B ≐ C → A ∪ B ≐ A ∪ C.
#U #A #B #C #eqBC #a @iff_or_l @eqBC qed.
+lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
+ A ≐ C → A - B ≐ C - B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+
+lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
+ B ≐ C → A - B ≐ A - C.
+#U #A #B #C #eqBC #a @iff_and_l /2/ qed.
+
+(* set equalities *)
+lemma union_empty_r: ∀U.∀A:U→Prop.
+ A ∪ ∅ ≐ A.
+#U #A #w % [* // normalize #abs @False_ind /2/ | /2/]
+qed.
+
+lemma union_comm : ∀U.∀A,B:U →Prop.
+ A ∪ B ≐ B ∪ A.
+#U #A #B #a % * /2/ qed.
+
+lemma union_assoc: ∀U.∀A,B,C:U → Prop.
+ A ∪ B ∪ C ≐ A ∪ (B ∪ C).
+#S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
+qed.
+
+(*distributivities *)
+
+lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
+ (A ∪ B) ∩ C ≐ (A ∩ C) ∪ (B ∩ C).
+#U #A #B #C #w % [* * /3/ | * * /3/]
+qed.
+
+lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
+ (A ∪ B) - C ≐ (A - C) ∪ (B - C).
+#U #A #B #C #w % [* * /3/ | * * /3/]
+qed.
+
(************************ Sets with decidable equality ************************)
(* We say that a property P:A → Prop over a datatype A is decidable when we have
*)
definition pre_concat_r ≝ λS:DeqSet.λi:pitem S.λe:pre S.
- match e with [ mk_Prod i1 b ⇒ 〈i · i1, b〉].
+ match e with [ pair 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.
+ A = B → A ≐ B.
#S #A #B #H >H #x % // qed.
-(* The behaviour of ◃ is summarized by the following, easy lemma: *)
+(* The behaoviour of ◃ is summarized by the following, easy lemma: *)
lemma sem_pre_concat_r : ∀S,i.∀e:pre S.
- \sem{i ◃ e} =1 \sem{i} · \sem{|fst \dots e|} ∪ \sem{e}.
+ \sem{i ◃ e} \doteq \sem{i} · \sem{|fst \dots e|} ∪ \sem{e}.
#S #i * #i1 #b1 cases b1 [2: @eq_to_ex_eq //]
->sem_pre_true >sem_cat >sem_pre_true /2/
+>sem_pre_true >sem_cat >sem_pre_true whd in match (fst ???); //
qed.
(* The definition of $•(-)$ (eclose) and ▹ (pre_concat_l) are mutually recursive.
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
+ [ pair i1 b1 ⇒ match b1 with
[ 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 \dots (f i), snd \dots (f i) ∨ b〉].
+ [ pair i b ⇒ 〈fst \dots (f i), snd \dots (f i) ∨ b〉].
definition preclose ≝ λS. lift S (eclose S).
interpretation "preclose" 'eclose x = (preclose ? x).
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/
// qed.
lemma sem_pcl_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 >pcl_false >sem_pre_false >sem_pre_false >sem_cat /2/
|#H >pcl_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 \dots e} =1 \sem{i} ∪ (A - {[ ]}).
+ \sem{e} ≐ \sem{i} ∪ A → \sem{fst \dots 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.
lemma erase_odot:∀S.∀e1,e2:pre S.
|fst \dots (e1 ⊙ e2)| = |fst \dots e1| · (|fst \dots e2|).
-#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot //
+#S * #i1 * * #i2 #b2 // >odot_true_b >erase_dot
+whd in match (fst ???) in ⊢ (???%); whd in match (fst ???) in ⊢ (???%);//
qed.
(* Let us come to the star operation: *)
definition lk ≝ λS:DeqSet.λe:pre S.
match e with
- [ mk_Prod i1 b1 ⇒
+ [ pair i1 b1 ⇒
match b1 with
[true ⇒ 〈(fst \dots (eclose ? i1))^*, true〉
|false ⇒ 〈i1^*,false〉
lemma erase_ostar: ∀S.∀e:pre S.
|fst \dots (e^⊛)| = |fst \dots e|^*.
-#S * #i * // qed.
+#S * #i * whd in match (fst ???) in ⊢ (???%); // 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 \dots (e1 ▹ i), snd \dots(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/
+ @eqP_union_l #w normalize % [/2/|* //]
|/2/
]
qed.
of ⊙ and ⊛. *)
lemma sem_odot:
- ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} =1 \sem{e1}· \sem{|fst \dots e2|} ∪ \sem{e2}.
+ ∀S.∀e1,e2: pre S. \sem{e1 ⊙ e2} ≐ \sem{e1}· \sem{|fst \dots 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 \dots e|}^*.
+ \sem{e^⊛} ≐ \sem{e} · \sem{|fst \dots 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 //]]]
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