X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fre_complete%2Fre.ma;fp=matita%2Fmatita%2Fre_complete%2Fre.ma;h=ff1f959934f732abac43f6b7f4337099498e912d;hb=0d481cc22ba8ada5781885da5398086a0b5662f3;hp=0000000000000000000000000000000000000000;hpb=0f5da967971b81ad7adc9fb1fedb9b2f24ca3ccf;p=helm.git

diff --git a/matita/matita/re_complete/re.ma b/matita/matita/re_complete/re.ma
new file mode 100644
index 000000000..ff1f95993
--- /dev/null
+++ b/matita/matita/re_complete/re.ma
@@ -0,0 +1,525 @@
+include "lang.ma".
+
+inductive re (S: DeqSet) : Type[0] ≝
+   z: re S
+ | e: re S
+ | s: S → re S
+ | c: re S → re S → re S
+ | o: re S → re S → re S
+ | k: re S → re S.
+
+interpretation "re epsilon" 'epsilon = (e ?).
+interpretation "re or" 'plus a b = (o ? a b).
+interpretation "re cat" 'middot a b = (c ? a b).
+interpretation "re star" 'star a = (k ? a).
+
+notation < "a" non associative with precedence 90 for @{ 'ps $a}.
+notation > "` term 90 a" non associative with precedence 90 for @{ 'ps $a}.
+interpretation "atom" 'ps a = (s ? a).
+
+notation "`∅" non associative with precedence 90 for @{ 'empty }.
+interpretation "empty" 'empty = (z ?).
+
+let rec in_l (S : DeqSet) (r : re S) on r : word S → Prop ≝ 
+match r with
+[ z ⇒ ∅
+| e ⇒ {ϵ}
+| s x ⇒ {[x]}
+| c r1 r2 ⇒ (in_l ? r1) · (in_l ? r2)
+| o r1 r2 ⇒ (in_l ? r1) ∪ (in_l ? r2)
+| k r1 ⇒ (in_l ? r1) ^*].
+
+notation "\sem{term 19 E}" non associative with precedence 75 for @{'in_l $E}.
+interpretation "in_l" 'in_l E = (in_l ? E).
+interpretation "in_l mem" 'mem w l = (in_l ? l w).
+
+lemma rsem_star : ∀S.∀r: re S. \sem{r^*} = \sem{r}^*.
+// qed.
+
+
+(* pointed items *)
+inductive pitem (S: DeqSet) : Type[0] ≝
+   pz: pitem S
+ | pe: pitem S
+ | ps: S → pitem S
+ | pp: S → pitem S
+ | pc: pitem S → pitem S → pitem S
+ | po: pitem S → pitem S → pitem S
+ | pk: pitem S → pitem S.
+ 
+definition pre ≝ λS.pitem S × bool.
+
+interpretation "pitem star" 'star a = (pk ? a).
+interpretation "pitem or" 'plus a b = (po ? a b).
+interpretation "pitem cat" 'middot a b = (pc ? a b).
+notation < ".a" non associative with precedence 90 for @{ 'pp $a}.
+notation > "`. term 90 a" non associative with precedence 90 for @{ 'pp $a}.
+interpretation "pitem pp" 'pp a = (pp ? a).
+interpretation "pitem ps" 'ps a = (ps ? a).
+interpretation "pitem epsilon" 'epsilon = (pe ?).
+interpretation "pitem empty" 'empty = (pz ?).
+
+let rec forget (S: DeqSet) (l : pitem S) on l: re 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)^* ].
+ 
+(* 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 ⇒ ∅
+| 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}^*  ].
+
+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}.
+  
+interpretation "in_prl mem" 'mem w l = (in_prl ? l w).
+interpretation "in_prl" 'in_l E = (in_prl ? E).
+
+lemma sem_pre_true : ∀S.∀i:pitem S. 
+  \sem{〈i,true〉} = \sem{i} ∪ {ϵ}. 
+// qed.
+
+lemma sem_pre_false : ∀S.∀i:pitem S. 
+  \sem{〈i,false〉} = \sem{i}. 
+// qed.
+
+lemma sem_cat: ∀S.∀i1,i2:pitem S. 
+  \sem{i1 · i2} = \sem{i1} · \sem{|i2|} ∪ \sem{i2}.
+// qed.
+
+lemma sem_cat_w: ∀S.∀i1,i2:pitem S.∀w.
+  \sem{i1 · i2} w = ((\sem{i1} · \sem{|i2|}) w ∨ \sem{i2} w).
+// qed.
+
+lemma sem_plus: ∀S.∀i1,i2:pitem S. 
+  \sem{i1 + i2} = \sem{i1} ∪ \sem{i2}.
+// qed.
+
+lemma sem_plus_w: ∀S.∀i1,i2:pitem S.∀w. 
