X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fre%2Fre.ma;h=8ba714b82592f01047735b685663a64ed200f3d5;hb=7c9d99dfb049d726491b71f07ba6a9b088b30166;hp=388c27e708f065adc1e09cc80a251843b86a5365;hpb=d5d5925101dd773efb2f90136adc5d714a530cb9;p=helm.git

diff --git a/matita/matita/lib/re/re.ma b/matita/matita/lib/re/re.ma
index 388c27e70..8ba714b82 100644
--- a/matita/matita/lib/re/re.ma
+++ b/matita/matita/lib/re/re.ma
@@ -12,81 +12,40 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "arithmetics/nat.ma".
-include "basics/list.ma".
+include "re/lang.ma".
+include "basics/core_notation/card_1.ma".
 
-interpretation "iff" 'iff a b = (iff a b).  
+(* The type re of regular expressions over an alphabet $S$ is the smallest 
+collection of objects generated by the following constructors: *)
 
-record 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).
-
-definition word ≝ λS:Alpha.list S.
-
-inductive re (S: Alpha) : Type[0] ≝
+inductive re (S: DeqSet) : Type[0] ≝
    z: re S
  | e: re S
  | s: S → re S
  | c: re S → re S → re S
  | 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 90 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:Alpha.∀p:re S.  re S →  re S   ≝ c  on _p : re ?  to ∀_:?.?.
-*)
+
+interpretation "re epsilon" 'epsilon = (e ?).
+interpretation "re or" 'plus a b = (o ? a b).
+interpretation "re cat" 'middot a b = (c ? a b).
+interpretation "re star" 'star a = (k ? a).
 
 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 "∅" non associative with precedence 90 for @{ 'empty }.
+notation "`∅" non associative with precedence 90 for @{ 'empty }.
 interpretation "empty" 'empty = (z ?).
 
-let rec flatten (S : Alpha) (l : list (word S)) on l : word S ≝ 
-match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
-
-let 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 ]. 
-// qed.
-
-definition empty_lang ≝ λS.λw:word S.False.
-notation "{}" non associative with precedence 90 for @{'empty_lang}.
-interpretation "empty lang" 'empty_lang = (empty_lang ?).
-
-definition sing_lang ≝ λS.λx,w:word S.x=w.
-(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}.*)
-interpretation "sing lang" 'singl x = (sing_lang ? x).
-
-definition 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).
-
-definition 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).
+(* The language sem{e} associated with the regular expression e is inductively 
+defined by the following function: *)
 
-definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l. 
-interpretation "star lang" 'pk l = (star ? l).
-
-let rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝ 
+let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝ 
 match r with
-[ z ⇒ {}
-| e ⇒ { [ ] }
-| s x ⇒ { [x] }
+[ z ⇒ ∅
+| e ⇒ {ϵ}
+| s x ⇒ {[x]}
 | c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
 | o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
 | k r1 ⇒ (in_l ? r1) ^*].
@@ -98,15 +57,38 @@ interpretation "in_l mem" 'mem w l = (in_l ? l w).
 lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
 // qed.
 
-notation "a || b" left associative with precedence 30 for @{'orb $a $b}.
-interpretation "orb" 'orb a b = (orb a b).
-
-definition 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).
-
-inductive pitem (S: Alpha) : Type[0] ≝
+(* 
+Pointed Regular expressions
+
+We now introduce pointed regular expressions, that are the main tool we shall 
+use for the construction of the automaton. 
+A pointed regular expression is just a regular expression internally labelled 
+with some additional points. Intuitively, points mark the positions inside the 
+regular expression which have been reached after reading some prefix of
+the input string, or better the positions where the processing of the remaining 
+string has to be started. Each pointed expression for $e$ represents a state of 
+the {\em deterministic} automaton associated with $e$; since we obviously have 
+only a finite number of possible labellings, the number of states of the automaton 
+is finite.
+
+Pointed regular expressions provide the tool for an algebraic revisitation of 
+McNaughton and Yamada's algorithm for position automata, making the proof of its 
+correctness, that is far from trivial, particularly clear and simple. In particular, 
+pointed expressions offer an appealing alternative to Brzozowski's derivatives, 
+avoiding their weakest point, namely the fact of being forced to quotient derivatives 
+w.r.t. a suitable notion of equivalence in order to get a finite number of states 
+(that is not essential for recognizing strings, but is crucial for comparing regular 
+expressions). 
+
+Our main data structure is the notion of pointed item, that is meant whose purpose
+is to encode a set of positions inside a regular expression. 
+The idea of formalizing pointers inside a data type by means of a labelled version 
+of the data type itself is probably one of the first, major lessons learned in the 
+formalization of the metatheory of programming languages. For our purposes, it is 
+enough to mark positions preceding individual characters, so we shall have two kinds 
+of characters •a (pp a) and a (ps a) according to the case a is pointed or not. *)
+
+inductive pitem (S: DeqSet) : Type[0] ≝
    pz: pitem S
  | pe: pitem S
  | ps: S → pitem S
@@ -115,41 +97,132 @@ inductive pitem (S: Alpha) : Type[0] ≝
  | po: pitem S → pitem S → pitem S
  | pk: pitem S → pitem S.
  
