X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter7.ma;h=e37ec25e6d3689c0be83211ace41b47a3bb320e8;hb=4112b9f87a555bfc4c3cd06bae652cd2382cad8b;hp=208db9e3fb27dd4d09c1030d62ec3cae5e3a5c1d;hpb=256ee9a806709bc4bede01cd80a62bbbfde908f3;p=helm.git

diff --git a/weblib/tutorial/chapter7.ma b/weblib/tutorial/chapter7.ma
index 208db9e3f..e37ec25e6 100644
--- a/weblib/tutorial/chapter7.ma
+++ b/weblib/tutorial/chapter7.ma
@@ -1,4 +1,6 @@
-(* We shall apply all the previous machinery to the study of regular languages 
+(* 
+h1Regular Expressions /h1
+We shall apply all the previous machinery to the study of regular languages 
 and the constructions of the associated finite automata. *)
 
 include "tutorial/chapter6.ma".
@@ -6,7 +8,7 @@ include "tutorial/chapter6.ma".
 (* The type re of regular expressions over an alphabet $S$ is the smallest 
 collection of objects generated by the following constructors: *)
 
-inductive re (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) : Type[0] ≝
+img class="anchor" src="icons/tick.png" id="re"inductive re (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) : Type[0] ≝
    z: re S                (* empty: ∅ *)
  | e: re S                (* epsilon: ϵ *)
  | s: S → re S            (* symbol: a *)
@@ -26,27 +28,29 @@ interpretation "atom" 'ps a = (s ? a).
 notation "`∅" non associative with precedence 90 for @{ 'empty }.
 interpretation "empty" 'empty = (z ?).
 
-(* The language $sem{e}$ associated with the regular expression e is inductively 
+(* The language sem{e} associated with the regular expression e is inductively 
 defined by the following function: *)
 
-let rec in_l (S : a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (r : a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S) on r : a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S → Prop ≝ 
+img class="anchor" src="icons/tick.png" id="in_l"let rec in_l (S : a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/aspan class="error" title="Parse error: RPAREN expected after [term] (in [arg])"/span) (r : a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S) on r : a href="cic:/matita/tutorial/chapter6/word.def(3)"word/aspan class="error" title="Parse error: SYMBOL '≝' expected (in [let_defs])"/span S → Prop ≝ 
 match r with
 [ z ⇒ a title="empty set" href="cic:/fakeuri.def(1)"∅/a
-| e ⇒ a title="singleton" href="cic:/fakeuri.def(1)"{/aa title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a}
-| s x ⇒ a title="singleton" href="cic:/fakeuri.def(1)"{/aa title="cons" href="cic:/fakeuri.def(1)"[/ax]}
+| e ⇒ a title="singleton" href="cic:/fakeuri.def(1)"{/aa title="epsilon" href="cic:/fakeuri.def(1)"ϵ/aa title="singleton" href="cic:/fakeuri.def(1)"}/a
+| s x ⇒ a title="singleton" href="cic:/fakeuri.def(1)"{/aspan class="error" title="Parse error: [ident] or [term level 19] expected after [sym{] (in [term])"/span (xa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a) a title="singleton" href="cic:/fakeuri.def(1)"}/a
 | c r1 r2 ⇒ (in_l ? r1) a title="cat lang" href="cic:/fakeuri.def(1)"·/a (in_l ? r2)
 | o r1 r2 ⇒ (in_l ? r1) a title="union" href="cic:/fakeuri.def(1)"∪/a (in_l ? r2)
-| k r1 ⇒ (in_l ? r1) a title="star lang" href="cic:/fakeuri.def(1)"^/a*].
+| k r1 ⇒ (in_l ? r1) a title="star lang" href="cic:/fakeuri.def(1)"^/aa title="star lang" href="cic:/fakeuri.def(1)"*/a].
 
 notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
 interpretation "in_l" 'in_l E = (in_l ? E).
 interpretation "in_l mem" 'mem w l = (in_l ? l w).
 
-lemma rsem_star : ∀S.∀r: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{ra title="re star" href="cic:/fakeuri.def(1)"^/a*} a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{r}a title="star lang" href="cic:/fakeuri.def(1)"^/a*.
+img class="anchor" src="icons/tick.png" id="rsem_star"lemma rsem_star : ∀S.∀r: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{ra title="re star" href="cic:/fakeuri.def(1)"^/aa title="re star" href="cic:/fakeuri.def(1)"*/aa title="in_l" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{ra title="in_l" href="cic:/fakeuri.def(1)"}/aa title="star lang" href="cic:/fakeuri.def(1)"^/aa title="star lang" href="cic:/fakeuri.def(1)"*/a.
 // qed.
 
