fixed notation

diff --git a/helm/software/matita/nlibrary/re/re.ma b/helm/software/matita/nlibrary/re/re.ma
index c66c73791e7304b155a152e3463843dcf2b7b08d..40947401f860939ceea720b2363c394a9671aef7 100644 (file)
--- a/helm/software/matita/nlibrary/re/re.ma
+++ b/helm/software/matita/nlibrary/re/re.ma
@@ -81,7 +81,7 @@ interpretation "cat lang" 'pc a b = (cat ? a b).
 ndefinition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l. 
 interpretation "star lang" 'pk l = (star ? l).
 
-notation > "𝐋 term 90 E" non associative with precedence 90 for @{in_l ? $E}.
+notation > "𝐋 term 70 E" non associative with precedence 75 for @{in_l ? $E}.
 nlet rec in_l (S : Alpha) (r : re S) on r : word S → Prop ≝ 
 match r with
 [ z ⇒ {}
@@ -90,7 +90,7 @@ match r with
 | c r1 r2 ⇒ 𝐋 r1 · 𝐋 r2
 | o r1 r2 ⇒  𝐋 r1 ∪ 𝐋 r2
 | k r1 ⇒ (𝐋 r1) ^*].
-notation "𝐋 term 90 E" non associative with precedence 90 for @{'in_l $E}.
+notation "𝐋 term 70 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).
 
@@ -135,7 +135,7 @@ nlet rec forget (S: Alpha) (l : pitem S) on l: re S ≝
   | pc E1 E2 ⇒ (|E1| · |E2|)
   | po E1 E2 ⇒ (|E1| + |E2|)
   | pk E ⇒ |E|^* ].
-notation < ".|term 19 e|" non associative with precedence 70 for @{'forget $e}.
+notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.
 interpretation "forget" 'forget a = (forget ? a).
 
 notation "\fst term 90 x" non associative with precedence 90 for @{'fst $x}.
@@ -143,18 +143,18 @@ interpretation "fst" 'fst x = (fst ? ? x).
 notation > "\snd term 90 x" non associative with precedence 90 for @{'snd $x}.
 interpretation "snd" 'snd x = (snd ? ? x).
 
-notation > "𝐋\p\ term 90 E" non associative with precedence 90 for @{in_pl ? $E}.
+notation > "𝐋\p\ term 70 E" non associative with precedence 75 for @{in_pl ? $E}.
 nlet rec in_pl (S : Alpha) (r : pitem S) on r : word S → Prop ≝ 
 match r with
 [ pz ⇒ {}
 | pe ⇒ {}
 | ps _ ⇒ {}
 | pp x ⇒ { [x] }
-| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋 .|r2| ∪ 𝐋\p\ r2
+| pc r1 r2 ⇒ 𝐋\p\ r1 · 𝐋  |r2| ∪ 𝐋\p\ r2
 | po r1 r2 ⇒ 𝐋\p\ r1 ∪ 𝐋\p\ r2
-| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (.|r1|^* ) ].
-notation > "𝐋\p term 90 E" non associative with precedence 90 for @{'in_pl $E}.
-notation "𝐋\sub(\p) term 90 E" non associative with precedence 90 for @{'in_pl $E}.
+| pk r1 ⇒ 𝐋\p\ r1 · 𝐋 (|r1|^* ) ].
+notation > "𝐋\p term 70 E" non associative with precedence 75 for @{'in_pl $E}.
+notation "𝐋\sub(\p) term 70 E" non associative with precedence 75 for @{'in_pl $E}.
 interpretation "in_pl" 'in_pl E = (in_pl ? E).
 interpretation "in_pl mem" 'mem w l = (in_pl ? l w).
 
@@ -164,7 +164,7 @@ interpretation "epsilon" 'epsilon = (epsilon ?).
 notation < "ϵ b" non associative with precedence 90 for @{'app_epsilon $b}.
 interpretation "epsilon lang" 'app_epsilon b = (epsilon ? b).
 
-ndefinition in_prl ≝ λS : Alpha.λp:pre S.  𝐋\p\ (\fst p) ∪ ϵ (\snd p).
+ndefinition in_prl ≝ λS : Alpha.λp:pre S.  𝐋\p (\fst p) ∪ ϵ (\snd p).
   
 interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
 interpretation "in_prl" 'in_pl E = (in_prl ? E).
@@ -182,7 +182,7 @@ nnormalize; *; ##[##2:*] nelim e;
 *; #w1; *; #w2; *; *; #defw1; nrewrite > (append_eq_nil … w1 w2 …); /3/ by {};//;
 nqed.
 
