+include "basics/logic.ma".
+(**** a subset of A is just an object of type A→Prop ****)
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "arithmetics/nat.ma".
-include "basics/lists/list.ma".
-include "basics/sets.ma".
-
-definition word ≝ λS:DeqSet.list S.
-
-notation "ϵ" non associative with precedence 90 for @{ 'epsilon }.
-interpretation "epsilon" 'epsilon = (nil ?).
-
-(* concatenation *)
-definition cat : ∀S,l1,l2,w.Prop ≝
- λS.λl1,l2.λw:word S.∃w1,w2.w1 @ w2 = w ∧ l1 w1 ∧ l2 w2.
-notation "a · b" non associative with precedence 60 for @{ 'middot $a $b}.
-interpretation "cat lang" 'middot a b = (cat ? a b).
-
-let rec flatten (S : DeqSet) (l : list (word S)) on l : word S ≝
-match l with [ nil ⇒ [ ] | cons w tl ⇒ w @ flatten ? tl ].
-
-let rec conjunct (S : DeqSet) (l : list (word S)) (r : word S → Prop) on l: Prop ≝
-match l with [ nil ⇒ True | cons w tl ⇒ r w ∧ conjunct ? tl r ].
-
-(* kleene's star *)
-definition star ≝ λS.λl.λw:word S.∃lw.flatten ? lw = w ∧ conjunct ? lw l.
-notation "a ^ *" non associative with precedence 90 for @{ 'star $a}.
-interpretation "star lang" 'star l = (star ? l).
-
-lemma cat_ext_l: ∀S.∀A,B,C:word S →Prop.
- A =1 C → A · B =1 C · B.
-#S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
-cases (H w1) /6/
+definition empty_set ≝ λA:Type[0].λa:A.False.
+notation "\emptyv" non associative with precedence 90 for @{'empty_set}.
+interpretation "empty set" 'empty_set = (empty_set ?).
+
+definition singleton ≝ λA.λx,a:A.x=a.
+(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
+interpretation "singleton" 'singl x = (singleton ? x).
+
+definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a ∨ Q a.
+interpretation "union" 'union a b = (union ? a b).
+
+definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a ∧ Q a.
+interpretation "intersection" 'intersects a b = (intersection ? a b).
+
+definition complement ≝ λU:Type[0].λA:U → Prop.λw.¬ A w.
+interpretation "complement" 'not a = (complement ? a).
+
+definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w ∧ ¬ B w.
+interpretation "substraction" 'minus a b = (substraction ? a b).
+
+definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
+interpretation "subset" 'subseteq a b = (subset ? a b).
+
+(* extensional equality *)
+definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a ↔ Q a.
+notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
+interpretation "extensional equality" 'eqP a b = (eqP ? a b).
+
+lemma eqP_sym: ∀U.∀A,B:U →Prop.
+ A =1 B → B =1 A.
+#U #A #B #eqAB #a @iff_sym @eqAB qed.
+
+lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
+ A =1 B → B =1 C → A =1 C.
+#U #A #B #C #eqAB #eqBC #a @iff_trans // qed.
+
+lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A ∪ B =1 C ∪ B.
+#U #A #B #C #eqAB #a @iff_or_r @eqAB qed.
+
+lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A ∪ B =1 A ∪ C.
+#U #A #B #C #eqBC #a @iff_or_l @eqBC qed.
+
+lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A ∩ B =1 C ∩ B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+
+lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A ∩ B =1 A ∩ C.
+#U #A #B #C #eqBC #a @iff_and_l @eqBC qed.
+
+lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A - B =1 C - B.
+#U #A #B #C #eqAB #a @iff_and_r @eqAB qed.
+
+lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A - B =1 A - C.
+#U #A #B #C #eqBC #a @iff_and_l /2/ qed.
+
+(* set equalities *)
+lemma union_empty_r: ∀U.∀A:U→Prop.
+ A ∪ ∅ =1 A.
+#U #A #w % [* // normalize #abs @False_ind /2/ | /2/]
qed.
-lemma cat_ext_r: ∀S.∀A,B,C:word S →Prop.
- B =1 C → A · B =1 A · C.
-#S #A #B #C #H #w % * #w1 * #w2 * * #eqw #inw1 #inw2
-cases (H w2) /6/
+lemma union_comm : ∀U.∀A,B:U →Prop.