 
-(* We now introduce pointed regular expressions, that are the main tool we shall 
+(* 
+h2Pointed Regular expressions /h2
+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 
@@ -74,7 +78,7 @@ formalization of the metatheory of programming languages. For our purposes, it i
 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: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) : Type[0] ≝
+img class="anchor" src="icons/tick.png" id="pitem"inductive pitem (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) : Type[0] ≝
    pz: pitem S                       (* empty *)
  | pe: pitem S                       (* epsilon *)
  | ps: S → pitem S                   (* symbol *)
@@ -89,7 +93,7 @@ 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.a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S a title="Product" href="cic:/fakeuri.def(1)"×/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a.
+img class="anchor" src="icons/tick.png" id="pre"definition pre ≝ λS.a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S a title="Product" href="cic:/fakeuri.def(1)"×/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a.
 
 interpretation "pitem star" 'star a = (pk ? a).
 interpretation "pitem or" 'plus a b = (po ? a b).
@@ -105,469 +109,196 @@ interpretation "pitem empty" 'empty = (pz ?).
 removing all the points. Similarly, the carrier of a pointed regular expression 
 is the carrier of its item. *)
 
-let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
+img class="anchor" src="icons/tick.png" id="forget"let rec forget (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (l : a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) on l: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S ≝
  match l with
-  [ 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)^* ].
+  [ pz ⇒ a href="cic:/matita/tutorial/chapter7/re.con(0,1,1)"z/a ? (* `∅ *)
+  | pe ⇒ a title="re epsilon" href="cic:/fakeuri.def(1)"ϵ/a
+  | ps x ⇒ a title="atom" href="cic:/fakeuri.def(1)"`/ax
+  | pp x ⇒ a title="atom" href="cic:/fakeuri.def(1)"`/ax
+  | pc E1 E2 ⇒ (forget ? E1) a title="re cat" href="cic:/fakeuri.def(1)"·/a (forget ? E2)
+  | po E1 E2 ⇒ (forget ? E1) a title="re or" href="cic:/fakeuri.def(1)"+/aspan class="error" title="Parse error: [term] expected after [sym+] (in [term])"/span (forget ? E2)
+  | pk E ⇒ (forget ? E)a title="re star" href="cic:/fakeuri.def(1)"^/aa title="re star" href="cic:/fakeuri.def(1)"*/a ].
  
 (* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
 interpretation "forget" 'norm a = (forget ? a).
 
-lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
+img class="anchor" src="icons/tick.png" id="erase_dot"lemma erase_dot : ∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. a title="forget" href="cic:/fakeuri.def(1)"|/ae1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/tutorial/chapter7/re.con(0,4,1)"c/a ? (a title="forget" href="cic:/fakeuri.def(1)"|/ae1a title="forget" href="cic:/fakeuri.def(1)"|/a) (a title="forget" href="cic:/fakeuri.def(1)"|/ae2a title="forget" href="cic:/fakeuri.def(1)"|/a).
 // qed.
 
-lemma erase_plus : ∀S.∀i1,i2:pitem S.
-  |i1 + i2| = |i1| + |i2|.
+img class="anchor" src="icons/tick.png" id="erase_plus"lemma erase_plus : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="forget" href="cic:/fakeuri.def(1)"|/ai1 a title="pitem or" href="cic:/fakeuri.def(1)"+/a i2a title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/ai1a title="forget" href="cic:/fakeuri.def(1)"|/a a title="re or" href="cic:/fakeuri.def(1)"+/a a title="forget" href="cic:/fakeuri.def(1)"|/ai2a title="forget" href="cic:/fakeuri.def(1)"|/a.
 // qed.
 
-lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
+img class="anchor" src="icons/tick.png" id="erase_star"lemma erase_star : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.a title="forget" href="cic:/fakeuri.def(1)"|/aia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/aa title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/aa title="re star" href="cic:/fakeuri.def(1)"^/aa title="re star" href="cic:/fakeuri.def(1)"*/a. 
 // qed.
 
