From e93fda0eb50a6761c1f91807db400f15d04b845e Mon Sep 17 00:00:00 2001
From: matitaweb <claudio.sacerdoticoen@unibo.it>
Date: Fri, 2 Mar 2012 09:42:52 +0000
Subject: [PATCH] commit by user andrea

---
 weblib/tutorial/chapter7.ma | 187 ++++++++++++++++++------------------
 1 file changed, 95 insertions(+), 92 deletions(-)

diff --git a/weblib/tutorial/chapter7.ma b/weblib/tutorial/chapter7.ma
index b9a35a8ad..2b3df9064 100644
--- a/weblib/tutorial/chapter7.ma
+++ b/weblib/tutorial/chapter7.ma
@@ -1,4 +1,5 @@
-(* <h1>Regular Expressions</h1>
+(* 
+h1Regular Expressions /h1
 We shall apply all the previous machinery to the study of regular languages 
 and the constructions of the associated finite automata. *)
 
@@ -30,11 +31,11 @@ interpretation "empty" 'empty = (z ?).
 (* 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/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 ≝ 
+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 ≝ 
 match r with
-[ z ⇒ a title="empty set" href="cic:/fakeuri.def(1)"∅/aspan class="error" title="Parse error: SYMBOL '|' or SYMBOL ']' expected (in [term])"/span
+[ 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)"{/aspan class="error" title="Parse error: [ident] or [term level 19] expected after [sym{] (in [term])"/spana title="cons" href="cic:/fakeuri.def(1)"[/aspan class="error" title="Parse error: [term] expected after [sym[] (in [term])"/spanx]}
+| s x ⇒ a title="singleton" href="cic:/fakeuri.def(1)"{/a (xa title="cons" href="cic:/fakeuri.def(1)":/a:a title="nil" 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*].
@@ -47,7 +48,8 @@ lemma rsem_star : ∀S.∀r: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1
 // qed.
 
 
-(* <h2>Pointed Regular expressions </h2>
+(* 
+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 
@@ -107,153 +109,155 @@ 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 ≝
+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)"+/a (forget ? E2)
+  | pk E ⇒ (forget ? E)a 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|).
+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 e2| 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)"|/ae1|) (a title="forget" href="cic:/fakeuri.def(1)"|/ae2|).
 // qed.
 
-lemma erase_plus : ∀S.∀i1,i2:pitem S.
-  |i1 + i2| = |i1| + |i2|.
+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 i2| a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/ai1| a title="re or" href="cic:/fakeuri.def(1)"+/a a title="forget" href="cic:/fakeuri.def(1)"|/ai2|.
 // qed.
 
-lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
+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)"^/a*| a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="forget" href="cic:/fakeuri.def(1)"|/ai|a title="re star" href="cic:/fakeuri.def(1)"^/a*. 
 // qed.
 
-(* <h2>Comparing items and pres<h2>
+(* 
+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. *)
 
-let rec beqitem S (i1,i2: pitem S) on i1 ≝ 
+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)"=/a=y2 | _ ⇒ 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)"=/a=y2 | _ ⇒ 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/a]
   | 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). 
+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).
+  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 ≔ S; 
-    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+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.
 
-(* <h2>Semantics of pointed regular expression<h2>
+(* 
+h2Semantics of pointed regular expression/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 ≝ 
+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)":/a:a title="nil" 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 ? r2} a title="union" href="cic:/fakeuri.def(1)"∪/a (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 ? r1}a 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}.
+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 p} 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/a} else a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a 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} ∪ {ϵ}. 
+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/a〉} a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i} 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/a}. 
 // qed.
 
-lemma sem_pre_false : ∀S.∀i:pitem S. 
-  \sem{〈i,false〉} = \sem{i}. 
+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/a〉} a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i}. 
 // qed.
 
-lemma sem_cat: ∀S.∀i1,i2:pitem S. 
-  \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
+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 i2} a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1} 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)"|/ai2|} a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2}.
 // qed.
 
-lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
-  \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} 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 i2} w a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a ((a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1} 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)"|/ai2|}) w a title="logical or" href="cic:/fakeuri.def(1)"∨/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2} w).
 // qed.
 
-lemma sem_plus: ∀S.∀i1,i2:pitem S. 
-  \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
+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 i2} a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1} a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2}.
 // qed.
 
-lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w. 
-  \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} 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 i2} w a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i1} w a title="logical or" href="cic:/fakeuri.def(1)"∨/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i2} w).
 // qed.
 
-lemma sem_star : ∀S.∀i:pitem S.
-  \sem{i^*} = \sem{i} · \sem{|i|}^*.
+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)"^/a*} a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i} 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)"|/ai|}a 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).
+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)"^/a*} w a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a (a title="exists" href="cic:/fakeuri.def(1)"∃/aw1,w2.w1 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{i} 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)"|/ai|}a title="star lang" href="cic:/fakeuri.def(1)"^/a* w2).
 // qed.
 
 (* Below are a few, simple, semantic properties of items. In particular:
@@ -264,38 +268,37 @@ lemma sem_star_w : ∀S.∀i:pitem S.∀w.
 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.
+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/
+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 epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
-#S * #i #b cases b // normalize #H @False_ind /2/ 
+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.
+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}-{[ ]}.
+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{i} a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{i}a title="substraction" href="cic:/fakeuri.def(1)"-/aa title="singleton" href="cic:/fakeuri.def(1)"{/aa title="nil" 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}-{[ ]}.
+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/a{a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e} a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e}a title="substraction" href="cic:/fakeuri.def(1)"-/aa title="singleton" href="cic:/fakeuri.def(1)"{/aa title="nil" 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/ | * // ]
+  [>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.
-
+qed.
\ No newline at end of file
-- 
2.39.2