+ A ∪ B =1 B ∪ A.
+#U #A #B #a % * /2/ qed.
+
+lemma union_assoc: ∀U.∀A,B,C:U → Prop.
+ A ∪ B ∪ C =1 A ∪ (B ∪ C).
+#S #A #B #C #w % [* [* /3/ | /3/] | * [/3/ | * /3/]
+qed.
+
+lemma cap_comm : ∀U.∀A,B:U →Prop.
+ A ∩ B =1 B ∩ A.
+#U #A #B #a % * /2/ qed.
+
+lemma union_idemp: ∀U.∀A:U →Prop.
+ A ∪ A =1 A.
+#U #A #a % [* // | /2/] qed.
+
+lemma cap_idemp: ∀U.∀A:U →Prop.
+ A ∩ A =1 A.
+#U #A #a % [* // | /2/] qed.
+
+(*distributivities *)
+
+lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
+ (A ∪ B) ∩ C =1 (A ∩ C) ∪ (B ∩ C).
+#U #A #B #C #w % [* * /3/ | * * /3/]
qed.
-lemma distr_cat_r: ∀S.∀A,B,C:word S →Prop.
- (A ∪ B) · C =1 A · C ∪ B · C.
-#S #A #B #C #w %
- [* #w1 * #w2 * * #eqw * /6/ |* * #w1 * #w2 * * /6/]
+lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
+ (A ∪ B) - C =1 (A - C) ∪ (B - C).
+#U #A #B #C #w % [* * /3/ | * * /3/]
qed.
-lemma espilon_in_star: ∀S.∀A:word S → Prop.
- A^* ϵ.
-#S #A @(ex_intro … [ ]) normalize /2/
+(* substraction *)
+lemma substract_def:∀U.∀A,B:U→Prop. A-B =1 A ∩ ¬B.
+#U #A #B #w normalize /2/
qed.
-lemma cat_to_star:∀S.∀A:word S → Prop.
- ∀w1,w2. A w1 → A^* w2 → A^* (w1@w2).
-#S #A #w1 #w2 #Aw * #l * #H #H1 @(ex_intro … (w1::l))
-% normalize /2/
-qed.
+include "basics/types.ma".
+include "basics/bool.ma".
-lemma fix_star: ∀S.∀A:word S → Prop.
- A^* =1 A · A^* ∪ {ϵ}.
-#S #A #w %
- [* #l generalize in match w; -w cases l [normalize #w * /2/]
- #w1 #tl #w * whd in ⊢ ((??%?)→?); #eqw whd in ⊢ (%→?); *
- #w1A #cw1 %1 @(ex_intro … w1) @(ex_intro … (flatten S tl))
- % /2/ whd @(ex_intro … tl) /2/
- |* [2: whd in ⊢ (%→?); #eqw <eqw //]
- * #w1 * #w2 * * #eqw <eqw @cat_to_star
- ]
-qed.
+(****** DeqSet: a set with a decidbale equality ******)
+
+record DeqSet : Type[1] ≝ { carr :> Type[0];
+ eqb: carr → carr → bool;
+ eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
+}.
-lemma star_fix_eps : ∀S.∀A:word S → Prop.
- A^* =1 (A - {ϵ}) · A^* ∪ {ϵ}.
-#S #A #w %
- [* #l elim l
- [* whd in ⊢ ((??%?)→?); #eqw #_ %2 <eqw //
- |* [#tl #Hind * #H * #_ #H2 @Hind % [@H | //]
- |#a #w1 #tl #Hind * whd in ⊢ ((??%?)→?); #H1 * #H2 #H3 %1
- @(ex_intro … (a::w1)) @(ex_intro … (flatten S tl)) %
- [% [@H1 | normalize % /2/] |whd @(ex_intro … tl) /2/]
- ]
- ]
- |* [* #w1 * #w2 * * #eqw * #H1 #_ <eqw @cat_to_star //
- | whd in ⊢ (%→?); #H <H //
- ]
+notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
+interpretation "eqb" 'eqb a b = (eqb ? a b).
+
+notation "\P H" non associative with precedence 90
+ for @{(proj1 … (eqb_true ???) $H)}.
+
+notation "\b H" non associative with precedence 90
+ for @{(proj2 … (eqb_true ???) $H)}.
+
+lemma eqb_false: ∀S:DeqSet.∀a,b:S.