-nlemma not_epsilon_lp : ∀S.∀e:pitem S. ¬ (𝐋\p e [ ]).
+nlemma not_epsilon_lp : ∀S:Alpha.∀e:pitem S. ¬ ((𝐋\p e) [ ]).
 #S e; nelim e; nnormalize; /2/ by nmk;
 ##[ #; @; #; ndestruct;
 ##| #r1 r2 n1 n2; @; *; /2/; *; #w1; *; #w2; *; *; #H;
@@ -265,7 +265,7 @@ nlemma odotEt :
   ∀S.∀e1,e2:pitem S.∀b2. 〈e1,true〉 ⊙ 〈e2,b2〉 = 〈e1 · \fst (•e2),b2 || \snd (•e2)〉.
 #S e1 e2 b2; nnormalize; ncases (•e2); //; nqed.
 
-nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) =  𝐋\p e1 · 𝐋 .|e2| ∪ 𝐋\p e2. //; nqed.
+nlemma LcatE : ∀S.∀e1,e2:pitem S.𝐋\p (e1 · e2) =  𝐋\p e1 · 𝐋 |e2| ∪ 𝐋\p e2. //; nqed.
 
 nlemma cup_dotD : ∀S.∀p,q,r:word S → Prop.(p ∪ q) · r = (p · r) ∪ (q · r). 
 #S p q r; napply extP; #w; nnormalize; @; 
@@ -276,14 +276,14 @@ nqed.
 nlemma cup0 :∀S.∀p:word S → Prop.p ∪ {} = p.
 #S p; napply extP; #w; nnormalize; @; /2/; *; //; *; nqed.
 
-nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 .|e1 · e2| =  𝐋 .|e1| · 𝐋 .|e2|.
+nlemma erase_dot : ∀S.∀e1,e2:pitem S.𝐋 |e1 · e2| =  𝐋 |e1| · 𝐋 |e2|.
 #S e1 e2; napply extP; nnormalize; #w; @; *; #w1; *; #w2; *; *; /7/ by ex_intro, conj;
 nqed.
 
-nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 .|e1 + e2| =  𝐋 .|e1| ∪ 𝐋 .|e2|.
+nlemma erase_plus : ∀S.∀e1,e2:pitem S.𝐋 |e1 + e2| =  𝐋 |e1| ∪ 𝐋 |e2|.
 #S e1 e2; napply extP; nnormalize; #w; @; *; /4/ by or_introl, or_intror; nqed.
 
-nlemma erase_star : ∀S.∀e1:pitem S.𝐋 .|e1|^* = 𝐋 .|e1^*|. //; nqed.
+nlemma erase_star : ∀S.∀e1:pitem S.𝐋 |e1|^* = 𝐋 |e1^*|. //; nqed.
 
 ndefinition substract := λS.λp,q:word S → Prop.λw.p w ∧ ¬ q w.
 interpretation "substract" 'minus a b = (substract ? a b).
@@ -300,18 +300,18 @@ nlemma subK : ∀S.∀a:word S → Prop. a - a = {}.
 nlemma subW : ∀S.∀a,b:word S → Prop.∀w.(a - b) w → a w.
 #S a b w; nnormalize; *; //; nqed.
 
-nlemma erase_bull : ∀S.∀a:pitem S. .|\fst (•a)| = .|a|.
+nlemma erase_bull : ∀S.∀a:pitem S. |\fst (•a)| = |a|.
 #S a; nelim a; // by {};
-##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| · .|e2|);
+##[ #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| · |e2|);
     nrewrite < IH1; nrewrite < IH2;  
     nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊙ 〈e2,false〉));
     ncases (•e1); #e3 b; ncases b; nnormalize;
     ##[ ncases (•e2); //; ##| nrewrite > IH2; //]
-##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (.|e1| + .|e2|);
+##| #e1 e2 IH1 IH2; nchange in ⊢ (???%) with (|e1| + |e2|);
     nrewrite < IH2; nrewrite < IH1;
     nchange in ⊢ (??(??%)?) with (\fst (•e1 ⊕ •e2));
     ncases (•e1); ncases (•e2); //;
-##| #e IH; nchange in ⊢ (???%) with (.|e|^* ); nrewrite < IH;
+##| #e IH; nchange in ⊢ (???%) with (|e|^* ); nrewrite < IH;
     nchange in ⊢ (??(??%)?) with (\fst (•e))^*; //; ##]
 nqed.
 
