X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter8.ma;h=0e3293c1e79186894105304f7d6d3b9eefe45e25;hb=cac0166656e08399eaaf1a1e19f0ccea28c36d39;hp=c0d4d91dc25594fffa820ea7f23ad4b5155abd6f;hpb=fc43587e06d59afb5b1c65904372843d928fdfe5;p=helm.git

diff --git a/weblib/tutorial/chapter8.ma b/weblib/tutorial/chapter8.ma
index c0d4d91dc..0e3293c1e 100644
--- a/weblib/tutorial/chapter8.ma
+++ b/weblib/tutorial/chapter8.ma
@@ -13,243 +13,303 @@ reached the end of a + ϵ and we must pursue broadcasting inside (b*a + b)b. Aga
 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〉
+               •((a + ϵ)(b^*a + b)b) = 〈(•a + ϵ)((•b)^*•a + •b)b, false〉
 *)
 
-
 include "tutorial/chapter7.ma".
 
-definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
+(* 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).
+*)
+
+img class="anchor" src="icons/tick.png" id="lo"definition lo ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λa,b:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a a a title="pitem or" href="cic:/fakeuri.def(1)"+/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a b,a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a a a title="boolean or" href="cic:/fakeuri.def(1)"∨/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
 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〉.
+img class="anchor" src="icons/tick.png" id="lo_def"lemma lo_def: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b1,b2. a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,b1a title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="oplus" href="cic:/fakeuri.def(1)"⊕/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai2,b2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1a title="pitem or" href="cic:/fakeuri.def(1)"+/ai2,b1a title="boolean or" href="cic:/fakeuri.def(1)"∨/ab2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
 // qed.
 
-definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
-  match e with [ mk_Prod 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:
+*)
+
+img class="anchor" src="icons/tick.png" id="pre_concat_r"definition pre_concat_r ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λi:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
+  match e with [ mk_Prod i1 b ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i1, ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a].
  
 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.
+img class="anchor" src="icons/tick.png" id="eq_to_ex_eq"lemma eq_to_ex_eq: ∀S.∀A,B:a href="cic:/matita/tutorial/chapter6/word.def(3)"word/a S → Prop. 
+  A a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a B → A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B. 
+#S #A #B #H >H #x % // qed.
+
+(* The behaviour of ◃ is summarized by the following, easy lemma: *)
 
-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/ 
+img class="anchor" src="icons/tick.png" id="sem_pre_concat_r"lemma sem_pre_concat_r : ∀S,i.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{i a title="pre_concat_r" href="cic:/fakeuri.def(1)"◃/a ea title="in_prl" 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)"}/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)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea 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_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_prl" href="cic:/fakeuri.def(1)"}/a.
+#S #i * #i1 #b1 cases b1 [2: @a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a //] 
+>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a /span class="autotactic"2span class="autotrace" trace /span/span/ 
 qed.
  
-definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.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. *)
+
+img class="anchor" src="icons/tick.png" id="pre_concat_l"definition pre_concat_l ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λbcast:∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S → a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λe1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λi2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
   match e1 with 
   [ mk_Prod i1 b1 ⇒ match b1 with 
-    [ true ⇒ (i1 ◃ (bcast ? i2)) 
-    | false ⇒ 〈i1 · i2,false〉
+    [ true ⇒ (i1 a title="pre_concat_r" href="cic:/fakeuri.def(1)"◃/a (bcast ? i2)) 
+    | false ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
     ]
   ].
 
 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).
 
+(* We are ready to give the formal definition of the broadcasting operation. *)
+
 notation "•" non associative with precedence 60 for @{eclose ?}.
 
-let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
+img class="anchor" src="icons/tick.png" id="eclose"let rec eclose (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (i: a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S) on i : a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a 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〉].
+  [ pz ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/a ?, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | pe ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem epsilon" href="cic:/fakeuri.def(1)"ϵ/a,  a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | ps x ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem pp" href="cic:/fakeuri.def(1)"`/aa title="pitem pp" href="cic:/fakeuri.def(1)"./ax, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | pp x ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a a title="pitem pp" href="cic:/fakeuri.def(1)"`/aa title="pitem pp" href="cic:/fakeuri.def(1)"./ax, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+  | po i1 i2 ⇒ •i1 a title="oplus" href="cic:/fakeuri.def(1)"⊕/a •i2
+  | pc i1 i2 ⇒ •i1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2
+  | pk i ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (•i))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a].
   