+(* A pointed regular expression (pre) is just a pointed item with an additional 
+boolean, that must be understood as the possibility to have a trailing point at 
+the end of the expression. As we shall see, pointed regular expressions can be 
+understood as states of a DFA, and the boolean indicates if
+the state is final or not. *)
+
 definition pre ≝ λS.pitem S × bool.
 
-interpretation "pstar" 'pk a = (pk ? a).
-interpretation "por" 'plus a b = (po ? a b).
-interpretation "pcat" 'pc a b = (pc ? a b).
+interpretation "pitem star" 'star a = (pk ? a).
+interpretation "pitem or" 'plus a b = (po ? a b).
+interpretation "pitem cat" 'middot 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 "pitem pp" 'pp a = (pp ? a).
+interpretation "pitem ps" 'ps a = (ps ? a).
+interpretation "pitem epsilon" 'epsilon = (pe ?).
+interpretation "pitem empty" 'empty = (pz ?).
 
-interpretation "patom" 'ps a = (ps ? a).
-interpretation "pepsilon" 'epsilon = (pe ?).
-interpretation "pempty" 'empty = (pz ?).
+(* The carrier $|i|$ of an item i is the regular expression obtained from i by 
+removing all the points. Similarly, the carrier of a pointed regular expression 
+is the carrier of its item. *)
 
-let rec forget (S: Alpha) (l : pitem S) on l: re S ≝
+let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
  match l with
-  [ pz ⇒ ∅
+  [ pz ⇒ `∅
   | pe ⇒ ϵ
   | ps x ⇒ `x
   | pp x ⇒ `x
   | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
   | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
   | pk E ⇒ (forget ? E)^* ].
-  
-(* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
-interpretation "forget" 'norm a = (forget ? a).
+ 
+interpretation "forget" 'card a = (forget ? a).
+
+lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
+// qed.
+
+lemma erase_plus : ∀S.∀i1,i2:pitem S.
+  |i1 + i2| = |i1| + |i2|.
+// qed.
+
+lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
+// qed.
+
+(* 
+Comparing items and pres
+
+Items and pres are very concrete datatypes: they can be effectively compared, 
+and enumerated. In particular, we can define a boolean equality beqitem and a proof
+beqitem_true that it refects propositional equality, enriching the set (pitem S)
+to a DeqSet. *)
+
+let rec beqitem S (i1,i2: pitem S) on i1 ≝ 
+  match i1 with
+  [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
+  | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
+  | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
+  | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
+  | po i11 i12 ⇒ match i2 with 
+    [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+    | _ ⇒ false]
+  | pc i11 i12 ⇒ match i2 with 
+    [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+    | _ ⇒ false]
+  | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
+  ].
+
+lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2). 
+#S #i1 elim i1
+  [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+  |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+  |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+    [>(\P H) // | @(\b (refl …))]
+  |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+    [>(\P H) // | @(\b (refl …))]
+  |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+   normalize #H destruct 
+    [cases (true_or_false (beqitem S i11 i21)) #H1
+      [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @False_ind /2/
+      ]
+    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    ]
+  |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+   normalize #H destruct 
+    [cases (true_or_false (beqitem S i11 i21)) #H1
+      [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @False_ind /2/
+      ]
+    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    ]
+  |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
+   normalize #H destruct 
+    [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
+  ]
+qed. 
+
+definition DeqItem ≝ λS.
+  mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
 
-let rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ 
+(* We also add a couple of unification hints to allow the type inference system 
+to look at (pitem S) as the carrier of a DeqSet, and at beqitem as if it was the 
+equality function of a DeqSet. *)
+
+unification hint  0 ≔ S; 
+    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢ 
+    pitem S ≡ carr X.
+    
+unification hint  0 ≔ S,i1,i2; 
+    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢ 
+    beqitem S i1 i2 ≡ eqb X i1 i2.
+
+(*
+Semantics of pointed regular expressions
+
+The intuitive semantic of a point is to mark the position where
+we should start reading the regular expression. The language associated
+to a pre is the union of the languages associated with its points. *)
+
+let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝ 
 match r with
-[ pz ⇒ {}
-| pe ⇒ {}
-| ps _ ⇒ {}
+[ pz ⇒ ∅
+| pe ⇒ ∅
+| ps _ ⇒ ∅
 | pp x ⇒ { [x] }
 | pc r1 r2 ⇒ (in_pl ? r1) · \sem{forget ? r2} ∪ (in_pl ? r2)
 | po r1 r2 ⇒ (in_pl ? r1) ∪ (in_pl ? r2)
@@ -158,20 +231,16 @@ match r with
 interpretation "in_pl" 'in_l E = (in_pl ? E).
 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
 
-definition 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).
-
-definition in_prl ≝ λS : Alpha.λp:pre S. 
-  if (\snd p) then \sem{\fst p} ∪ { ([ ] : word S) } else \sem{\fst p}.
+definition in_prl ≝ λS : DeqSet.λp:pre S. 
+  if (\snd p) then \sem{\fst p} ∪ {ϵ} else \sem{\fst p}.
   
 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
 interpretation "in_prl" 'in_l E = (in_prl ? E).
 
+(* The following, trivial lemmas are only meant for rewriting purposes. *)
+
 lemma sem_pre_true : ∀S.∀i:pitem S. 
-  \sem{〈i,true〉} = \sem{i} ∪ { ([ ] : word S) }. 
+  \sem{〈i,true〉} = \sem{i} ∪ {ϵ}. 
 // qed.
 
 lemma sem_pre_false : ∀S.∀i:pitem S. 
@@ -202,10 +271,18 @@ lemma sem_star_w : ∀S.∀i:pitem S.∀w.
   \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
 // qed.
 
-lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = [ ] → w1 = [ ].
+(* Below are a few, simple, semantic properties of items. In particular:
+- not_epsilon_item : ∀S:DeqSet.∀i:pitem S. ¬ (\sem{i} ϵ).
+- epsilon_pre : ∀S.∀e:pre S. (\sem{i} ϵ) ↔ (\snd e = true).
+- minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{[ ]}.
+- minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} ≐ \sem{e}-{[ ]}.
+The first property is proved by a simple induction on $i$; the other
+results are easy corollaries. We need an auxiliary lemma first. *)
+
+lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
 
-lemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ e).
+lemma not_epsilon_lp : ∀S:DeqSet.∀e:pitem S. ¬ (ϵ ∈ e).
 #S #e elim e normalize /2/  
   [#r1 #r2 * #n1 #n2 % * /2/ * #w1 * #w2 * * #H 
    >(append_eq_nil …H…) /2/
@@ -214,200 +291,173 @@ lemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ([ ] ∈ e).
   ]
 qed.
 
-(* lemma 12 *)
-lemma epsilon_to_true : ∀S.∀e:pre S. [ ] ∈ e → \snd e = true.
+lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
 #S * #i #b cases b // normalize #H @False_ind /2/ 
 qed.
 
-lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → [ ] ∈ e.
+lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
 #S * #i #b #btrue normalize in btrue; >btrue %2 // 
 qed.
 
-definition lo ≝ λS:Alpha.λa,b:pre S.〈\fst a + \fst b,\snd a || \snd b〉.
-notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
+lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{[ ]}.
+#S #i #w % 
+  [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
+  |* //
+  ]
+qed.
+
+lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} ≐ \sem{e}-{[ ]}.
+#S * #i * 
+  [>sem_pre_true normalize in ⊢ (??%?); #w % 
+    [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
+  |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
+  ]
+qed.
+
+(*
+Broadcasting points
+
+Intuitively, a regular expression e must be understood as a pointed expression with a single 
+point in front of it. Since however we only allow points before symbols, we must broadcast 
+this initial point inside e traversing all nullable subexpressions, that essentially corresponds 
+to the ϵ-closure operation on automata. We use the notation •(_) to denote such an operation;
+its definition is the expected one: let us start discussing an example.
+
+Example
+Let us broadcast a point inside (a + ϵ)(b*a + b)b. We start working in parallel on the 
+first occurrence of a (where the point stops), and on ϵ that gets traversed. We have hence 
+reached the end of a + ϵ and we must pursue broadcasting inside (b*a + b)b. Again, we work in 
+parallel on the two additive subterms b^*a and b; the first point is allowed to both enter the 
+star, and to traverse it, stopping in front of a; the second point just stops in front of b. 
+No point reached that end of b^*a + b hence no further propagation is possible. In conclusion: 
+               •((a + ϵ)(b^*a + b)b) = 〈(•a + ϵ)((•b)^*•a + •b)b, false〉
+*)
+
+(* Broadcasting a point inside an item generates a pre, since the point could possibly reach 
+the end of the expression. 
+Broadcasting inside a i1+i2 amounts to broadcast in parallel inside i1 and i2.
+If we define
+                 〈i1,b1〉 ⊕ 〈i2,b2〉 = 〈i1 + i2, b1 ∨ b2〉
+then, we just have •(i1+i2) = •(i1)⊕ •(i2).
+*)
+
+definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
+notation "a ⊕ b" left associative with precedence 65 for @{'oplus $a $b}.
 interpretation "oplus" 'oplus a b = (lo ? a b).
 
-lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1||b2〉.
+lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
 // qed.
 
-definition pre_concat_r ≝ λS:Alpha.λi:pitem S.λe:pre S.
-  match e with [ pair i1 b ⇒ 〈i · i1, b〉].
+(*
+Concatenation is a bit more complex. In order to broadcast a point inside i1 · i2 
+we should start broadcasting it inside i1 and then proceed into i2 if and only if a 
+point reached the end of i1. This suggests to define •(i1 · i2) as •(i1) ▹ i2, where 
+e ▹ i is a general operation of concatenation between a pre and an item, defined by 
+cases on the boolean in e: 
+
+       〈i1,true〉 ▹ i2  = i1 ◃ •(i_2)
+       〈i1,false〉 ▹ i2 = i1 · i2
+       
+In turn, ◃ says how to concatenate an item with a pre, that is however extremely simple:
+
+        i1 ◃ 〈i1,b〉  = 〈i_1 · i2, b〉
+
+Let us come to the formalized definitions:
+*)
+
+definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
+  match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
  
-notation "i ◂ e" left associative with precedence 60 for @{'ltrif $i $e}.
-interpretation "pre_concat_r" 'ltrif i e = (pre_concat_r ? i e).
-  
-definition 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}.
-interpretation "eq f1" 'eq_f1 a b = (eq_f1 ? a b).
+notation "i ◃ e" left associative with precedence 65 for @{'lhd $i $e}.
+interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
+
+(* The behaviour of ◃ is summarized by the following, easy lemma: *)
 
 lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
-  A = B → A =1 B. 
+  A = B → A ≐ B. 
 #S #A #B #H >H /2/ qed.
 
