X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fre%2Fre.ma;h=5894af561fc9126882111daff83772dec7da9ae2;hb=2fef56731b0d38913967105495b96754e327efab;hp=3f58c2c4838344070027c766ecf58744fa39b15b;hpb=927bda3b4b7fe5f521ae73eb008a746e8606a0b4;p=helm.git

diff --git a/matita/matita/lib/re/re.ma b/matita/matita/lib/re/re.ma
index 3f58c2c48..5894af561 100644
--- a/matita/matita/lib/re/re.ma
+++ b/matita/matita/lib/re/re.ma
@@ -82,10 +82,81 @@ let rec forget (S: DeqSet) (l : pitem S) on l: re S ≝
   | pc E1 E2 ⇒ (forget ? E1) · (forget ? E2)
   | po E1 E2 ⇒ (forget ? E1) + (forget ? E2)
   | pk E ⇒ (forget ? E)^* ].
-  
+ 
 (* notation < "|term 19 e|" non associative with precedence 70 for @{'forget $e}.*)
 interpretation "forget" 'norm a = (forget ? a).
 
+lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
+// qed.
+
+lemma erase_plus : ∀S.∀i1,i2:pitem S.
+  |i1 + i2| = |i1| + |i2|.
+// qed.
+
+lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
+// qed.
+
+(* boolean equality *)
+let rec beqitem S (i1,i2: pitem S) on i1 ≝ 
+  match i1 with
+  [ pz ⇒ match i2 with [ pz ⇒ true | _ ⇒ false]
+  | pe ⇒ match i2 with [ pe ⇒ true | _ ⇒ false]
+  | ps y1 ⇒ match i2 with [ ps y2 ⇒ y1==y2 | _ ⇒ false]
+  | pp y1 ⇒ match i2 with [ pp y2 ⇒ y1==y2 | _ ⇒ false]
+  | po i11 i12 ⇒ match i2 with 
+    [ po i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+    | _ ⇒ false]
+  | pc i11 i12 ⇒ match i2 with 
+    [ pc i21 i22 ⇒ beqitem S i11 i21 ∧ beqitem S i12 i22
+    | _ ⇒ false]
+  | pk i11 ⇒ match i2 with [ pk i21 ⇒ beqitem S i11 i21 | _ ⇒ false]
+  ].
+
+lemma beqitem_true: ∀S,i1,i2. iff (beqitem S i1 i2 = true) (i1 = i2). 
+#S #i1 elim i1
+  [#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+  |#i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % // normalize #H destruct
+  |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+    [>(\P H) // | @(\b (refl …))]
+  |#x #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] % normalize #H destruct
+    [>(\P H) // | @(\b (refl …))]
+  |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+   normalize #H destruct 
+    [cases (true_or_false (beqitem S i11 i21)) #H1
+      [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @False_ind /2/
+      ]
+    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    ]
+  |#i11 #i12 #Hind1 #Hind2 #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i3] %
+   normalize #H destruct 
+    [cases (true_or_false (beqitem S i11 i21)) #H1
+      [>(proj1 … (Hind1 i21) H1) >(proj1 … (Hind2 i22)) // >H1 in H; #H @H
+      |>H1 in H; normalize #abs @False_ind /2/
+      ]
+    |>(proj2 … (Hind1 i21) (refl …)) >(proj2 … (Hind2 i22) (refl …)) //
+    ]
+  |#i3 #Hind #i2 cases i2 [||#a|#a|#i21 #i22| #i21 #i22|#i4] %
+   normalize #H destruct 
+    [>(proj1 … (Hind i4) H) // |>(proj2 … (Hind i4) (refl …)) //]
+  ]
+qed. 
+
+definition DeqItem ≝ λS.
+  mk_DeqSet (pitem S) (beqitem S) (beqitem_true S).
+  
+unification hint  0 ≔ S; 
+    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢ 
+    pitem S ≡ carr X.
+    
+unification hint  0 ≔ S,i1,i2; 
+    X ≟ mk_DeqSet (pitem S) (beqitem S) (beqitem_true S)
+(* ---------------------------------------- *) ⊢ 
+    beqitem S i1 i2 ≡ eqb X i1 i2.
+
+(* semantics *)
+
 let rec in_pl (S : DeqSet) (r : pitem S) on r : word S → Prop ≝ 
 match r with
 [ pz ⇒ ∅
@@ -158,6 +229,21 @@ lemma true_to_epsilon : ∀S.∀e:pre S. \snd e = true → ϵ ∈ e.
 #S * #i #b #btrue normalize in btrue; >btrue %2 // 
 qed.
 
+lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
+#S #i #w % 
+  [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
+  |* //
+  ]
+qed.
+
+lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
+#S * #i * 
+  [>sem_pre_true normalize in ⊢ (??%?); #w % 
+    [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
+  |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
+  ]
+qed.
+
 definition lo ≝ λS:DeqSet.λa,b:pre S.〈\fst a + \fst b,\snd a ∨ \snd b〉.
 notation "a ⊕ b" left associative with precedence 60 for @{'oplus $a $b}.
 interpretation "oplus" 'oplus a b = (lo ? a b).
@@ -168,46 +254,29 @@ lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2
 definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
   match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
  
-notation "i ◂ e" left associative with precedence 60 for @{'ltrif $i $e}.
-interpretation "pre_concat_r" 'ltrif i e = (pre_concat_r ? i e).
+notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
+interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
 
 lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
   A = B → A =1 B. 
 #S #A #B #H >H /2/ qed.
 
 lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
-  \sem{i ◂ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+  \sem{i ◃ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
 #S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //] 
 >sem_pre_true >sem_cat >sem_pre_true /2/ 
 qed.
  
-definition lc ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
+definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
   match e1 with 
   [ mk_Prod i1 b1 ⇒ match b1 with 
-    [ true ⇒ (i1 ◂ (bcast ? i2)) 
+    [ true ⇒ (i1 ◃ (bcast ? i2)) 
     | false ⇒ 〈i1 · i2,false〉
     ]
   ].
-        
-definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
-  match e with 
-  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
 
-notation "a ▸ b" left associative with precedence 60 for @{'lc eclose $a $b}.
-interpretation "lc" 'lc op a b = (lc ? op a b).
-
-definition lk ≝ λS:DeqSet.λbcast:∀S:DeqSet.∀E:pitem S.pre S.λe:pre S.
-  match e with 
-  [ mk_Prod i1 b1 ⇒
-    match b1 with 
-    [true ⇒ 〈(\fst (bcast ? i1))^*, true〉
-    |false ⇒ 〈i1^*,false〉
-    ]
-  ]. 
-
-(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $op $a}.*)
-interpretation "lk" 'lk op a = (lk ? op a).
-notation "a^⊛" non associative with precedence 90 for @{'lk eclose $a}.
+notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
+interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
 
 notation "•" non associative with precedence 60 for @{eclose ?}.
 
@@ -218,7 +287,7 @@ let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
   | ps x ⇒ 〈 `.x, false〉
   | pp x ⇒ 〈 `.x, false 〉
   | po i1 i2 ⇒ •i1 ⊕ •i2
-  | pc i1 i2 ⇒ •i1 ▸ i2
+  | pc i1 i2 ⇒ •i1 ▹ i2
   | pk i ⇒ 〈(\fst (•i))^*,true〉].
   
 notation "• x" non associative with precedence 60 for @{'eclose $x}.
@@ -229,15 +298,19 @@ lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
 // qed.
 
 lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
-  •(i1 · i2) = •i1 ▸ i2.
+  •(i1 · i2) = •i1 ▹ i2.
 // qed.
 
 lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
   •i^* = 〈(\fst(•i))^*,true〉.
 // qed.
 
-definition reclose ≝ λS. lift S (eclose S). 
-interpretation "reclose" 'eclose x = (reclose ? x).
+definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
+  match e with 
+  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
+  
+definition preclose ≝ λS. lift S (eclose S). 
+interpretation "preclose" 'eclose x = (preclose ? x).
 
 (* theorem 16: 2 *)
 lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
@@ -250,33 +323,23 @@ qed.
 
 lemma odot_true : 
   ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▸ i2 = i1 ◂ (•i2).
+  〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
 // qed.
 
 lemma odot_true_bis : 
   ∀S.∀i1,i2:pitem S.
-  〈i1,true〉 ▸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+  〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
 #S #i1 #i2 normalize cases (•i2) // qed.
 
 lemma odot_false: 
   ∀S.∀i1,i2:pitem S.
-  〈i1,false〉 ▸ i2 = 〈i1 · i2, false〉.
+  〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
 // qed.
 
 lemma LcatE : ∀S.∀e1,e2:pitem S.
   \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
 // qed.
 
-lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
-// qed.
-
-lemma erase_plus : ∀S.∀i1,i2:pitem S.
-  |i1 + i2| = |i1| + |i2|.
-// qed.
-
-lemma erase_star : ∀S.∀i:pitem S.|i^*| = |i|^*. 
-// qed.
-
 lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
 #S #i elim i // 
   [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
@@ -286,15 +349,17 @@ lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
   | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
   ]
 qed.
-  
+
+(*
 lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
   \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
 /2/ qed.
+*)
 
 (* theorem 16: 1 → 3 *)
 lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
    \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
-   \sem{e1 ▸ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+   \sem{e1 ▹ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
 #S * #i1 #b1 #i2 cases b1
   [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
   |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
@@ -303,33 +368,18 @@ lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
     @eqP_trans [|@eqP_sym @union_assoc ] /3/ 
   ]
 qed.
-
-lemma sem_fst: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
-#S * #i * 
-  [>sem_pre_true normalize in ⊢ (??%?); #w % 
-    [/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
-  |>sem_pre_false normalize in ⊢ (??%?); #w % [ /3/ | * // ]
-  ]
-qed.
-
-lemma item_eps: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
-#S #i #w % 
-  [#H whd % // normalize @(not_to_not … (not_epsilon_lp …i)) //
-  |* //
-  ]
-qed.
   