 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.
+(* Here are a few simple properties of ▹ and •(-) *)
+
+img class="anchor" src="icons/tick.png" id="pcl_true"lemma pcl_true : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a i1 a title="pre_concat_r" href="cic:/fakeuri.def(1)"◃/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2).
 // qed.
 
-lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 · i2) = •i1 ▹ i2.
+img class="anchor" src="icons/tick.png" id="pcl_true_bis"lemma pcl_true_bis : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2)a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
+#S #i1 #i2 normalize cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) // qed.
+
+img class="anchor" src="icons/tick.png" id="pcl_false"lemma pcl_false: ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a  i2  a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
+// qed.
+
+img class="anchor" src="icons/tick.png" id="eclose_plus"lemma eclose_plus: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="eclose" href="cic:/fakeuri.def(1)"•/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="eclose" href="cic:/fakeuri.def(1)"•/ai1 a title="oplus" href="cic:/fakeuri.def(1)"⊕/a a title="eclose" href="cic:/fakeuri.def(1)"•/ai2.
+// qed.
+
+img class="anchor" src="icons/tick.png" id="eclose_dot"lemma eclose_dot: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="eclose" href="cic:/fakeuri.def(1)"•/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="eclose" href="cic:/fakeuri.def(1)"•/ai1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2.
 // qed.
 
-lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
-  •i^* = 〈(\fst(•i))^*,true〉.
+img class="anchor" src="icons/tick.png" id="eclose_star"lemma eclose_star: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  a title="eclose" href="cic:/fakeuri.def(1)"•/aia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a(a title="eclose" href="cic:/fakeuri.def(1)"•/ai))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
 // qed.
 
-definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
+(* The definition of •(-) (eclose) can then be lifted from items to pres
+in the obvious way. *)
+
+img class="anchor" src="icons/tick.png" id="lift"definition lift ≝ λS.λf:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S →a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
   match e with 
-  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
+  [ mk_Prod i b ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (f i), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (f i) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a].
   
-definition preclose ≝ λS. lift S (eclose S). 
+img class="anchor" src="icons/tick.png" id="preclose"definition preclose ≝ λS. a href="cic:/matita/tutorial/chapter8/lift.def(2)"lift/a S (a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"eclose/a 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/
+(* Obviously, broadcasting does not change the carrier of the item,
+as it is easily proved by structural induction. *)
+
+img class="anchor" src="icons/tick.png" id="erase_bull"lemma erase_bull : ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S. a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai)a 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)"|/a.
+#S #i elim i // 
+  [ #i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"erase_dot/a <IH1 >a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"eclose_dot/a
+    cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai1) #i11 #b1 cases b1 // <IH2 >a href="cic:/matita/tutorial/chapter8/pcl_true_bis.def(5)"pcl_true_bis/a //
+  | #i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter8/eclose_plus.def(5)"eclose_plus/a >(a href="cic:/matita/tutorial/chapter7/erase_plus.def(4)"erase_plus/a … i1) <IH1 <IH2
+    cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai1) #i11 #b1 cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) #i21 #b2 //  
+  | #i #IH >a href="cic:/matita/tutorial/chapter8/eclose_star.def(5)"eclose_star/a >(a href="cic:/matita/tutorial/chapter7/erase_star.def(4)"erase_star/a … i) <IH cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai) //
   ]
 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.
+(* We are now ready to state the main semantic properties of ⊕, ◃ and •(-):
 
-lemma odot_false: 
-  ∀S.∀i1,i2:pitem S.
-  〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
-// qed.
+sem_oplus:     \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2} 
+sem_pcl:       \sem{e1 ▹ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}
+sem_bullet     \sem{•i} =1 \sem{i} ∪ \sem{|i|}
 
