X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter4.ma;h=7f28c5ddb9cb070a6104d375a2944fbd53f26c77;hb=613d8642b1154dde0c026cbdcd96568910198251;hp=a0a5a938d1785d58bf1422016caad88e5089c7f7;hpb=e8e18e32e06c0406c12a873a01eb489a35b828c8;p=helm.git

diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma
index a0a5a938d..7f28c5ddb 100644
--- a/weblib/tutorial/chapter4.ma
+++ b/weblib/tutorial/chapter4.ma
@@ -1,126 +1,331 @@
+(* 
+h1 class="section"Naif Set Theory/h1
+*)
+include "basics/types.ma".
+include "basics/bool.ma".
+(* 
+In this Chapter we shall develop a naif theory of sets represented as 
+characteristic predicates over some universe codeA/code, that is as objects of type 
+A→Prop. 
+For instance the empty set is defined by the always false function: *)
 
+img class="anchor" src="icons/tick.png" id="empty_set"definition empty_set ≝ λA:Type[0].λa:A.a href="cic:/matita/basics/logic/False.ind(1,0,0)"False/a.
+notation "\emptyv" non associative with precedence 90 for @{'empty_set}.
+interpretation "empty set" 'empty_set = (empty_set ?).
 
-(**************************************************************************)
-(*       ___                                                              *)
-(*      ||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/ 
+(* Similarly, a singleton set contaning containing an element a, is defined
+by by the characteristic function asserting equality with a *)
+
+img class="anchor" src="icons/tick.png" id="singleton"definition singleton ≝ λA.λx,a:A.xa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aspan class="error" title="Parse error: [term] expected after [sym=] (in [term])"/spana.
+(* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
+interpretation "singleton" 'singl x = (singleton ? x).
+
+(* The membership relation between an element of type A and a set S:A →Prop is
+simply the predicate resulting from the application of S to a.
+The operations of union, intersection, complement and substraction 
+are easily defined in terms of the propositional connectives of dijunction,
+conjunction and negation *)
+
+img class="anchor" src="icons/tick.png" id="union"definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a a title="logical or" href="cic:/fakeuri.def(1)"∨/a Q a.
+interpretation "union" 'union a b = (union ? a b).
+
+img class="anchor" src="icons/tick.png" id="intersection"definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a a title="logical and" href="cic:/fakeuri.def(1)"∧/aspan class="error" title="Parse error: [term] expected after [sym∧] (in [term])"/span Q a.
+interpretation "intersection" 'intersects a b = (intersection ? a b).
+
+img class="anchor" src="icons/tick.png" id="complement"definition complement ≝ λU:Type[0].λA:U → Prop.λw.a title="logical not" href="cic:/fakeuri.def(1)"¬/a A w.
+interpretation "complement" 'not a = (complement ? a).
+
+img class="anchor" src="icons/tick.png" id="substraction"definition substraction := λU:Type[0].λA,B:U → Prop.λw.A w a title="logical and" href="cic:/fakeuri.def(1)"∧/a a title="logical not" href="cic:/fakeuri.def(1)"¬/a B w.
+interpretation "substraction" 'minus a b = (substraction ? a b).
+
+(* Finally, we use implication to define the inclusion relation between
+sets *)
+
+img class="anchor" src="icons/tick.png" id="subset"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).
+
+(* 
+h2 class="section"Set Equality/h2
+Two sets are equals if and only if they have the same elements, that is,
+if the two characteristic functions are extensionally equivalent: *) 
+
+img class="anchor" src="icons/tick.png" id="eqP"definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a a title="iff" href="cic:/fakeuri.def(1)"↔/aspan class="error" title="Parse error: [term] expected after [sym↔] (in [term])"/span Q a.
+notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
+interpretation "extensional equality" 'eqP a b = (eqP ? a b).
+
+(*
+This notion of equality is different from the intensional equality of
+functions; the fact it defines an equivalence relation must be explicitly 
+proved: *)
+
+img class="anchor" src="icons/tick.png" id="eqP_sym"lemma eqP_sym: ∀U.∀A,B:U →Prop. 
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B → B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #B #eqAB #a @a href="cic:/matita/basics/logic/iff_sym.def(2)"iff_sym/a @eqAB qed.
+ 
+img class="anchor" src="icons/tick.png" id="eqP_trans"lemma eqP_trans: ∀U.∀A,B,C:U →Prop. 
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B → B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C → A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C.
+#U #A #B #C #eqAB #eqBC #a @a href="cic:/matita/basics/logic/iff_trans.def(2)"iff_trans/a // qed.
+