-lemma ext_eq_trans: ∀S.∀A,B,C:word S → Prop. 
-  A =1 B → B =1 C → A =1 C. 
-#S #A #B #C #eqAB #eqBC #w cases (eqAB w) cases (eqBC w) /4/
-qed.   
-
-lemma union_assoc: ∀S.∀A,B,C:word S → Prop. 
-  A ∪ B ∪ C =1 A ∪ (B ∪ C).
-#S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
-qed.   
-
 lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
-  \sem{i ◂ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
-#S #i * #i1 #b1 cases b1 /2/
+  \sem{i ◃ e} ≐ \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //] 
 >sem_pre_true >sem_cat >sem_pre_true /2/ 
 qed.
- 
-definition lc ≝ λS:Alpha.λbcast:∀S:Alpha.pitem S → pre S.λe1:pre S.λi2:pitem S.
+
+(* The definition of $•(-)$ (eclose) and ▹ (pre_concat_l) are mutually recursive.
+In this situation, a viable alternative that is usually simpler to reason about, 
+is to abstract one of the two functions with respect to the other. In particular
+we abstract pre_concat_l with respect to an input bcast function from items to
+pres. *)
+
+definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
   match e1 with 
-  [ pair i1 b1 ⇒ match b1 with 
-    [ true ⇒ (i1 ◂ (bcast ? i2)) 
+  [ mk_Prod i1 b1 ⇒ match b1 with 
+    [ true ⇒ (i1 ◃ (bcast ? i2)) 
     | false ⇒ 〈i1 · i2,false〉
     ]
   ].
-        
-definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
-  match e with 
-  [ pair i b ⇒ 〈\fst (f i), \snd (f i) || b〉].
 
-notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}.
-interpretation "lc" 'lc op a b = (lc ? op a b).
+notation "a ▹ b" left associative with precedence 65 for @{'tril eclose $a $b}.
+interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
 
-definition lk ≝ λS:Alpha.λbcast:∀S:Alpha.∀E:pitem S.pre S.λe:pre S.
-  match e with 
-  [ pair i1 b1 ⇒
-    match b1 with 
-    [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
-    |false ⇒ 〈i1^*,false〉
-    ]
-  ]. 
-
-(*
-lemma oplus_tt : ∀S: Alpha.∀i1,i2:pitem S. 
-  〈i1,true〉 ⊕ 〈i2,true〉  = 〈i1 + i2,true〉.
-// qed.
-
-lemma oplus_tf : ∀S: Alpha.∀i1,i2:pitem S. 
-  〈i1,true〉 ⊕ 〈i2,false〉  = 〈i1 + i2,true〉.
-// qed.
-
-lemma oplus_ft : ∀S: Alpha.∀i1,i2:pitem S. 
-  〈i1,false〉 ⊕ 〈i2,true〉  = 〈i1 + i2,true〉.
-// qed.
-
-lemma oplus_ff : ∀S: Alpha.∀i1,i2:pitem S. 
-  〈i1,false〉 ⊕ 〈i2,false〉  = 〈i1 + i2,false〉.
-// qed. *)
-
-(* 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}.
+notation "•" non associative with precedence 65 for @{eclose ?}.
 
-notation "•" non associative with precedence 60 for @{eclose ?}.
+(* We are ready to give the formal definition of the broadcasting operation. *)
 
-let rec eclose (S: Alpha) (i: pitem S) on i : pre S ≝
+let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
  match i with
-  [ pz ⇒ 〈 ∅, false 〉
+  [ pz ⇒ 〈 `∅, false 〉
   | pe ⇒ 〈 ϵ,  true 〉
   | ps x ⇒ 〈 `.x, false〉
   | pp x ⇒ 〈 `.x, false 〉
   | po i1 i2 ⇒ •i1 ⊕ •i2
-  | pc i1 i2 ⇒ •i1 ▸ i2
+  | pc i1 i2 ⇒ •i1 ▹ i2
   | pk i ⇒ 〈(\fst (•i))^*,true〉].
   
-notation "• x" non associative with precedence 60 for @{'eclose $x}.
+notation "• x" non associative with precedence 65 for @{'eclose $x}.
 interpretation "eclose" 'eclose x = (eclose ? x).
 
-lemma eclose_plus: ∀S:Alpha.∀i1,i2:pitem S.
+(* Here are a few simple properties of ▹ and •(-) *)
+
+lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
   •(i1 + i2) = •i1 ⊕ •i2.
 // qed.
 
-lemma eclose_dot: ∀S:Alpha.∀i1,i2:pitem S.
-  •(i1 · i2) = •i1 ▸ i2.
+lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
+  •(i1 · i2) = •i1 ▹ i2.
 // qed.
 
-lemma eclose_star: ∀S:Alpha.∀i:pitem S.
+lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
   •i^* = 〈(\fst(•i))^*,true〉.
 // qed.
 