-lemma LcatE : ∀S.∀e1,e2:pitem S.
-  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
-// qed.
+The proof of sem_oplus is straightforward. *)
 
-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) //
+img class="anchor" src="icons/tick.png" id="sem_oplus"lemma sem_oplus: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="oplus" href="cic:/fakeuri.def(1)"⊕/a e2a title="in_prl" 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{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a. 
+#S * #i1 #b1 * #i2 #b2 #w %
+  [cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace /span/span/ * /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/Or.con(0,2,2)"or_intror/a/span/span/ * /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/Or.con(0,2,2)"or_intror/a/span/span/
+  |cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace /span/span/ * /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/Or.con(0,2,2)"or_intror/a/span/span/ * /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/Or.con(0,2,2)"or_intror/a/span/span/
   ]
 qed.
 
-(*
-lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
-  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
-/2/ qed.
-*)
+(* For the others, we proceed as follow: we first prove the following 
+auxiliary lemma, that assumes sem_bullet:
 
-(* theorem 16: 1 → 3 *)
-lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
+sem_pcl_aux: 
    \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
    \sem{e1 ▹ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+
+Then, using the previous result, we prove sem_bullet by induction 
+on i. Finally, sem_pcl_aux and sem_bullet give sem_pcl. *)
+
+img class="anchor" src="icons/tick.png" id="LcatE"lemma LcatE : ∀S.∀e1,e2: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{e1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a e2a 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{e1a 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)"|/ae2a 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{e2a title="in_pl" href="cic:/fakeuri.def(1)"}/a. 
+// qed.
+
+img class="anchor" src="icons/tick.png" id="sem_pcl_aux"lemma sem_pcl_aux : ∀S.∀e1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i2: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="eclose" href="cic:/fakeuri.def(1)"•/ai2a title="in_prl" 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{i2a title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="union" 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="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i2a title="in_prl" 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{e1a title="in_prl" 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.
 #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/ 
+  [2:#th >a href="cic:/matita/tutorial/chapter8/pcl_false.def(5)"pcl_false/a >a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/
+  |#H >a href="cic:/matita/tutorial/chapter8/pcl_true.def(5)"pcl_true/a >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a @(a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a … (a href="cic:/matita/tutorial/chapter8/sem_pre_concat_r.def(10)"sem_pre_concat_r/a …))
+   >a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"erase_bull/a @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@(a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a … H)]
+    @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a[|@a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"union_comm/a ]]
+    @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a ] /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a, a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a/span/span/ 
   ]
 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 - {[ ]}).
+img class="anchor" src="icons/tick.png" id="minus_eps_pre_aux"lemma minus_eps_pre_aux: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀A. 
+ a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_prl" 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)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a A → a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{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_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 a title="substraction" href="cic:/fakeuri.def(1)"-/a a 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 #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 //
+@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter7/minus_eps_pre.def(10)"minus_eps_pre/a]
+@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter7/minus_eps_item.def(9)"minus_eps_item/a]]
+@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/distribute_substract.def(3)"distribute_substract/a] 
+@a href="cic:/matita/tutorial/chapter4/eqP_substract_r.def(3)"eqP_substract_r/a //
 qed.
 
-(* theorem 16: 1 *)
-theorem sem_bull: ∀S:DeqSet. ∀i:pitem S.  \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
+img class="anchor" src="icons/tick.png" id="sem_bull"theorem sem_bull: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a. ∀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="eclose" href="cic:/fakeuri.def(1)"•/aia title="in_prl" 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)"}/a a title="union" 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)"}/a.
 #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 
+  [#w normalize % [/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | * //]
+  |/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/ 
+  |#x normalize #w % [ /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | * [@a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a | //]]
+  |#x normalize #w % [ /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | * // ] 
+  |#i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter8/eclose_dot.def(5)"eclose_dot/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"sem_pcl_aux/a //] >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a
+     [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a
+       [|@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@(a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a … IH1)] @a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a //
+  |#i1 #i2 #IH1 #IH2 >a href="cic:/matita/tutorial/chapter8/eclose_plus.def(5)"eclose_plus/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_oplus.def(9)"sem_oplus/a] >a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"sem_plus/a >a href="cic:/matita/tutorial/chapter7/erase_plus.def(4)"erase_plus/a 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@(a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a … IH2)]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a [|@a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"union_comm/a]]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a/span/span/
+  |#i #H >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a >a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"erase_bull/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a [|@a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"minus_eps_pre_aux/a //]]]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a >a href="cic:/matita/tutorial/chapter7/erase_star.def(4)"erase_star/a 
+   @a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/star_fix_eps.def(7)"star_fix_eps/a 
   ]
 qed.
 
