(**************************************************************************)
include "re/lang.ma".
+include "basics/core_notation/card_1.ma".
(* The type re of regular expressions over an alphabet $S$ is the smallest
collection of objects generated by the following constructors: *)
| 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).
+interpretation "forget" 'card a = (forget ? a).
lemma erase_dot : ∀S.∀e1,e2:pitem S. |e1 · e2| = c ? (|e1|) (|e2|).
// qed.
(* Below are a few, simple, semantic properties of items. In particular:
- not_epsilon_item : ∀S:DeqSet.∀i:pitem S. ¬ (\sem{i} ϵ).
- epsilon_pre : ∀S.∀e:pre S. (\sem{i} ϵ) ↔ (\snd e = true).
-- minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
-- minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} =1 \sem{e}-{[ ]}.
+- minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \sem{i}-{[ ]}.
+- minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} ≐ \sem{e}-{[ ]}.
The first property is proved by a simple induction on $i$; the other
results are easy corollaries. We need an auxiliary lemma first. *)
#S * #i #b #btrue normalize in btrue; >btrue %2 //
qed.
-lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} =1 \sem{i}-{[ ]}.
+lemma minus_eps_item: ∀S.∀i:pitem S. \sem{i} ≐ \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}-{[ ]}.
+lemma minus_eps_pre: ∀S.∀e:pre S. \sem{\fst e} ≐ \sem{e}-{[ ]}.
#S * #i *
[>sem_pre_true normalize in ⊢ (??%?); #w %
[/3/ | * * // #H1 #H2 @False_ind @(absurd …H1 H2)]
*)
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}.
+notation "a ⊕ b" left associative with precedence 65 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〉.
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}.
+notation "i ◃ e" left associative with precedence 65 for @{'lhd $i $e}.
interpretation "pre_concat_r" 'lhd i e = (pre_concat_r ? i e).
(* The behaviour of ◃ is summarized by the following, easy lemma: *)
lemma eq_to_ex_eq: ∀S.∀A,B:word S → Prop.
- A = B → A =1 B.
+ A = B → A ≐ 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} ≐ \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.
]
].
-notation "a ▹ b" left associative with precedence 60 for @{'tril eclose $a $b}.
+notation "a ▹ b" left associative with precedence 65 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 ?}.
+notation "•" non associative with precedence 65 for @{eclose ?}.
(* We are ready to give the formal definition of the broadcasting operation. *)
| pc i1 i2 ⇒ •i1 ▹ i2
| pk i ⇒ 〈(\fst (•i))^*,true〉].
-notation "• x" non associative with precedence 60 for @{'eclose $x}.
+notation "• x" non associative with precedence 65 for @{'eclose $x}.
interpretation "eclose" 'eclose x = (eclose ? x).
(* Here are a few simple properties of ▹ and •(-) *)
(* We are now ready to state the main semantic properties of ⊕, ◃ and •(-):
-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|}
+sem_oplus: \sem{e1 ⊕ e2} ≐ \sem{e1} ∪ \sem{e2}
+sem_pcl: \sem{e1 ▹ i2} ≐ \sem{e1} · \sem{|i2|} ∪ \sem{i2}
+sem_bullet \sem{•i} ≐ \sem{i} ∪ \sem{|i|}
The proof of sem_oplus is straightforward. *)
lemma sem_oplus: ∀S:DeqSet.∀e1,e2:pre S.
- \sem{e1 ⊕ e2} =1 \sem{e1} ∪ \sem{e2}.
+ \sem{e1 ⊕ e2} ≐ \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/
auxiliary lemma, that assumes sem_bullet:
sem_pcl_aux:
- \sem{•i2} =1 \sem{i2} ∪ \sem{|i2|} →
- \sem{e1 ▹ i2} =1 \sem{e1} · \sem{|i2|} ∪ \sem{i2}.
+ \sem{•i2} ≐ \sem{i2} ∪ \sem{|i2|} →
+ \sem{e1 ▹ i2} ≐ \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. *)
// qed.
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{•i2} ≐ \sem{i2} ∪ \sem{|i2|} →
+ \sem{e1 ▹ i2} ≐ \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 …))
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 - {[ ]}).
+ \sem{e} ≐ \sem{i} ∪ A → \sem{\fst e} ≐ \sem{i} ∪ (A - {[ ]}).
#S #e #i #A #seme
@eqP_trans [|@minus_eps_pre]
@eqP_trans [||@eqP_union_r [|@eqP_sym @minus_eps_item]]
@eqP_substract_r //
qed.
-theorem sem_bull: ∀S:DeqSet. ∀i:pitem S. \sem{•i} =1 \sem{i} ∪ \sem{|i|}.
+theorem sem_bull: ∀S:DeqSet. ∀i:pitem S. \sem{•i} ≐ \sem{i} ∪ \sem{|i|}.
#S #e elim e
[#w normalize % [/2/ | * //]
|/2/
#S #e elim e normalize //
qed.
-lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} =1 ∅.
+lemma sem_blank: ∀S.∀e:re S.\sem{blank S e} ≐ ∅.
#S #e elim e
[1,2:@eq_to_ex_eq //
|#s @eq_to_ex_eq //
qed.
theorem re_embedding: ∀S.∀e:re S.
- \sem{•(blank S e)} =1 \sem{e}.
+ \sem{•(blank S e)} ≐ \sem{e}.
#S #e @eqP_trans [|@sem_bull] >forget_blank
@eqP_trans [|@eqP_union_r [|@sem_blank]]
@eqP_trans [|@union_comm] @union_empty_r.
#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〉} ≐ \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
of ⊙ and ⊛. *)
lemma sem_odot:
- ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
+ ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} ≐ \sem{e1}· \sem{|\fst e2|} ∪ \sem{e2}.
#S #e1 * #i2 *
[>sem_pre_true
@eqP_trans [|@sem_odot_true]
qed.
theorem sem_ostar: ∀S.∀e:pre S.
- \sem{e^⊛} =1 \sem{e} · \sem{|\fst e|}^*.
+ \sem{e^⊛} ≐ \sem{e} · \sem{|\fst e|}^*.
#S * #i #b cases b
[(* lhs = \sem{〈i,true〉^⊛} *)
>sem_pre_true (* >sem_pre_true *)
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