-definition reclose ≝ λS. lift S (eclose S). 
-interpretation "reclose" 'eclose x = (reclose ? x).
-
-lemma epsilon_or : ∀S:Alpha.∀b1,b2. epsilon S (b1 || b2) =1 ϵ b1 ∪ ϵ b2. 
-#S #b1 #b2 #w % cases b1 cases b2 normalize /2/ * /2/ * ;
-qed.
-
-(*
-lemma 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 *)
-lemma sem_oplus: ∀S:Alpha.∀e1,e2:pre S.
-  \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}. 
-#S * #i1 #b1 * #i2 #b2 #w %
-  [cases b1 cases b2 normalize /2/ * /3/ * /3/
-  |cases b1 cases b2 normalize /2/ * /3/ * /3/
-  ]
-qed.
-
 lemma odot_true : 
   ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▸ i2 = i1 ◂ (•i2).
+  〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
 // qed.
 
 lemma odot_true_bis : 
   ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+  〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
 #S #i1 #i2 normalize cases (•i2) // qed.
 
 lemma odot_false: 
   ∀S.∀i1,i2:pitem S.
-  〈i1,false〉 ▸ i2 = 〈i1 · i2, false〉.
+  〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
 // qed.
 
-lemma LcatE : ∀S.∀e1,e2:pitem S.
-  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
-// qed.
-
-(*
-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.*)
+(* The definition of •(-) (eclose) can then be lifted from items to pres
+in the obvious way. *)
 
-lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
-// qed.
-
-lemma erase_plus : ∀S.∀i1,i2:pitem S.
-  |i1 + i2| = |i1| + |i2|.
-// qed.
-
-lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
-// qed.
-
-definition 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.
+definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
+  match e with 
+  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
+  
+definition preclose ≝ λS. lift S (eclose S). 
+interpretation "preclose" 'eclose x = (preclose ? x).
 
-nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
-#S a b w; nnormalize; *; //; nqed. *)
+(* Obviously, broadcasting does not change the carrier of the item,
+as it is easily proved by structural induction. *)
 
 lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
 #S #i elim i // 
@@ -419,271 +469,252 @@ lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
   ]
 qed.
 
-axiom eq_ext_sym: ∀S.∀A,B:word S →Prop. 
-  A =1 B → B =1 A.
-
-axiom union_ext_l: ∀S.∀A,B,C:word S →Prop. 
-  A =1 C  → A ∪ B =1 C ∪ B.
-  
-axiom union_ext_r: ∀S.∀A,B,C:word S →Prop. 
-  B =1 C → A ∪ B =1 A ∪ C.
-  
-axiom union_comm : ∀S.∀A,B:word S →Prop. 
-  A ∪ B =1 B ∪ A.
-
-axiom union_idemp: ∀S.∀A:word S →Prop. 
-  A ∪ A =1 A.
-
-axiom cat_ext_l: ∀S.∀A,B,C:word S →Prop. 
-  A =1 C  → A · B =1 C · B.
-  
-axiom cat_ext_r: ∀S.∀A,B,C:word S →Prop. 
-  B =1 C → A · B =1 A · C.
-  
-lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
-  (A ∪ B) · C =1  A · C ∪ B · C. 
-#S #A #B #C #w %
-  [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/] 
-qed.
-
-axiom fix_star: ∀S.∀A:word S → Prop. 
-  A^* =1 A · A^* ∪ { [ ] }.
+(* We are now ready to state the main semantic properties of ⊕, ◃ and •(-):
 
-axiom star_epsilon: ∀S:Alpha.∀A:word S → Prop.
-  A^* ∪ { [ ] } =1 A^*.
+sem_oplus:     \sem{e1 ⊕ e2} ≐ \sem{e1} ∪ \sem{e2} 
+sem_pcl:       \sem{e1 ▹ i2} ≐  \sem{e1} · \sem{|i2|} ∪ \sem{i2}
+sem_bullet     \sem{•i} ≐ \sem{i} ∪ \sem{|i|}
 
-lemma sem_eclose_star: ∀S:Alpha.∀i:pitem S.
-  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ { [ ] }.
-/2/ qed.
+The proof of sem_oplus is straightforward. *)
 
-(*
-lemma sem_eclose_star: ∀S:Alpha.∀i:pitem S.
-  \sem{〈i^*,true〉} =1 \sem{〈i,true〉}·\sem{|i|}^* ∪ { [ ] }.
-/2/ qed.
-
-#S #i #b cases b 
-  [>sem_pre_true >sem_star
-  |/2/
-  ] *)
-
-(* this kind of results are pretty bad for automation;
-   better not index them *)
-lemma epsilon_cat_r: ∀S.∀A:word S →Prop.
-   A · { [ ] } =1  A. 
-#S #A #w %
-  [* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
-  |#inA @(ex_intro … w) @(ex_intro … [ ]) /3/
-  ]
-qed-.
-
-lemma epsilon_cat_l: ∀S.∀A:word S →Prop.
-   { [ ] } · A =1  A. 
-#S #A #w %
-  [* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
-  |#inA @(ex_intro … [ ]) @(ex_intro … w) /3/
+lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
+  \sem{e1 ⊕ e2} ≐ \sem{e1} ∪ \sem{e2}. 
+#S * #i1 #b1 * #i2 #b2 #w %
+  [cases b1 cases b2 normalize /2/ * /3/ * /3/
+  |cases b1 cases b2 normalize /2/ * /3/ * /3/
   ]
-qed-.
-
-
-lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop.
-  (A ∪ { [ ] }) · C =1  A · C ∪ C. 
-#S #A #C @ext_eq_trans [|@distr_cat_r |@union_ext_r @epsilon_cat_l]
 qed.
 