-(* Items and pres are a very concrete datatype: they can be effectively compared, 
-and enumerated. In particular, we can define a boolean equality beqitem and 
-\vebeqitem_true+ enriching the set \verb+(pitem S)+
-to a \verb+DeqSet+. boolean equality *)
-let rec beqitem S (i1,i2: pitem S) on i1 ≝ 
+(* 
+h2Comparing items and pres/h2
+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. *)
+
+img class="anchor" src="icons/tick.png" id="beqitem"let rec beqitem S (i1,i2: a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a 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]
+  [ pz ⇒ match i2 with [ pz ⇒ a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]
+  | pe ⇒ match i2 with [ pe ⇒ a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]
+  | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ay2 | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]
+  | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/ay2 | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]
   | po i11 i12 ⇒ match i2 with 
-    [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
-    | _ ⇒ false]
+    [ po i21 i22 ⇒ beqitem S i11 i21 a title="boolean and" href="cic:/fakeuri.def(1)"∧/a beqitem S i12 i22
+    | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aspan class="error" title="Parse error: SYMBOL '|' or SYMBOL ']' expected (in [term])"/span]
   | 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]
+    [ pc i21 i22 ⇒ beqitem S i11 i21 a title="boolean and" href="cic:/fakeuri.def(1)"∧/a beqitem S i12 i22
+    | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]
+  | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a]
   ].
 
-lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2). 
+img class="anchor" src="icons/tick.png" id="beqitem_true"lemma beqitem_true: ∀S,i1,i2. a href="cic:/matita/basics/logic/iff.def(1)"iff/a (a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S i1 i2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) (i1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a 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 …))]
+    [>(\P H) // | @(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …))]
   |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
-    [>(\P H) // | @(\b (refl …))]
+    [>(\P H) // | @(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …))]
   |#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/
+    [cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S i11 i21)) #H1
+      [>(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hind1 i21) H1) >(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
       ]
-    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    |>(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (Hind1 i21) (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …)) >(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (Hind2 i22) (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …)) //
     ]
   |#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/
+    [cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S i11 i21)) #H1
+      [>(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hind1 i21) H1) >(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
       ]
-    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    |>(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (Hind1 i21) (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …)) >(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (Hind2 i22) (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …)) //
     ]
   |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
    normalize #H destruct 
-    [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
+    [>(a href="cic:/matita/basics/logic/proj1.def(2)"proj1/a … (Hind i4) H) // |>(a href="cic:/matita/basics/logic/proj2.def(2)"proj2/a … (Hind i4) (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a …)) //]
   ]
 qed. 
 
-definition DeqItem ≝ λS.
-  mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
-  
-unification hint  0 ≔ S; 
-    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+img class="anchor" src="icons/tick.png" id="DeqItem"definition DeqItem ≝ λS.
+  a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a (a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) (a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S) (a href="cic:/matita/tutorial/chapter7/beqitem_true.def(5)"beqitem_true/a S).
+
+(* 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 a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"≔/a S; 
+    X ≟ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a (a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) (a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S) (a href="cic:/matita/tutorial/chapter7/beqitem_true.def(5)"beqitem_true/a S)
 (* ---------------------------------------- *) ⊢ 
-    pitem S ≡ carr X.
+    a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S ≡ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a X.
     
-unification hint  0 ≔ S,i1,i2; 
-    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+unification hint  0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"≔/a S,i1,i2; 
+    X ≟ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a (a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) (a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S) (a href="cic:/matita/tutorial/chapter7/beqitem_true.def(5)"beqitem_true/a S)
 (* ---------------------------------------- *) ⊢ 
-    beqitem S i1 i2 ≡ eqb X i1 i2.
+    a href="cic:/matita/tutorial/chapter7/beqitem.fix(0,1,4)"beqitem/a S i1 i2 ≡ a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a X i1 i2.
 
