| 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 ⇒ ∅
#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).
definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
-notation "i â\97\82 e" left associative with precedence 60 for @{'ltrif $i $e}.
-interpretation "pre_concat_r" 'ltrif i e = (pre_concat_r ? i e).
+notation "i â\97\83 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 â\97\82 e} =1 \sem{i} · \sem{|\fst e|} ∪ \sem{e}.
+ \sem{i â\97\83 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 â\87\92 (i1 â\97\82 (bcast ? i2))
+ [ true â\87\92 (i1 â\97\83 (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 ?}.
| ps x ⇒ 〈 `.x, false〉
| pp x ⇒ 〈 `.x, false 〉
| po i1 i2 ⇒ •i1 ⊕ •i2
- | pc i1 i2 â\87\92 â\80¢i1 â\96¸ i2
+ | pc i1 i2 â\87\92 â\80¢i1 â\96¹ i2
| pk i ⇒ 〈(\fst (•i))^*,true〉].
notation "• x" non associative with precedence 60 for @{'eclose $x}.
// qed.
lemma eclose_dot: ∀S:DeqSet.∀i1,i2:pitem S.
- â\80¢(i1 · i2) = â\80¢i1 â\96¸ i2.
+ â\80¢(i1 · i2) = â\80¢i1 â\96¹ 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.
lemma odot_true :
∀S.∀i1,i2:pitem S.
- â\8c©i1,trueâ\8cª â\96¸ i2 = i1 â\97\82 (•i2).
+ â\8c©i1,trueâ\8cª â\96¹ i2 = i1 â\97\83 (•i2).
// qed.
lemma odot_true_bis :
∀S.∀i1,i2:pitem S.
- â\8c©i1,trueâ\8cª â\96¸ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
+ â\8c©i1,trueâ\8cª â\96¹ i2 = 〈i1 · \fst (•i2), \snd (•i2)〉.
#S #i1 #i2 normalize cases (•i2) // qed.
lemma odot_false:
∀S.∀i1,i2:pitem S.
- â\8c©i1,falseâ\8cª â\96¸ i2 = 〈i1 · i2, false〉.
+ â\8c©i1,falseâ\8cª â\96¹ 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
| #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 â\96¸ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+ \sem{e1 â\96¹ 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 …))
@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/
@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}.
#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.
#S * #i * // qed.
lemma sem_odot_true: ∀S:DeqSet.∀e1:pre S.∀i.
- \sem{e1 â\8a\99 â\8c©i,trueâ\8cª} =1 \sem{e1 â\96¸ i} ∪ { [ ] }.
+ \sem{e1 â\8a\99 â\8c©i,trueâ\8cª} =1 \sem{e1 â\96¹ i} ∪ { [ ] }.
#S #e1 #i
-cut (e1 â\8a\99 â\8c©i,trueâ\8cª = â\8c©\fst (e1 â\96¸ i), \snd(e1 â\96¸ i) ∨ true〉) [//]
-#H >H cases (e1 â\96¸ i) #i1 #b1 cases b1
+cut (e1 â\8a\99 â\8c©i,trueâ\8cª = â\8c©\fst (e1 â\96¹ i), \snd(e1 â\96¹ i) ∨ true〉) [//]
+#H >H cases (e1 â\96¹ 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 â\8a\99 â\8c©i,falseâ\8cª = e1 â\96¸ i.
+ e1 â\8a\99 â\8c©i,falseâ\8cª = e1 â\96¹ i.
#S #e1 #i
-cut (e1 â\8a\99 â\8c©i,falseâ\8cª = â\8c©\fst (e1 â\96¸ i), \snd(e1 â\96¸ i) ∨ false〉) [//]
-cases (e1 â\96¸ i) #i1 #b1 cases b1 #H @H
+cut (e1 â\8a\99 â\8c©i,falseâ\8cª = â\8c©\fst (e1 â\96¹ i), \snd(e1 â\96¹ i) ∨ false〉) [//]
+cases (e1 â\96¹ i) #i1 #b1 cases b1 #H @H
qed.
lemma sem_odot:
\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
|>sem_pre_false >sem_pre_false >sem_star /2/
]
qed.
-
+
projects / helm.git / blobdiff