@@ -325,8 +325,8 @@ napply Hw2; nqed.
 
 (* theorem 16: 1 → 3 *)
 nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
-      𝐋\p (•(\fst e2)) =  𝐋\p (\fst e2) ∪ 𝐋 .|\fst e2| → 
-         𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
+      𝐋\p (•(\fst e2)) =  𝐋\p (\fst e2) ∪ 𝐋 |\fst e2| → 
+         𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
 #S e1 e2 th1; ncases e1; #e1' b1'; ncases b1';
 ##[ nwhd in ⊢ (??(??%)?); nletin e2' ≝ (\fst e2); nletin b2' ≝ (\snd e2); 
     nletin e2'' ≝ (\fst (•(\fst e2))); nletin b2'' ≝ (\snd (•(\fst e2)));
@@ -335,8 +335,8 @@ nlemma odot_dot_aux : ∀S.∀e1,e2: pre S.
     nchange in match (𝐋\p 〈?,?〉) with (?∪?);
     nrewrite > (epsilon_or …); nrewrite > (cupC ? (ϵ ?)…);
     nrewrite > (cupA …);nrewrite < (cupA ?? (ϵ?)…);
-    nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 .|e2'|); ##[##2:
-      nchange with (𝐋\p 〈e2'',b2''〉 =  𝐋\p e2' ∪ 𝐋 .|e2'|); 
+    nrewrite > (?: 𝐋\p e2'' ∪ ϵ b2'' = 𝐋\p e2' ∪ 𝐋 |e2'|); ##[##2:
+      nchange with (𝐋\p 〈e2'',b2''〉 =  𝐋\p e2' ∪ 𝐋 |e2'|); 
       ngeneralize in match th1;
       nrewrite > (eta_lp…); #th1; nrewrite > th1; //;##]
     nrewrite > (eta_lp ? e2); 
@@ -385,7 +385,7 @@ alias symbol "pc" (instance 13) = "cat lang".
 alias symbol "in_pl" (instance 23) = "in_pl".
 alias symbol "in_pl" (instance 5) = "in_pl".
 alias symbol "eclose" (instance 21) = "eclose".
-ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 𝐋 .|e|.
+ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 𝐋 |e|.
 #S e; nelim e; //;
   ##[ #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl, or_intror;
   ##| #a; napply extP; #w; nnormalize; @; *; /3/ by or_introl; *;
@@ -405,22 +405,22 @@ ntheorem bull_cup : ∀S:Alpha. ∀e:pitem S.  𝐋\p (•e) =  𝐋\p e ∪ 
   ##| #e; nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e)); #IH;
       nchange in match (𝐋\p ?) with  (𝐋\p 〈e'^*,true〉);
       nchange in match (𝐋\p ?) with (𝐋\p (e'^* ) ∪ {[ ]});
-      nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
+      nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
       nrewrite > (erase_bull…e);
       nrewrite > (erase_star …);
-      nrewrite > (?: 𝐋\p e' =  𝐋\p e ∪ (𝐋 .|e| - ϵ b')); ##[##2:
+      nrewrite > (?: 𝐋\p e' =  𝐋\p e ∪ (𝐋 |e| - ϵ b')); ##[##2:
         nchange in IH : (??%?) with (𝐋\p e' ∪ ϵ b'); ncases b' in IH; 
         ##[ #IH; nrewrite > (cup_sub…); //; nrewrite < IH; 
             nrewrite < (cup_sub…); //; nrewrite > (subK…); nrewrite > (cup0…);//;
         ##| nrewrite > (sub0 …); #IH; nrewrite < IH; nrewrite > (cup0 …);//; ##]##]
       nrewrite > (cup_dotD…); nrewrite > (cupA…); 
-      nrewrite > (?: ((?·?)∪{[]} = 𝐋 .|e^*|)); //;
-      nchange in match (𝐋 .|e^*|) with ((𝐋. |e|)^* ); napply sub_dot_star;##]
+      nrewrite > (?: ((?·?)∪{[]} = 𝐋 |e^*|)); //;
+      nchange in match (𝐋 |e^*|) with ((𝐋 |e|)^* ); napply sub_dot_star;##]
  nqed.
 