-lemma sem_fst_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
+lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
  \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
 #S #e #i #A #seme
-@eqP_trans [|@sem_fst]
-@eqP_trans [||@eqP_union_r [|@eqP_sym @item_eps]]
+@eqP_trans [|@minus_eps_pre]
+@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
 @eqP_trans [||@distribute_substract] 
 @eqP_substract_r //
 qed.
 
 (* theorem 16: 1 *)
-theorem sem_bull: ∀S:DeqSet. ∀e:pitem S.  \sem{•e} =1 \sem{e} ∪ \sem{|e|}.
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S.  \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
 #S #e elim e 
   [#w normalize % [/2/ | * //]
   |/2/ 
@@ -352,15 +402,53 @@ theorem sem_bull: ∀S:DeqSet. ∀e:pitem S.  \sem{•e} =1 \sem{e} ∪ \sem{|e|
    @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 [|@sem_fst_aux //]]]
+   @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
    @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
    @eqP_trans [|@union_assoc] @eqP_union_l >erase_star 
    @eqP_sym @star_fix_eps 
   ]
 qed.
 
+(* blank item *)
+let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
+ match i with
+  [ z ⇒ `∅
+  | e ⇒ ϵ
+  | s y ⇒ `y
+  | o e1 e2 ⇒ (blank S e1) + (blank S e2) 
+  | c e1 e2 ⇒ (blank S e1) · (blank S e2)
+  | k e ⇒ (blank S e)^* ].
+  
+lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
+#S #e elim e normalize //
+qed.
+
+lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
+#S #e elim e 
+  [1,2:@eq_to_ex_eq // 
+  |#s @eq_to_ex_eq //
+  |#e1 #e2 #Hind1 #Hind2 >sem_cat 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
+  |#e1 #e2 #Hind1 #Hind2 >sem_plus 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
+  |#e #Hind >sem_star
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
+  ]
+qed.
+   
+theorem re_embedding: ∀S.∀e:re S. 
+  \sem{•(blank S e)} =1 \sem{e}.
+#S #e @eqP_trans [|@sem_bull] >forget_blank 
+@eqP_trans [|@eqP_union_r [|@sem_blank]]
+@eqP_trans [|@union_comm] @union_empty_r.
+qed.
+
+(* lefted operations *)
 definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
-  lift S (lc S eclose e).
+  lift S (pre_concat_l S eclose e).
 
 notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
 
@@ -381,6 +469,20 @@ lemma erase_odot:∀S.∀e1,e2:pre S.
 #S * #i1 * * #i2 #b2 // >odot_true_b // 
 qed.
 
+definition lk ≝ λS:DeqSet.λe:pre S.
+  match e with 
+  [ mk_Prod i1 b1 ⇒
+    match b1 with 
+    [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
+    |false ⇒ 〈i1^*,false〉
+    ]
+  ]. 
+
+(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
+interpretation "lk" 'lk a = (lk ? a).
+notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
+
+
 lemma ostar_true: ∀S.∀i:pitem S.
   〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
 // qed.
@@ -394,10 +496,10 @@ lemma erase_ostar: ∀S.∀e:pre S.
 #S * #i * // qed.
 
 lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
-  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▸ i} ∪ { [ ] }.
+  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
 #S #e1 #i 
-cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ true〉) [//]
-#H >H cases (e1 ▸ i) #i1 #b1 cases b1 
+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/
@@ -405,10 +507,10 @@ cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ true〉) [//]
 qed.
 
 lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
-  e1 ⊙ 〈i,false〉 = e1 ▸ i.
+  e1 ⊙ 〈i,false〉 = e1 ▹ i.
 #S #e1 #i  
-cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▸ i), \snd(e1 ▸ i) ∨ false〉) [//]
-cases (e1 ▸ i) #i1 #b1 cases b1 #H @H
+cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
+cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
 qed.
 
 lemma sem_odot: 
@@ -426,7 +528,7 @@ theorem sem_ostar: ∀S.∀e:pre S.
   \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
 #S * #i #b cases b
   [>sem_pre_true >sem_pre_true >sem_star >erase_bull
-   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@sem_fst_aux //]]]
+   @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
@@ -434,4 +536,4 @@ theorem sem_ostar: ∀S.∀e:pre S.
   |>sem_pre_false >sem_pre_false >sem_star /2/
   ]
 qed.
-  
+