-(* semantics *)
+(* 
+h2Semantics of pointed regular expressions/h2
+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 ≝ 
+img class="anchor" src="icons/tick.png" id="in_pl"let rec in_pl (S : a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (r : a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) on r : a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S → Prop ≝ 
 match r with
-[ 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)
-| pk r1 ⇒ (in_pl ? r1) · \sem{forget ? r1}^*  ].
+[ pz ⇒ a title="empty set" href="cic:/fakeuri.def(1)"∅/a
+| pe ⇒ a title="empty set" href="cic:/fakeuri.def(1)"∅/a
+| ps _ ⇒ a title="empty set" href="cic:/fakeuri.def(1)"∅/a
+| pp x ⇒ a title="singleton" href="cic:/fakeuri.def(1)"{/a (xa title="cons" href="cic:/fakeuri.def(1)":/aa title="cons" href="cic:/fakeuri.def(1)":/aa title="nil" href="cic:/fakeuri.def(1)"[/aa title="nil" href="cic:/fakeuri.def(1)"]/a) a title="singleton" href="cic:/fakeuri.def(1)"}/a
+| pc r1 r2 ⇒ (in_pl ? r1) a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a href="cic:/matita/tutorial/chapter7/forget.fix(0,1,3)"forget/a ? r2a title="in_l" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/aspan class="error" title="Parse error: [term] expected after [sym∪] (in [term])"/span (in_pl ? r2)
+| po r1 r2 ⇒ (in_pl ? r1) a title="union" href="cic:/fakeuri.def(1)"∪/a (in_pl ? r2)
+| pk r1 ⇒ (in_pl ? r1) a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a href="cic:/matita/tutorial/chapter7/forget.fix(0,1,3)"forget/a ? r1a title="in_l" href="cic:/fakeuri.def(1)"}/aa title="star lang" href="cic:/fakeuri.def(1)"^/aa title="star lang" href="cic:/fakeuri.def(1)"*/a  ].
 
 interpretation "in_pl" 'in_l E = (in_pl ? E).
 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
 
-definition in_prl ≝ λS : DeqSet.λp:pre S. 
-  if (\snd p) then \sem{\fst p} ∪ {ϵ} else \sem{\fst p}.
+img class="anchor" src="icons/tick.png" id="in_prl"definition in_prl ≝ λS : a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λp:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+  if (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p) then a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a pa title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="singleton" href="cic:/fakeuri.def(1)"{/aa title="epsilon" href="cic:/fakeuri.def(1)"ϵ/aa title="singleton" href="cic:/fakeuri.def(1)"}/a else a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a pa title="in_pl" href="cic:/fakeuri.def(1)"}/a.
   
 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
 interpretation "in_prl" 'in_l E = (in_prl ? E).
 
-lemma sem_pre_true : ∀S.∀i:pitem S. 
-  \sem{〈i,true〉} = \sem{i} ∪ {ϵ}. 
+(* The following, trivial lemmas are only meant for rewriting purposes. *)
+
+img class="anchor" src="icons/tick.png" id="sem_pre_true"lemma sem_pre_true : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. 
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="singleton" href="cic:/fakeuri.def(1)"{/aa title="epsilon" href="cic:/fakeuri.def(1)"ϵ/aa title="singleton" href="cic:/fakeuri.def(1)"}/a. 
 // qed.
 
-lemma sem_pre_false : ∀S.∀i:pitem S. 
-  \sem{〈i,false〉} = \sem{i}. 
+img class="anchor" src="icons/tick.png" id="sem_pre_false"lemma sem_pre_false : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. 
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a. 
 // qed.
 
-lemma sem_cat: ∀S.∀i1,i2:pitem S. 
-  \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
+img class="anchor" src="icons/tick.png" id="sem_cat"lemma sem_cat: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. 
+  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/ai2a title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a.
 // qed.
 
-lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
-  \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
+img class="anchor" src="icons/tick.png" id="sem_cat_w"lemma sem_cat_w: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀w.
+  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a w a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ((a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/ai2a title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/a) w a title="logical or" href="cic:/fakeuri.def(1)"∨/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a w).
 // qed.
 
-lemma sem_plus: ∀S.∀i1,i2:pitem S. 
-  \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
+img class="anchor" src="icons/tick.png" id="sem_plus"lemma sem_plus: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. 
+  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1 a title="pitem or" href="cic:/fakeuri.def(1)"+/a i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a.
 // qed.
 
-lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w. 
-  \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
+img class="anchor" src="icons/tick.png" id="sem_plus_w"lemma sem_plus_w: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀w. 
+  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1 a title="pitem or" href="cic:/fakeuri.def(1)"+/a i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a w a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1a title="in_pl" href="cic:/fakeuri.def(1)"}/a w a title="logical or" href="cic:/fakeuri.def(1)"∨/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a w).
 // qed.
 
-lemma sem_star : ∀S.∀i:pitem S.
-  \sem{i^*} = \sem{i} · \sem{|i|}^*.
+img class="anchor" src="icons/tick.png" id="sem_star"lemma sem_star : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/aa title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="cat lang" href="cic:/fakeuri.def(1)"·/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/aa title="star lang" href="cic:/fakeuri.def(1)"^/aa title="star lang" href="cic:/fakeuri.def(1)"*/a.
 // qed.
 
