < defw; nrewrite < flx; @; //;
+          @; //; 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: @[]; @; //] 
+                    @; //; @; //; @; *;##]##]##]##]##]
+nqed.
+
+(* theorem 16: 3 *)      
+nlemma odot_dot: 
+  ∀S.∀e1,e2: pre S.  𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
+#S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
+
+nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} =  𝐋 e^*.
+#S e; napply extP; #w; nnormalize; @;
+##[ *; ##[##2: #H; nrewrite < H; @[]; /3/] *; #w1; *; #w2; 
+    *; *; #defw Hw1; *; #wl; *; #defw2 Hwl; @(w1 :: wl);
+    nrewrite < defw; nrewrite < defw2; @; //; @;//;
+##| *; #wl; *; #defw Hwl; ncases wl in defw Hwl; ##[#defw; #; @2; nrewrite < defw; //]
+    #x xs defw; *; #Hx Hxs; @; @x; @(flatten ? xs); nrewrite < defw;
+    @; /2/; @xs; /2/;##]
+ nqed.
+
+nlemma nil_star : ∀S.∀e:re S. [ ] ∈ e^*.
+#S e; @[]; /2/; nqed.
+
+nlemma cupID : ∀S.∀l:word S → Prop.l ∪ l = l.
+#S l; napply extP; #w; @; ##[*]//; #; @; //; nqed.
+
+nlemma cup_star_nil : ∀S.∀l:word S → Prop. l^* ∪ {[]} = l^*.
+#S a; napply extP; #w; @; ##[*; //; #H; nrewrite < H; @[]; @; //] #;@; //;nqed.
+
+naxiom Admit : False.
+
+nlemma key_id : ∀S.∀e:pitem S. 𝐋\p e · 𝐋 .|e|^* ∪ {[]} = 𝐋\p e · 𝐋 .|e|^* ∪ 𝐋 .|e|^*.
+#S e; napply extP; #w; @;##[##2:*]
+##[*; #w1; *; #w2; *; *; #defw Hw1 Hw2;@; @w1; @w2; /3/;
+##|*; #wl; *; #H; nrewrite < H;
+(*
+   ngeneralize in match e;
+   nelim wl;##[#e;#_;@2;//] #x xs IH e;*; #Hx Hxs; ncases (IH Hxs); 
+   ##[##2: #H; nnormalize; nrewrite < H; nrewrite > (append_nil…);
+
+   ncases wl; ##[#_;@2; //] #x xs; *; #Hx Hxs; @; @x; @(flatten ? xs); @;
+   ##[@;//;##|@xs; @; //]
+   ngeneralize in match Hx; ngeneralize in match x; nelim e; nnormalize; //;
+   ##[#e1 e2 IH1 IH2 x; *; #w1; *; #w2; *; *; #defx Hw1 Hw2;
+      @; @w1; @w2; /4/ by conj;
+   ##|#e1 e2 IH1 IH2 y;*; #; ##[@|@2] /2/;
+   ##|#e IH y; *; #wl; *; #delwl Hw2; nrewrite < delwl;
+      nelim wl in Hw2; ##[#_;@[];@[];@;//; 
+*)
+   ncases Admit;
+##|*;##[##2: #H; nrewrite < H; @2; //] *; #w1; *; #w2; *; *; #defw Hw1 Hw2;
+   @; @w1; @w2; /3/;##]
 nqed.
 
+(* theorem 16: 4 *)      
+nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 .|\fst e|)^*.
+#S e; ncases e; #e' b'; ncases b';
+##[ nchange in match (〈e',true〉^⊛) with 〈?,?〉;
+    nletin e'' ≝ (\fst (•e'));
+    nchange in ⊢ (??%?) with (?∪?);
+    nchange in ⊢ (??(??%?)?) with (𝐋\p e'' · 𝐋 .|e''|^* );
+    nrewrite > (?:  𝐋\p e'' ·  𝐋.|e''|^* ∪ {[]} =  (𝐋\p e' ∪ 𝐋.|e'|) · 𝐋.|e''|^* ∪ {[]}); ##[##2:
+      nrewrite < (bull_cup…); nchange in ⊢ (???(??(??%?)?)) with (?∪?);
+      nchange in match e'' with e'';
+      ncases (\snd (•e')); ##[##2: nrewrite > (cup0…); //]
+      nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…);
+      nrewrite > (cupA…); nrewrite > (cup_star_nil…);
+      napply key_id;##]
+    nrewrite > (cup_dotD…); nrewrite > (cupA…);
+    nrewrite > (?: ?·? ∪ {[]} = 𝐋.|e'|^* ); ##[##2:
+      nrewrite > (erase_bull…); nrewrite > (dot_star_epsilon…); //]
+    nrewrite > (erase_bull…);  
+    nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+    nrewrite > (cup_dotD…);nrewrite > (epsilon_dot…); //;
+##| nwhd in match (〈e',false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
+    nrewrite > (cup0…);
+    nchange in ⊢ (??%?) with (𝐋\p e' · 𝐋 .|e'|^* );
+    nrewrite < (cup0 ? (𝐋\p e')); //;##]
+nqed.
+
+STOP
+
 notation > "\move term 90 x term 90 E" 
 non associative with precedence 60 for @{move ? $x $E}.
 nlet rec move (S: Alpha) (x:S) (E: pitem S) on E : pre S ≝
-- 
2.39.2