+  \sem{i1 + i2} w = (\sem{i1} w ∨ \sem{i2} w).
+// qed.
+
+lemma sem_star : ∀S.∀i:pitem S.
+  \sem{i^*} = \sem{i} · \sem{|i|}^*.
+// qed.
+
+lemma sem_star_w : ∀S.∀i:pitem S.∀w.
+  \sem{i^*} w = (∃w1,w2.w1 @ w2 = w ∧ \sem{i} w1 ∧ \sem{|i|}^* w2).
+// qed.
+
+lemma append_eq_nil : ∀S.∀w1,w2:word S. w1 @ w2 = ϵ → w1 = ϵ.
+#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/
+  ]
+qed.
+
+(* lemma 12 *)
+lemma epsilon_to_true : ∀S.∀e:pre S. ϵ ∈ e → \snd e = true.
+#S * #i #b cases b // normalize #H @False_ind /2/ 
+qed.
+
+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).
+
+lemma lo_def: ∀S.∀i1,i2:pitem S.∀b1,b2. 〈i1,b1〉⊕〈i2,b2〉=〈i1+i2,b1∨b2〉.
+// qed.
+
+definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
+  match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
+ 
+notation "i ◃ e" left associative with precedence 60 for @{'lhd $i $e}.
+interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
+
+lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop. 
+  A = B → A =1 B. 
+#S #A #B #H >H /2/ qed.
+
+lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
+  \sem{i ◃ e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //] 
+>sem_pre_true >sem_cat >sem_pre_true /2/ 
+qed.
+ 
+definition pre_concat_l ≝ λS:DeqSet.λbcast:∀S:DeqSet.pitem S → pre S.λe1:pre S.λi2:pitem S.
+  match e1 with 
+  [ mk_Prod i1 b1 ⇒ match b1 with 
+    [ true ⇒ (i1 ◃ (bcast ? i2)) 
+    | false ⇒ 〈i1 · i2,false〉
+    ]
+  ].
+
+notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
+interpretation "item-pre concat" 'tril op a b = (pre_concat_l ? op a b).
+
+notation "•" non associative with precedence 60 for @{eclose ?}.
+
+let rec eclose (S: DeqSet) (i: pitem S) on i : pre S ≝
+ match i with
+  [ pz ⇒ 〈 `∅, false 〉
+  | pe ⇒ 〈 ϵ,  true 〉
+  | ps x ⇒ 〈 `.x, false〉
+  | pp x ⇒ 〈 `.x, false 〉
+  | po i1 i2 ⇒ •i1 ⊕ •i2
+  | pc i1 i2 ⇒ •i1 ▹ i2
+  | pk i ⇒ 〈(\fst (•i))^*,true〉].
+  
+notation "• x" non associative with precedence 60 for @{'eclose $x}.
+interpretation "eclose" 'eclose x = (eclose ? x).
+
+lemma eclose_plus: ∀S:DeqSet.∀i1,i2:pitem S.
+  •(i1 + i2) = •i1 ⊕ •i2.
+// qed.
+
+lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
+  •(i1 · i2) = •i1 ▹ i2.
+// qed.
+
+lemma eclose_star: ∀S:DeqSet.∀i:pitem S.
+  •i^* = 〈(\fst(•i))^*,true〉.
+// qed.
+
+definition lift ≝ λS.λf:pitem S →pre S.λe:pre S. 
+  match e with 
+  [ mk_Prod i b ⇒ 〈\fst (f i), \snd (f i) ∨ b〉].
+  
+definition preclose ≝ λS. lift S (eclose S). 
+interpretation "preclose" 'eclose x = (preclose ? x).
+
+(* theorem 16: 2 *)
+lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
+  \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}. 
+#S * #i1 #b1 * #i2 #b2 #w %
+  [cases b1 cases b2 normalize /2/ * /3/ * /3/
+  |cases b1 cases b2 normalize /2/ * /3/ * /3/
+  ]
+qed.
+
+lemma odot_true : 
+  ∀S.∀i1,i2:pitem S.
+  〈i1,true〉 ▹ i2 = i1 ◃ (•i2).
+// qed.
+
+lemma odot_true_bis : 
+  ∀S.∀i1,i2:pitem S.
+  〈i1,true〉 ▹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+#S #i1 #i2 normalize cases (•i2) // qed.
+
+lemma odot_false: 
+  ∀S.∀i1,i2:pitem S.
+  〈i1,false〉 ▹ i2 = 〈i1 · i2, false〉.
+// qed.
+
+lemma LcatE : ∀S.∀e1,e2:pitem S.
+  \sem{e1 · e2} = \sem{e1} · \sem{|e2|} ∪ \sem{e2}. 
+// qed.
+
+lemma erase_bull : ∀S.∀i:pitem S. |\fst (•i)| = |i|.