+ (eqb ? a b) = false ↔ a ≠ b.
+#S #a #b % #H
+ [@(not_to_not … not_eq_true_false) #H1 <H @sym_eq @(\b H1)
+ |cases (true_or_false (eqb ? a b)) // #H1 @False_ind @(absurd … (\P H1) H)
]
-qed.
-
-lemma star_epsilon: ∀S:DeqSet.∀A:word S → Prop.
- A^* ∪ {ϵ} =1 A^*.
-#S #A #w % /2/ * //
qed.
+
+notation "\Pf H" non associative with precedence 90
+ for @{(proj1 … (eqb_false ???) $H)}.
+
+notation "\bf H" non associative with precedence 90
+ for @{(proj2 … (eqb_false ???) $H)}.
-lemma epsilon_cat_r: ∀S.∀A:word S →Prop.
- A · {ϵ} =1 A.
-#S #A #w %
- [* #w1 * #w2 * * #eqw #inw1 normalize #eqw2 <eqw //
- |#inA @(ex_intro … w) @(ex_intro … [ ]) /3/
- ]
+lemma dec_eq: ∀S:DeqSet.∀a,b:S. a = b ∨ a ≠ b.
+#S #a #b cases (true_or_false (eqb ? a b)) #H
+ [%1 @(\P H) | %2 @(\Pf H)]
qed.
-lemma epsilon_cat_l: ∀S.∀A:word S →Prop.
- {ϵ} · A =1 A.
-#S #A #w %
- [* #w1 * #w2 * * #eqw normalize #eqw2 <eqw <eqw2 //
- |#inA @(ex_intro … ϵ) @(ex_intro … w) /3/
- ]
+definition beqb ≝ λb1,b2.
+ match b1 with [ true ⇒ b2 | false ⇒ notb b2].
+
+notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
+lemma beqb_true: ∀b1,b2. iff (beqb b1 b2 = true) (b1 = b2).
+#b1 #b2 cases b1 cases b2 normalize /2/
+qed.
+
+definition DeqBool ≝ mk_DeqSet bool beqb beqb_true.
+
+unification hint 0 ≔ ;
+ X ≟ mk_DeqSet bool beqb beqb_true
+(* ---------------------------------------- *) ⊢
+ bool ≡ carr X.
+
+unification hint 0 ≔ b1,b2:bool;
+ X ≟ mk_DeqSet bool beqb beqb_true
+(* ---------------------------------------- *) ⊢
+ beqb b1 b2 ≡ eqb X b1 b2.
+
+example exhint: ∀b:bool. (b == false) = true → b = false.
+#b #H @(\P H).
qed.
-lemma distr_cat_r_eps: ∀S.∀A,C:word S →Prop.
- (A ∪ {ϵ}) · C =1 A · C ∪ C.
-#S #A #C @eqP_trans [|@distr_cat_r |@eqP_union_l @epsilon_cat_l]
+(* pairs *)
+definition eq_pairs ≝
+ λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
+
+lemma eq_pairs_true: ∀A,B:DeqSet.∀p1,p2:A×B.
+ eq_pairs A B p1 p2 = true ↔ p1 = p2.
+#A #B * #a1 #b1 * #a2 #b2 %
+ [#H cases (andb_true …H) #eqa #eqb >(\P eqa) >(\P eqb) //
+ |#H destruct normalize >(\b (refl … a2)) >(\b (refl … b2)) //
+ ]
qed.
+definition DeqProd ≝ λA,B:DeqSet.
+ mk_DeqSet (A×B) (eq_pairs A B) (eq_pairs_true A B).
+
+unification hint 0 ≔ C1,C2;
+ T1 ≟ carr C1,
+ T2 ≟ carr C2,
+ X ≟ DeqProd C1 C2
+(* ---------------------------------------- *) ⊢
+ T1×T2 ≡ carr X.
+
+unification hint 0 ≔ T1,T2,p1,p2;
+ X ≟ DeqProd T1 T2
+(* ---------------------------------------- *) ⊢
+ eq_pairs T1 T2 p1 p2 ≡ eqb X p1 p2.
+
+example hint2: ∀b1,b2.
+ 〈b1,true〉==〈false,b2〉=true → 〈b1,true〉=〈false,b2〉.
+#b1 #b2 #H @(\P H).
\ No newline at end of file
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