 (* theorem 16: 3 *)      
 nlemma odot_dot: 
-  ∀S.∀e1,e2: pre S.  𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 .|\fst e2| ∪ 𝐋\p e2.
+  ∀S.∀e1,e2: pre S.  𝐋\p (e1 ⊙ e2) =  𝐋\p e1 · 𝐋 |\fst e2| ∪ 𝐋\p e2.
 #S e1 e2; napply odot_dot_aux; napply (bull_cup S (\fst e2)); nqed.
 
 nlemma dot_star_epsilon : ∀S.∀e:re S.𝐋 e · 𝐋 e^* ∪ {[]} =  𝐋 e^*.
@@ -449,13 +449,13 @@ nlemma rcanc_sing : ∀S.∀A,C:word S → Prop.∀b:word S .
 nqed.
 
 (* theorem 16: 4 *)      
-nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 .|\fst e|)^*.
+nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 |\fst e|)^*.
 #S p; ncases p; #e b; ncases b;
 ##[ nchange in match (〈e,true〉^⊛) with 〈?,?〉;
     nletin e' ≝ (\fst (•e)); nletin b' ≝ (\snd (•e));
     nchange in ⊢ (??%?) with (?∪?);
-    nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 .|e'|^* );
-    nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 .|e| - ϵ b' )); ##[##2:
+    nchange in ⊢ (??(??%?)?) with (𝐋\p e' · 𝐋 |e'|^* );
+    nrewrite > (?: 𝐋\p e' = 𝐋\p e ∪ (𝐋 |e| - ϵ b' )); ##[##2:
       nlapply (bull_cup ? e); #bc;
       nchange in match (𝐋\p (•e)) in bc with (?∪?);
       nchange in match b' in bc with b';
@@ -467,7 +467,7 @@ nlemma star_dot: ∀S.∀e:pre S. 𝐋\p (e^⊛) = 𝐋\p e · (𝐋 .|\fst e|)^
     nrewrite > (cup_dotD…); nrewrite > (epsilon_dot…); //;    
 ##| nwhd in match (〈e,false〉^⊛); nchange in match (𝐋\p 〈?,?〉) with (?∪?);
     nrewrite > (cup0…);
-    nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 .|e|^* );
+    nchange in ⊢ (??%?) with (𝐋\p e · 𝐋 |e|^* );
     nrewrite < (cup0 ? (𝐋\p e)); //;##]
 nqed.
 
@@ -496,7 +496,7 @@ nlemma Lp_pre_of_re : ∀S.∀e:re S. 𝐋\p (pre_of_re ? e) = {}.
     nrewrite > H1; napply dot0; ##]
 nqed.
 
-nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 .|pre_of_re S e| = 𝐋 e.
+nlemma erase_pre_of_reK : ∀S.∀e. 𝐋 |pre_of_re S e| = 𝐋 e.
 #S A; nelim A; //; 
 ##[ #e1 e2 H1 H2; nchange in match (𝐋 (e1 · e2)) with (𝐋 e1·?);
     nrewrite < H1; nrewrite < H2; //
@@ -516,12 +516,11 @@ nlemma Pext : ∀S.∀f,g:word S → Prop. f = g → ∀w.f w → g w.
 #S f g H; nrewrite > H; //; nqed.
  
 (* corollary 18 *)
-ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ .|e|.
+ntheorem bull_true_epsilon : ∀S.∀e:pitem S. \snd (•e) = true ↔ [ ] ∈ |e|.
 #S e; @;
 ##[ #defsnde; nlapply (bull_cup ? e); nchange in match (𝐋\p (•e)) with (?∪?);
     nrewrite > defsnde; #H; 
     nlapply (Pext ??? H [ ] ?); ##[ @2; //] *; //;
-    E MO?
 
 STOP
 
@@ -543,7 +542,7 @@ interpretation "move" 'move x E = (move ? x E).
 ndefinition rmove ≝ λS:Alpha.λx:S.λe:pre S. \move x (\fst e).
 interpretation "rmove" 'move x E = (rmove ? x E).
 
-nlemma XXz :  ∀S:Alpha.∀w:word S. w .∈ ∅ → False.
+nlemma XXz :  ∀S:Alpha.∀w:word S. w ∈ ∅ → False.
 #S w abs; ninversion abs; #; ndestruct;
 nqed.