+#S #i elim i // 
+  [ #i1 #i2 #IH1 #IH2 >erase_dot <IH1 >eclose_dot
+    cases (•i1) #i11 #b1 cases b1 // <IH2 >odot_true_bis //
+  | #i1 #i2 #IH1 #IH2 >eclose_plus >(erase_plus … i1) <IH1 <IH2
+    cases (•i1) #i11 #b1 cases (•i2) #i21 #b2 //  
+  | #i #IH >eclose_star >(erase_star … i) <IH cases (•i) //
+  ]
+qed.
+
+(*
+lemma sem_eclose_star: ∀S:DeqSet.∀i:pitem S.
+  \sem{〈i^*,true〉} =1 \sem{〈i,false〉}·\sem{|i|}^* ∪ {ϵ}.
+/2/ qed.
+*)
+
+(* theorem 16: 1 → 3 *)
+lemma odot_dot_aux : ∀S.∀e1:pre S.∀i2:pitem S.
+   \sem{•i2} =1  \sem{i2} ∪ \sem{|i2|} →
+   \sem{e1 ▹ i2} =1  \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+#S * #i1 #b1 #i2 cases b1
+  [2:#th >odot_false >sem_pre_false >sem_pre_false >sem_cat /2/
+  |#H >odot_true >sem_pre_true @(eqP_trans … (sem_pre_concat_r …))
+   >erase_bull @eqP_trans [|@(eqP_union_l … H)]
+    @eqP_trans [|@eqP_union_l[|@union_comm ]]
+    @eqP_trans [|@eqP_sym @union_assoc ] /3/ 
+  ]
+qed.
+  
+lemma minus_eps_pre_aux: ∀S.∀e:pre S.∀i:pitem S.∀A. 
+ \sem{e} =1 \sem{i} ∪ A → \sem{\fst e} =1 \sem{i} ∪ (A - {[ ]}).
+#S #e #i #A #seme
+@eqP_trans [|@minus_eps_pre]
+@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
+@eqP_trans [||@distribute_substract] 
+@eqP_substract_r //
+qed.
+
+(* theorem 16: 1 *)
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S.  \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
+#S #e elim e 
+  [#w normalize % [/2/ | * //]
+  |/2/ 
+  |#x normalize #w % [ /2/ | * [@False_ind | //]]
+  |#x normalize #w % [ /2/ | * // ] 
+  |#i1 #i2 #IH1 #IH2 >eclose_dot
+   @eqP_trans [|@odot_dot_aux //] >sem_cat 
+   @eqP_trans
+     [|@eqP_union_r
+       [|@eqP_trans [|@(cat_ext_l … IH1)] @distr_cat_r]]
+   @eqP_trans [|@union_assoc]
+   @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_union_l //
+  |#i1 #i2 #IH1 #IH2 >eclose_plus
+   @eqP_trans [|@sem_oplus] >sem_plus >erase_plus 
+   @eqP_trans [|@(eqP_union_l … IH2)]
+   @eqP_trans [|@eqP_sym @union_assoc]
+   @eqP_trans [||@union_assoc] @eqP_union_r
+   @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_trans [||@eqP_union_l [|@union_comm]]
+   @eqP_trans [||@union_assoc] /2/
+  |#i #H >sem_pre_true >sem_star >erase_bull >sem_star
+   @eqP_trans [|@eqP_union_r [|@cat_ext_l [|@minus_eps_pre_aux //]]]
+   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+   @eqP_trans [|@union_assoc] @eqP_union_l >erase_star 
+   @eqP_sym @star_fix_eps 
+  ]
+qed.
+
+(* blank item *)
+let rec blank (S: DeqSet) (i: re S) on i :pitem S ≝
+ match i with
+  [ z ⇒ `∅
+  | e ⇒ ϵ
+  | s y ⇒ `y
+  | o e1 e2 ⇒ (blank S e1) + (blank S e2) 
+  | c e1 e2 ⇒ (blank S e1) · (blank S e2)
+  | k e ⇒ (blank S e)^* ].
+  
+lemma forget_blank: ∀S.∀e:re S.|blank S e| = e.
+#S #e elim e normalize //
+qed.
+
+lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
+#S #e elim e 
+  [1,2:@eq_to_ex_eq // 
+  |#s @eq_to_ex_eq //
+  |#e1 #e2 #Hind1 #Hind2 >sem_cat 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind1
+  |#e1 #e2 #Hind1 #Hind2 >sem_plus 
+   @eqP_trans [||@(union_empty_r … ∅)] 
+   @eqP_trans [|@eqP_union_l[|@Hind2]] @eqP_union_r @Hind1
+  |#e #Hind >sem_star
+   @eqP_trans [||@(cat_empty_l … ?)] @cat_ext_l @Hind
+  ]
+qed.