-(* axiom eplison_cut_l: ∀S.∀A:word S →Prop. 
-   { [ ] } · A =1  A.
-   
- axiom eplison_cut_r: ∀S.∀A:word S →Prop.
-   A · { [ ] } =1  A. *)
+(* For the others, we proceed as follow: we first prove the following 
+auxiliary lemma, that assumes sem_bullet:
 
-(*
-lemma eta_lp : ∀S.∀p:pre S.𝐋\p p = 𝐋\p 〈\fst p, \snd p〉.
-#S p; ncases p; //; nqed.
+sem_pcl_aux: 
+   \sem{•i2} ≐  \sem{i2} ∪ \sem{|i2|} →
+   \sem{e1 ▹ i2} ≐  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
 
-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.*)
+Then, using the previous result, we prove sem_bullet by induction 
+on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
+
+lemma LcatE : ∀S.∀e1,e2:pitem S.
+  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
+// qed.
 
-(* theorem 16: 1 → 3 *)
 lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
-   \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
-   \sem{e1 ▸ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+   \sem{•i2} ≐  \sem{i2} ∪ \sem{|i2|} →
+   \sem{e1 ▹ i2} ≐  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
 #S * #i1 #b1 #i2 cases b1
   [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
-  |#H >odot_true >sem_pre_true @(ext_eq_trans … (sem_pre_concat_r …))
-   >erase_bull @ext_eq_trans [|@(union_ext_r … H)]
-    @ext_eq_trans [|@union_ext_r [|@union_comm ]]
-    @ext_eq_trans [|@eq_ext_sym @union_assoc ] /3/ 
+  |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
+   >erase_bull @eqP_trans [|@(eqP_union_l … H)]
+    @eqP_trans [|@eqP_union_l[|@union_comm ]]
+    @eqP_trans [|@eqP_sym @union_assoc ] /3/ 
   ]
 qed.
-
-axiom star_fix : 
-  ∀S.∀X:word S → Prop.(X - {[ ]}) · X^* ∪ {[ ]} =1 X^*.
   
-axiom sem_fst: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
-
-axiom sem_fst_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
- \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
+lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
+ \sem{e} ≐ \sem{i} ∪ A → \sem{\fst e} ≐ \sem{i} ∪ (A - {[ ]}).
+#S #e #i #A #seme
+@eqP_trans [|@minus_eps_pre]
+@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
+@eqP_trans [||@distribute_substract] 
+@eqP_substract_r //
+qed.
 
-(* theorem 16: 1 *)
-theorem sem_bull: ∀S:Alpha. ∀e:pitem S.  \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S.  \sem{•i} ≐ \sem{i} ∪ \sem{|i|}.
 #S #e elim e 
   [#w normalize % [/2/ | * //]
   |/2/ 
   |#x normalize #w % [ /2/ | * [@False_ind | //]]
   |#x normalize #w % [ /2/ | * // ] 
-  |#i1 #i2 #IH1 #IH2 >eclose_dot
-   @ext_eq_trans [|@odot_dot_aux //] >sem_cat 
-   @ext_eq_trans
-     [|@union_ext_l 
-       [|@ext_eq_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
-   @ext_eq_trans [|@union_assoc]
-   @ext_eq_trans [||@eq_ext_sym @union_assoc]
-   @union_ext_r //
+  |#i1 #i2 #IH1 #IH2 
+   (* lhs = \sem{•(i1 ·i2)} *)
+   >eclose_dot
+   (* lhs =\sem{•(i1) ▹ i2)} *) 
+   @eqP_trans [|@odot_dot_aux //] 
+   (* lhs = \sem{•(i1)·\sem{|i2|}∪\sem{i2} *)
+   @eqP_trans
+     [|@eqP_union_r
+       [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
+   (* lhs = \sem{i1}·\sem{|i2|}∪\sem{|i1|}·\sem{|i2|}∪\sem{i2} *) 
+   @eqP_trans [|@union_assoc]
+   (* lhs = \sem{i1}·\sem{|i2|}∪(\sem{|i1|}·\sem{|i2|}∪\sem{i2}) *) 
+   (* Now we work on the rhs that is 
+      rhs = \sem{i1·i2} ∪ \sem{|i1·i2|} *)
+   >sem_cat 
+   (* rhs = \sem{i1}·\sem{|i2|} ∪ \sem{i2} ∪ \sem{|i1·i2|} *)
+   @eqP_trans [||@eqP_sym @union_assoc]
+   (* rhs = \sem{i1}·\sem{|i2|}∪ (\sem{i2} ∪ \sem{|i1·i2|}) *)
+   @eqP_union_l @union_comm 
   |#i1 #i2 #IH1 #IH2 >eclose_plus
-   @ext_eq_trans [|@sem_oplus] >sem_plus >erase_plus 
-   @ext_eq_trans [|@(union_ext_r … IH2)]
-   @ext_eq_trans [|@eq_ext_sym @union_assoc]
-   @ext_eq_trans [||@union_assoc] @union_ext_l
-   @ext_eq_trans [||@eq_ext_sym @union_assoc]
-   @ext_eq_trans [||@union_ext_r [|@union_comm]]
-   @ext_eq_trans [||@union_assoc] /3/
+   @eqP_trans [|@sem_oplus] >sem_plus >erase_plus 
+   @eqP_trans [|@(eqP_union_l … IH2)]
+   @eqP_trans [|@eqP_sym @union_assoc]
+   @eqP_trans [||@union_assoc] @eqP_union_r
+   @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_trans [||@eqP_union_l [|@union_comm]]
+   @eqP_trans [||@union_assoc] /2/
   |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
-   @ext_eq_trans [|@union_ext_l [|@cat_ext_l [|@sem_fst_aux //]]]
-   @ext_eq_trans [|@union_ext_l [|@distr_cat_r]]
-   @ext_eq_trans [|@union_assoc] @union_ext_r >erase_star @star_fix 
+   @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
+   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+   @eqP_trans [|@union_assoc] @eqP_union_l >erase_star 
+   @eqP_sym @star_fix_eps 
   ]
 qed.
 