-(* blank item *)
-let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
+(*
+h2Blank item/h2 
+
+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. *)
+
+img class="anchor" src="icons/tick.png" id="blank"let rec blank (S: a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a) (i: a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S) on i :a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a 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)^* ].
+  [ z ⇒ a href="cic:/matita/tutorial/chapter7/pitem.con(0,1,1)"pz/a ?
+  | e ⇒ a title="pitem epsilon" href="cic:/fakeuri.def(1)"ϵ/a
+  | s y ⇒ a title="pitem ps" href="cic:/fakeuri.def(1)"`/ay
+  | o e1 e2 ⇒ (blank S e1) a title="pitem or" href="cic:/fakeuri.def(1)"+/a (blank S e2) 
+  | c e1 e2 ⇒ (blank S e1) a title="pitem cat" href="cic:/fakeuri.def(1)"·/a (blank S e2)
+  | k e ⇒ (blank S e)a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a ].
   
-lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
+img class="anchor" src="icons/tick.png" id="forget_blank"lemma forget_blank: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S.a title="forget" href="cic:/fakeuri.def(1)"|/aa href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S ea title="forget" href="cic:/fakeuri.def(1)"|/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a e.
 #S #e elim e normalize //
 qed.
 
-lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
+img class="anchor" src="icons/tick.png" id="sem_blank"lemma sem_blank: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S.a title="in_pl" href="cic:/fakeuri.def(1)"\sem/a{a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S ea title="in_pl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="empty set" href="cic:/fakeuri.def(1)"∅/a.
 #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
+  [1,2:@a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a // 
+  |#s @a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a //
+  |#e1 #e2 #Hind1 #Hind2 >a href="cic:/matita/tutorial/chapter7/sem_cat.def(8)"sem_cat/a 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"union_empty_r/a … a title="empty set" href="cic:/fakeuri.def(1)"∅/a)] 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a[|@Hind2]] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter6/cat_empty_l.def(5)"cat_empty_l/a … ?)] @a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a @Hind1
+  |#e1 #e2 #Hind1 #Hind2 >a href="cic:/matita/tutorial/chapter7/sem_plus.def(8)"sem_plus/a 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"union_empty_r/a … a title="empty set" href="cic:/fakeuri.def(1)"∅/a)] 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a[|@Hind2]] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a @Hind1
+  |#e #Hind >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@(a href="cic:/matita/tutorial/chapter6/cat_empty_l.def(5)"cat_empty_l/a … ?)] @a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a @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.
+img class="anchor" src="icons/tick.png" id="re_embedding"theorem re_embedding: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/re.ind(1,0,1)"re/a S. 
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{a title="eclose" href="cic:/fakeuri.def(1)"•/a(a href="cic:/matita/tutorial/chapter8/blank.fix(0,1,3)"blank/a S e)a title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_l" href="cic:/fakeuri.def(1)"\sem/a{ea title="in_l" href="cic:/fakeuri.def(1)"}/a.
+#S #e @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_bull.def(12)"sem_bull/a] >a href="cic:/matita/tutorial/chapter8/forget_blank.def(4)"forget_blank/a 
+@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter8/sem_blank.def(9)"sem_blank/a]]
+@a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_comm.def(3)"union_comm/a] @a href="cic:/matita/tutorial/chapter4/union_empty_r.def(3)"union_empty_r/a.
 qed.
 