+   
+theorem re_embedding: ∀S.∀e:re S. 
+  \sem{•(blank S e)} =1 \sem{e}.
+#S #e @eqP_trans [|@sem_bull] >forget_blank 
+@eqP_trans [|@eqP_union_r [|@sem_blank]]
+@eqP_trans [|@union_comm] @union_empty_r.
+qed.
+
+(* lefted operations *)
+definition lifted_cat ≝ λS:DeqSet.λe:pre S. 
+  lift S (pre_concat_l S eclose e).
+
+notation "e1 ⊙ e2" left associative with precedence 70 for @{'odot $e1 $e2}.
+
+interpretation "lifted cat" 'odot e1 e2 = (lifted_cat ? e1 e2).
+
+lemma odot_true_b : ∀S.∀i1,i2:pitem S.∀b. 
+  〈i1,true〉 ⊙ 〈i2,b〉 = 〈i1 · (\fst (•i2)),\snd (•i2) ∨ b〉.
+#S #i1 #i2 #b normalize in ⊢ (??%?); cases (•i2) // 
+qed.
+
+lemma odot_false_b : ∀S.∀i1,i2:pitem S.∀b.
+  〈i1,false〉 ⊙ 〈i2,b〉 = 〈i1 · i2 ,b〉.
+// 
+qed.
+  
+lemma erase_odot:∀S.∀e1,e2:pre S.
+  |\fst (e1 ⊙ e2)| = |\fst e1| · (|\fst e2|).
+#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //  
+qed.
+
+definition lk ≝ λS:DeqSet.λe:pre S.
+  match e with 
+  [ mk_Prod i1 b1 ⇒
+    match b1 with 
+    [true ⇒ 〈(\fst (eclose ? i1))^*, true〉
+    |false ⇒ 〈i1^*,false〉
+    ]
+  ]. 
+
+(* notation < "a \sup ⊛" non associative with precedence 90 for @{'lk $a}.*)
+interpretation "lk" 'lk a = (lk ? a).
+notation "a^⊛" non associative with precedence 90 for @{'lk $a}.
+
+
+lemma ostar_true: ∀S.∀i:pitem S.
+  〈i,true〉^⊛ = 〈(\fst (•i))^*, true〉.
+// qed.
+
+lemma ostar_false: ∀S.∀i:pitem S.
+  〈i,false〉^⊛ = 〈i^*, false〉.
+// qed.
+  
+lemma erase_ostar: ∀S.∀e:pre S.
+  |\fst (e^⊛)| = |\fst e|^*.
+#S * #i * // qed.
+
+lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i. 
+  \sem{e1 ⊙ 〈i,true〉} =1 \sem{e1 ▹ i} ∪ { [ ] }.
+#S #e1 #i 
+cut (e1 ⊙ 〈i,true〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ true〉) [//]
+#H >H cases (e1 ▹ i) #i1 #b1 cases b1 
+  [>sem_pre_true @eqP_trans [||@eqP_sym @union_assoc]
+   @eqP_union_l /2/ 
+  |/2/
+  ]
+qed.
+
+lemma eq_odot_false: ∀S:DeqSet.∀e1:pre S.∀i. 
+  e1 ⊙ 〈i,false〉 = e1 ▹ i.
+#S #e1 #i  
+cut (e1 ⊙ 〈i,false〉 = 〈\fst (e1 ▹ i), \snd(e1 ▹ i) ∨ false〉) [//]
+cases (e1 ▹ i) #i1 #b1 cases b1 #H @H
+qed.
+
+lemma sem_odot: 
+  ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
+#S #e1 * #i2 * 
+  [>sem_pre_true 
+   @eqP_trans [|@sem_odot_true]
+   @eqP_trans [||@union_assoc] @eqP_union_r @odot_dot_aux //
+  |>sem_pre_false >eq_odot_false @odot_dot_aux //
+  ]
+qed.
+
+(* theorem 16: 4 *)      
+theorem sem_ostar: ∀S.∀e:pre S. 
+  \sem{e^⊛} =1  \sem{e} · \sem{|\fst e|}^*.
+#S * #i #b cases b
+  [>sem_pre_true >sem_pre_true >sem_star >erase_bull
+   @eqP_trans [|@eqP_union_r[|@cat_ext_l [|@minus_eps_pre_aux //]]]
+   @eqP_trans [|@eqP_union_r [|@distr_cat_r]]
+   @eqP_trans [||@eqP_sym @distr_cat_r]
+   @eqP_trans [|@union_assoc] @eqP_union_l
+   @eqP_trans [||@eqP_sym @epsilon_cat_l] @eqP_sym @star_fix_eps 
+  |>sem_pre_false >sem_pre_false >sem_star /2/
+  ]
+qed.
+