-lemma sem_star_w : ∀S.∀i:pitem S.∀w.
-  \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
+img class="anchor" src="icons/tick.png" id="sem_star_w"lemma sem_star_w : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀w.
+  a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/aa title="in_pl" href="cic:/fakeuri.def(1)"}/a w a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a title="exists" href="cic:/fakeuri.def(1)"∃/aw1,w2a title="exists" href="cic:/fakeuri.def(1)"./aw1 a title="append" href="cic:/fakeuri.def(1)"@/a w2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a w a title="logical and" href="cic:/fakeuri.def(1)"∧/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a w1 a title="logical and" href="cic:/fakeuri.def(1)"∧/a a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{a title="forget" href="cic:/fakeuri.def(1)"|/aia title="forget" href="cic:/fakeuri.def(1)"|/aa title="in_l" href="cic:/fakeuri.def(1)"}/aa title="star lang" href="cic:/fakeuri.def(1)"^/aa title="star lang" href="cic:/fakeuri.def(1)"*/a w2).
 // qed.
 
-lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
+(* 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} =1 \sem{i}-{[ ]}.
+- minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \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. *)
+
+img class="anchor" src="icons/tick.png" id="append_eq_nil"lemma append_eq_nil : ∀S.∀w1,w2:a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S. w1 a title="append" href="cic:/fakeuri.def(1)"@/a w2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a → w1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a.
 #S #w1 #w2 cases w1 // #a #tl normalize #H destruct qed.
 
-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/
-  |#r1 #r2 #n1 #n2 % * /2/
-  |#r #n % * #w1 * #w2 * * #H >(append_eq_nil …H…) /2/
+img class="anchor" src="icons/tick.png" id="not_epsilon_lp"lemma not_epsilon_lp : ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. a title="logical not" href="cic:/fakeuri.def(1)"¬/a (a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a a title="in_pl mem" href="cic:/fakeuri.def(1)"∈/a e).
+#S #e elim e normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Not.con(0,1,1)"nmk/a/span/span/  
+  [#r1 #r2 * #n1 #n2 % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ * #w1 * #w2 * * #H 
+   >(a href="cic:/matita/tutorial/chapter7/append_eq_nil.def(4)"append_eq_nil/a …H…) /span class="autotactic"2span class="autotrace" trace /span/span/
+  |#r1 #r2 #n1 #n2 % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
+  |#r #n % * #w1 * #w2 * * #H >(a href="cic:/matita/tutorial/chapter7/append_eq_nil.def(4)"append_eq_nil/a …H…) /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/
   ]
 qed.
 
-(* lemma 12 *)
-lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
-#S * #i #b cases b // normalize #H @False_ind /2/ 
+img class="anchor" src="icons/tick.png" id="epsilon_to_true"lemma epsilon_to_true : ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e → a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a.
+#S * #i #b cases b // normalize #H @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a/span/span/ 
 qed.
 
-lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
+img class="anchor" src="icons/tick.png" id="true_to_epsilon"lemma true_to_epsilon : ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a e a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → a title="epsilon" href="cic:/fakeuri.def(1)"ϵ/a a title="in_prl mem" href="cic:/fakeuri.def(1)"∈/a e.
 #S * #i #b #btrue normalize in btrue; >btrue %2 // 
 qed.
 
-lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
+img class="anchor" src="icons/tick.png" id="minus_eps_item"lemma minus_eps_item: ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{ia title="in_pl" href="cic:/fakeuri.def(1)"}/aa title="substraction" href="cic:/fakeuri.def(1)"-/aa title="singleton" href="cic:/fakeuri.def(1)"{/aa title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/aa title="singleton" href="cic:/fakeuri.def(1)"}/a.
 #S #i #w % 
-  [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
+  [#H whd % // normalize @(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … (a href="cic:/matita/tutorial/chapter7/not_epsilon_lp.def(8)"not_epsilon_lp/a …i)) //
   |* //
   ]
 qed.
 
-lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
+img class="anchor" src="icons/tick.png" id="minus_eps_pre"lemma minus_eps_pre: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. a title="in_pl" href="cic:/fakeuri.def(1)"\sem/aspan class="error" title="Parse error: [sym{] expected after [sym\sem ] (in [term])"/span{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_prl" href="cic:/fakeuri.def(1)"}/aa title="substraction" href="cic:/fakeuri.def(1)"-/aa title="singleton" href="cic:/fakeuri.def(1)"{/aa title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/aa title="singleton" href="cic:/fakeuri.def(1)"}/a.
 #S * #i * 
-  [>sem_pre_true normalize in ⊢ (??%?); #w % 
-    [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
-  |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
-  ]
-qed.
-
-definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
-notation "a ⊕ b" left associative with precedence 60 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〉.
-// qed.
-
-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 @{'lhd $i $e}.
-interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
-
-lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
-  A = B → A =1 B. 
-#S #A #B #H >H /2/ 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: @eq_to_ex_eq //] 
->sem_pre_true >sem_cat >sem_pre_true /2/ 
-qed.
- 
-definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
-  match e1 with 
-  [ mk_Prod i1 b1 ⇒ match b1 with 
-    [ true ⇒ (i1 ◃ (bcast ? i2)) 
-    | false ⇒ 〈i1 · i2,false〉
-    ]
-  ].
-
-notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
-interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
-
-notation "•" non associative with precedence 60 for @{eclose ?}.
-
-let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
- match i with
-  [ pz ⇒ 〈 `∅, false 〉
-  | pe ⇒ 〈 ϵ,  true 〉
-  | ps x ⇒ 〈 `.x, false〉
-  | pp x ⇒ 〈 `.x, false 〉
-  | po i1 i2 ⇒ •i1 ⊕ •i2
-  | pc i1 i2 ⇒ •i1 ▹ i2
-  | pk i ⇒ 〈(\fst (•i))^*,true〉].
-  
-notation "• x" non associative with precedence 60 for @{'eclose $x}.
-interpretation "eclose" 'eclose x = (eclose ? x).
-
-lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 + i2) = •i1 ⊕ •i2.
-// qed.
-
-lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 · i2) = •i1 ▹ i2.
-// qed.
-
-lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
-  •i^* = 〈(\fst(•i))^*,true〉.
-// qed.
-
-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).
-
-(* theorem 16: 2 *)
-lemma sem_oplus: ∀S:DeqSet.∀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).
-// qed.
-
-lemma odot_true_bis : 
-  ∀S.∀i1,i2:pitem S.
-  〈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〉.
-// qed.
-
-lemma LcatE : ∀S.∀e1,e2:pitem S.
-  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
-// qed.
-
-lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
-#S #i elim i // 
-  [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
-    cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
-  | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
-    cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //  
-  | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
-  ]
-qed.
-
-(*
-lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
-  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
-/2/ 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}.
-#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 @(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.
-  
-lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
- \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \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:DeqSet. ∀i:pitem S.  \sem{•i} =1 \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
-   @eqP_trans [|@odot_dot_aux //] >sem_cat 
-   @eqP_trans
-     [|@eqP_union_r
-       [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
-   @eqP_trans [|@union_assoc]
-   @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_union_l //
-  |#i1 #i2 #IH1 #IH2 >eclose_plus
-   @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
-   @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.
-
-(* blank item *)
-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} =1 ∅.
-#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)} =1 \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.
-
-(* lefted operations *)
-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 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 >erase_dot //  
-qed.
-
-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〉} =1 \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 @eqP_trans [||@eqP_sym @union_assoc]
-   @eqP_union_l /2/ 
-  |/2/
-  ]
-qed.
-
-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
-qed.
-
-lemma sem_odot: 
-  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
-#S #e1 * #i2 * 
-  [>sem_pre_true 
-   @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.
-
-(* theorem 16: 4 *)      
-theorem sem_ostar: ∀S.∀e:pre S. 
-  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
-#S * #i #b cases b
-  [>sem_pre_true >sem_pre_true >sem_star >erase_bull
-   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
-   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
-   @eqP_trans [||@eqP_sym @distr_cat_r]
-   @eqP_trans [|@union_assoc] @eqP_union_l
-   @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps 
-  |>sem_pre_false >sem_pre_false >sem_star /2/
+  [>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a normalize in ⊢ (??%?); #w % 
+    [/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a, a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a/span/span/ | * * // #H1 #H2 @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a @(a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a …H1 H2)]
+  |>a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a normalize in ⊢ (??%?); #w % [ /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a, a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a/span/span/ | * // ]
   ]
 qed.
\ No newline at end of file