-(* lefted operations *)
-definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
-  lift S (pre_concat_l S eclose e).
+(*
+h2Lifted Operators/h2 
+
+Plus and bullet have been already lifted from items to pres. We can now 
+do a similar job for concatenation ⊙ and Kleene's star ⊛.*)
+
+img class="anchor" src="icons/tick.png" id="lifted_cat"definition lifted_cat ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+  a href="cic:/matita/tutorial/chapter8/lift.def(2)"lift/a S (a href="cic:/matita/tutorial/chapter8/pre_concat_l.def(3)"pre_concat_l/a S a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"eclose/a 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) // 
+img class="anchor" src="icons/tick.png" id="odot_true_b"lemma odot_true_b : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b. 
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai2,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a (a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2)),a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
+#S #i1 #i2 #b normalize in ⊢ (??%?); cases (a title="eclose" href="cic:/fakeuri.def(1)"•/ai2) // 
 qed.
 
-lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
-  〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
+img class="anchor" src="icons/tick.png" id="odot_false_b"lemma odot_false_b : ∀S.∀i1,i2:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.∀b.
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai2,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1 a title="pitem cat" href="cic:/fakeuri.def(1)"·/a i2 ,ba title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
 // 
 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 //  
+img class="anchor" src="icons/tick.png" id="erase_odot"lemma erase_odot:∀S.∀e1,e2:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
+  a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a e2)a 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)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e1a title="forget" href="cic:/fakeuri.def(1)"|/a a title="re cat" href="cic:/fakeuri.def(1)"·/a (a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a title="forget" href="cic:/fakeuri.def(1)"|/a).
+#S * #i1 * * #i2 #b2 // >a href="cic:/matita/tutorial/chapter8/odot_true_b.def(6)"odot_true_b/a >a href="cic:/matita/tutorial/chapter7/erase_dot.def(4)"erase_dot/a //  
 qed.
 
-definition lk ≝ λS:DeqSet.λe:pre S.
+(* Let us come to the star operation: *)
+
+img class="anchor" src="icons/tick.png" id="lk"definition lk ≝ λS:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λe:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
   match e with 
   [ mk_Prod i1 b1 ⇒
     match b1 with 
-    [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
-    |false ⇒ 〈i1^*,false〉
+    [true ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a href="cic:/matita/tutorial/chapter8/eclose.fix(0,1,4)"eclose/a ? i1))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a, a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
+    |false ⇒ a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ai1a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a,a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a
     ]
   ]. 
 
@@ -257,57 +317,58 @@ definition lk ≝ λS:DeqSet.λe:pre S.
 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〉.
+img class="anchor" src="icons/tick.png" id="ostar_true"lemma ostar_true: ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  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="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/a(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (a title="eclose" href="cic:/fakeuri.def(1)"•/ai))a title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a, a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
 // qed.
 
-lemma ostar_false: ∀S.∀i:pitem S.
-  〈i,false〉^⊛ = 〈i^*, false〉.
+img class="anchor" src="icons/tick.png" id="ostar_false"lemma ostar_false: ∀S.∀i:a href="cic:/matita/tutorial/chapter7/pitem.ind(1,0,1)"pitem/a S.
+  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="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aia title="pitem star" href="cic:/fakeuri.def(1)"^/aa title="pitem star" href="cic:/fakeuri.def(1)"*/a, a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
 // qed.
   
-lemma erase_ostar: ∀S.∀e:pre S.
-  |\fst (e^⊛)| = |\fst e|^*.
+img class="anchor" src="icons/tick.png" id="erase_ostar"lemma erase_ostar: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.
+  a title="forget" href="cic:/fakeuri.def(1)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (ea title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/a)a 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)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea 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.
 #S * #i * // qed.
 
-lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
-  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
+img class="anchor" src="icons/tick.png" id="sem_odot_true"lemma sem_odot_true: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i. 
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/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="extensional equality" href="cic:/fakeuri.def(1)"=/a1 a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a ia title="in_prl" href="cic:/fakeuri.def(1)"}/a a title="union" href="cic:/fakeuri.def(1)"∪/a a title="singleton" href="cic:/fakeuri.def(1)"{/a a title="nil" href="cic:/fakeuri.def(1)"[/a a title="nil" href="cic:/fakeuri.def(1)"]/a a title="singleton" href="cic:/fakeuri.def(1)"}/a.
 #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/
+cut (e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/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)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a(e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) [//]
+#H >H cases (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) #i1 #b1 cases b1 
+  [>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a/span/span/ 
+  |/span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/
   ]
 qed.
 
-lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
-  e1 ⊙ 〈i,false〉 = e1 ▹ i.
+img class="anchor" src="icons/tick.png" id="eq_odot_false"lemma eq_odot_false: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀e1:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S.∀i. 
+  e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/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)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a 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 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/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)"〉/a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i), a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a(e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a i) a title="boolean or" href="cic:/fakeuri.def(1)"∨/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/a) [//]
+cases (e1 a title="item-pre concat" href="cic:/fakeuri.def(1)"▹/a 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}.
+(* We conclude this section with the proof of the main semantic properties
+of ⊙ and ⊛. *)
+
+img class="anchor" src="icons/tick.png" id="sem_odot"lemma sem_odot: 
+  ∀S.∀e1,e2: a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{e1 a title="lifted cat" href="cic:/fakeuri.def(1)"⊙/a e2a title="in_prl" 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{e1a title="in_prl" href="cic:/fakeuri.def(1)"}/aa 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)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a e2a 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_prl" href="cic:/fakeuri.def(1)"\sem/a{e2a title="in_prl" href="cic:/fakeuri.def(1)"}/a.
 #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 //
+  [>a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a 
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter8/sem_odot_true.def(10)"sem_odot_true/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a @a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"sem_pcl_aux/a //
+  |>a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter8/eq_odot_false.def(6)"eq_odot_false/a @a href="cic:/matita/tutorial/chapter8/sem_pcl_aux.def(11)"sem_pcl_aux/a //
   ]
 qed.
 
-(* theorem 16: 4 *)      
-theorem sem_ostar: ∀S.∀e:pre S. 
-  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
+img class="anchor" src="icons/tick.png" id="sem_ostar"theorem sem_ostar: ∀S.∀e:a href="cic:/matita/tutorial/chapter7/pre.def(1)"pre/a S. 
+  a title="in_prl" href="cic:/fakeuri.def(1)"\sem/a{ea title="lk" href="cic:/fakeuri.def(1)"^/aa title="lk" href="cic:/fakeuri.def(1)"⊛/aa title="in_prl" 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)"}/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)"|/aa title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a ea 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.
 #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 >a href="cic:/matita/tutorial/chapter7/sem_pre_true.def(9)"sem_pre_true/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a >a href="cic:/matita/tutorial/chapter8/erase_bull.def(6)"erase_bull/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a[|@a href="cic:/matita/tutorial/chapter6/cat_ext_l.def(5)"cat_ext_l/a [|@a href="cic:/matita/tutorial/chapter8/minus_eps_pre_aux.def(11)"minus_eps_pre_aux/a //]]]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/eqP_union_r.def(3)"eqP_union_r/a [|@a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/distr_cat_r.def(5)"distr_cat_r/a]
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [|@a href="cic:/matita/tutorial/chapter4/union_assoc.def(3)"union_assoc/a] @a href="cic:/matita/tutorial/chapter4/eqP_union_l.def(3)"eqP_union_l/a
+   @a href="cic:/matita/tutorial/chapter4/eqP_trans.def(3)"eqP_trans/a [||@a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/epsilon_cat_l.def(5)"epsilon_cat_l/a] @a href="cic:/matita/tutorial/chapter4/eqP_sym.def(3)"eqP_sym/a @a href="cic:/matita/tutorial/chapter6/star_fix_eps.def(7)"star_fix_eps/a 
+  |>a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_pre_false.def(9)"sem_pre_false/a >a href="cic:/matita/tutorial/chapter7/sem_star.def(8)"sem_star/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/tutorial/chapter8/eq_to_ex_eq.def(4)"eq_to_ex_eq/a/span/span/
   ]
-qed.
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+qed. 
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