-definition lifted_cat ≝ λS:Alpha.λe:pre S. 
-  lift S (lc S eclose e).
+(*
+Blank item
+ 
+As a corollary of theorem sem_bullet, given a regular expression e, we can easily 
+find an item with the same semantics of $e$: it is enough to get an item (blank e) 
+having e as carrier and no point, and then broadcast a point in it. The semantics of
+(blank e) is obviously the empty language: from the point of view of the automaton,
+it corresponds with the pit state. *)
+
+let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
+ match i with
+  [ z ⇒ `∅
+  | e ⇒ ϵ
+  | s y ⇒ `y
+  | o e1 e2 ⇒ (blank S e1) + (blank S e2) 
+  | c e1 e2 ⇒ (blank S e1) · (blank S e2)
+  | k e ⇒ (blank S e)^* ].
+  
+lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
+#S #e elim e normalize //
+qed.
+
+lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} ≐ ∅.
+#S #e elim e 
+  [1,2:@eq_to_ex_eq // 
+  |#s @eq_to_ex_eq //
+  |#e1 #e2 #Hind1 #Hind2 >sem_cat 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
+  |#e1 #e2 #Hind1 #Hind2 >sem_plus 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
+  |#e #Hind >sem_star
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
+  ]
+qed.
+   
+theorem re_embedding: ∀S.∀e:re S. 
+  \sem{•(blank S e)} ≐ \sem{e}.
+#S #e @eqP_trans [|@sem_bull] >forget_blank 
+@eqP_trans [|@eqP_union_r [|@sem_blank]]
+@eqP_trans [|@union_comm] @union_empty_r.
+qed.
+
+(*
+Lifted Operators
+ 
+Plus and bullet have been already lifted from items to pres. We can now 
+do a similar job for concatenation ⊙ and Kleene's star ⊛. *)
+
+definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
+  lift S (pre_concat_l S eclose e).
 
 notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
 
 interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
 
-lemma sem_odot_true: ∀S:Alpha.∀e1:pre S.∀i. 
-  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▸ i} ∪ { [ ] }.
+lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b. 
+  〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
+#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) // 
+qed.
+
+lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
+  〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
+// 
+qed.
+  
+lemma erase_odot:∀S.∀e1,e2:pre S.
+  |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
+#S * #i1 * * #i2 #b2 // >odot_true_b // 
+qed.
+
+(* Let us come to the star operation: *)
+
+definition lk ≝ λS:DeqSet.λe:pre S.
+  match e with 
+  [ mk_Prod i1 b1 ⇒
+    match b1 with 
+    [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
+    |false ⇒ 〈i1^*,false〉
+    ]
+  ]. 
+
+(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
+interpretation "lk" 'lk a = (lk ? a).
+notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
+
+
+lemma ostar_true: ∀S.∀i:pitem S.
+  〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
+// qed.
+
+lemma ostar_false: ∀S.∀i:pitem S.
+  〈i,false〉^⊛ = 〈i^*, false〉.
+// qed.
+  
+lemma erase_ostar: ∀S.∀e:pre S.
+  |\fst (e^⊛)| = |\fst e|^*.
+#S * #i * // qed.
+
+lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
+  \sem{e1 ⊙ 〈i,true〉} ≐ \sem{e1 ▹ i} ∪ { [ ] }.
 #S #e1 #i 
-cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) || true〉) [//]
-#H >H cases (e1 ▸ i) #i1 #b1 cases b1 
-  [>sem_pre_true @ext_eq_trans [||@eq_ext_sym @union_assoc]
-   @union_ext_r /2/ 
+cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
+#H >H cases (e1 ▹ i) #i1 #b1 cases b1 
+  [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_union_l /2/ 
   |/2/
   ]
 qed.
 
-lemma eq_odot_false: ∀S:Alpha.∀e1:pre S.∀i. 
-  e1 ⊙ 〈i,false〉 = e1 ▸ i.
+lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
+  e1 ⊙ 〈i,false〉 = e1 ▹ i.
 #S #e1 #i  
-cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) || false〉) [//]
-cases (e1 ▸ i) #i1 #b1 cases b1 #H @H
+cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
+cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
 qed.
 
+(* We conclude this section with the proof of the main semantic properties
+of ⊙ and ⊛. *)
+
 lemma sem_odot: 
-  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
+  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} ≐ \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
 #S #e1 * #i2 * 
   [>sem_pre_true 
-   @ext_eq_trans [|@sem_odot_true]
-   @ext_eq_trans [||@union_assoc] @union_ext_l @odot_dot_aux //
+   @eqP_trans [|@sem_odot_true]
+   @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
   |>sem_pre_false >eq_odot_false @odot_dot_aux //
   ]
 qed.
-
-(*
-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.
-
-nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
-  ¬ (A b) → A ∪ { (b) } = C → A = C - { (b) }.
-#S A C b nbA defC; nrewrite < defC; napply extP; #w; @;
-##[ #Aw; /3/| *; *; //; #H nH; ncases nH; #abs; nlapply (abs H); *]
-nqed.
-*)
-
-(* theorem 16: 4 *)      
+      
 theorem sem_ostar: ∀S.∀e:pre S. 
-  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
+  \sem{e^⊛} ≐  \sem{e} · \sem{|\fst e|}^*.
 #S * #i #b cases b
-  [>sem_pre_true >sem_pre_true >sem_star >erase_bull
-   @ext_eq_trans [|@union_ext_l [|@cat_ext_l [|@sem_fst_aux //]]]
-   @ext_eq_trans [|@union_ext_l [|@distr_cat_r]]
-   @ext_eq_trans [||@eq_ext_sym @distr_cat_r]
-   @ext_eq_trans [|@union_assoc] @union_ext_r 
-   @ext_eq_trans [||@eq_ext_sym @epsilon_cat_l] @star_fix 
+  [(* lhs = \sem{〈i,true〉^⊛} *)
+   >sem_pre_true (* >sem_pre_true *) 
+   (* lhs = \sem{(\fst (•i))^*}∪{ϵ} *)
+   >sem_star >erase_bull
+   (* lhs = \sem{\fst (•i)}·(\sem{|i|)^*∪{ϵ} *)
+   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
+   (* lhs = (\sem{i}∪(\sem{|i|}-{ϵ})·(\sem{|i|)^*∪{ϵ} *)
+   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+   (* lhs = (\sem{i}·(\sem{|i|)^*∪(\sem{|i|}-{ϵ})·(\sem{|i|)^*∪{ϵ} *)
+   @eqP_trans [|@union_assoc]
+   (* lhs = (\sem{i}·(\sem{|i|)^*∪((\sem{|i|}-{ϵ})·(\sem{|i|)^*∪{ϵ}) *)
+   @eqP_trans [|@eqP_union_l[|@eqP_sym @star_fix_eps]]
+   (* lhs = (\sem{i}·(\sem{|i|)^*∪(\sem{|i|)^* *)
+   (* now we work on the right hand side, that is
+      rhs = \sem{〈i,true〉}·(\sem{|i|}^* *)
+   @eqP_trans [||@eqP_sym @distr_cat_r]
+   (* rhs = (\sem{i}·(\sem{|i|)^*∪{ϵ}·(\sem{|i|)^* *)
+   @eqP_union_l @eqP_sym @epsilon_cat_l
   |>sem_pre_false >sem_pre_false >sem_star /2/
   ]
 qed.
-  
-(*
-nlet rec pre_of_re (S : Alpha) (e : re S) on e : pitem S ≝ 
-  match e with 
-  [ z ⇒ pz ?
-  | e ⇒ pe ?
-  | s x ⇒ ps ? x
-  | c e1 e2 ⇒ pc ? (pre_of_re ? e1) (pre_of_re ? e2)
-  | o e1 e2 ⇒ po ? (pre_of_re ? e1) (pre_of_re ? e2)
-  | k e1 ⇒ pk ? (pre_of_re ? e1)].
-
-nlemma notFalse : ¬False. @; //; nqed.
-
-nlemma dot0 : ∀S.∀A:word S → Prop. {} · A = {}.
-#S A; nnormalize; napply extP; #w; @; ##[##2: *]
-*; #w1; *; #w2; *; *; //; nqed.
-
-nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
-#S e; nelim e; ##[##1,2,3: //]
-##[ #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1 e2))) with (?∪?);
-    nrewrite > H1; nrewrite > H2; nrewrite > (dot0…); nrewrite > (cupID…);//
-##| #e1 e2 H1 H2; nchange in match (𝐋\p (pre_of_re S (e1+e2))) with (?∪?);
-    nrewrite > H1; nrewrite > H2; nrewrite > (cupID…); //
-##| #e1 H1; nchange in match (𝐋\p (pre_of_re S (e1^* ))) with (𝐋\p (pre_of_re ??) · ?);
-    nrewrite > H1; napply dot0; ##]
-nqed.
-
-nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
-#S A; nelim A; //; 
-##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
-    nrewrite < H1; nrewrite < H2; //
-##| #e1 e2 H1 H2; nchange in match (𝐋 (e1 + e2)) with (𝐋 e1 ∪ ?);
-    nrewrite < H1; nrewrite < H2; //
-##| #e1 H1; nchange in match (𝐋  (e1^* )) with ((𝐋 e1)^* );
-    nrewrite < H1; //]
-nqed.     
-
-(* corollary 17 *)
-nlemma L_Lp_bull : ∀S.∀e:re S.𝐋 e = 𝐋\p (•pre_of_re ? e).
-#S e; nrewrite > (bull_cup…); nrewrite > (Lp_pre_of_re…);
-nrewrite > (cupC…); nrewrite > (cup0…); nrewrite > (erase_pre_of_reK…); //;
-nqed.
-
-nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
-#S f g H; nrewrite > H; //; nqed.
- 
-(* corollary 18 *)
-ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
-#S e; @;
-##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
-    nrewrite > defsnde; #H; 
-    nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
-
-*)
-