+(* For the same reason, we must also prove that all the operations behave well
+with respect to eqP: *)
+
+img class="anchor" src="icons/tick.png" id="eqP_union_r"lemma eqP_union_r: ∀U.∀A,B,C:U →Prop. 
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/aspan class="error" title="Parse error: NUMBER '1' or [term] expected after [sym=] (in [term])"/span1 C  → A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="union" href="cic:/fakeuri.def(1)"∪/a B.
+#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_or_r.def(2)"iff_or_r/a @eqAB qed.
+  
+img class="anchor" src="icons/tick.png" id="eqP_union_l"lemma eqP_union_l: ∀U.∀A,B,C:U →Prop. 
+  B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="union" href="cic:/fakeuri.def(1)"∪/a C.
+#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_or_l.def(2)"iff_or_l/a @eqBC qed.
+  
+img class="anchor" src="icons/tick.png" id="eqP_intersect_r"lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop. 
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="intersection" href="cic:/fakeuri.def(1)"∩/a B.
+#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed.
+  
+img class="anchor" src="icons/tick.png" id="eqP_intersect_l"lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop. 
+  B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="intersection" href="cic:/fakeuri.def(1)"∩/aspan class="error" title="Parse error: [term] expected after [sym∩] (in [term])"/span B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C.
+#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_and_l.def(2)"iff_and_l/a @eqBC qed.
+
+img class="anchor" src="icons/tick.png" id="eqP_substract_r"lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop. 
+  A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="substraction" href="cic:/fakeuri.def(1)"-/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C a title="substraction" href="cic:/fakeuri.def(1)"-/a B.
+#U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed.
+  
+img class="anchor" src="icons/tick.png" id="eqP_substract_l"lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop. 
+  B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 C  → A a title="substraction" href="cic:/fakeuri.def(1)"-/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="substraction" href="cic:/fakeuri.def(1)"-/a C.
+#U #A #B #C #eqBC #a @a href="cic:/matita/basics/logic/iff_and_l.def(2)"iff_and_l/a /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/iff_not.def(4)"iff_not/a/span/span/ qed.
+
+(* 
+h2 class="section"Simple properties of sets/h2
+We can now prove several properties of the previous set-theoretic operations. 
+In particular, union is commutative and associative, and the empty set is an 
+identity element: *)
+
+img class="anchor" src="icons/tick.png" id="union_empty_r"lemma union_empty_r: ∀U.∀A:U→Prop. 
+  A a title="union" href="cic:/fakeuri.def(1)"∪/a a title="empty set" href="cic:/fakeuri.def(1)"∅/a a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #w % [* // normalize #abs @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a /span class="autotactic"2span class="autotrace" trace /span/span/ | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/]
 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/ 
+img class="anchor" src="icons/tick.png" id="union_comm"lemma union_comm : ∀U.∀A,B:U →Prop. 
+  A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B a title="union" href="cic:/fakeuri.def(1)"∪/a A.
+#U #A #B #a % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ qed. 
+
+img class="anchor" src="icons/tick.png" id="union_assoc"lemma union_assoc: ∀U.∀A,B,C:U → Prop. 
+  A a title="union" href="cic:/fakeuri.def(1)"∪/a B a title="union" href="cic:/fakeuri.def(1)"∪/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="union" href="cic:/fakeuri.def(1)"∪/a C).
+#S #A #B #C #w % [* [* /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ | /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/ ] | * [/span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a/span/span/ | * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/]
+qed.   
+
+(* In the same way we prove commutativity and associativity for set 
+interesection *)
+
+img class="anchor" src="icons/tick.png" id="cap_comm"lemma cap_comm : ∀U.∀A,B:U →Prop. 
+  A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 B a title="intersection" href="cic:/fakeuri.def(1)"∩/a A.
+#U #A #B #a % * /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ qed. 
+
+img class="anchor" src="icons/tick.png" id="cap_assoc"lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
+  A a title="intersection" href="cic:/fakeuri.def(1)"∩/a (B a title="intersection" href="cic:/fakeuri.def(1)"∩/a C) a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="intersection" href="cic:/fakeuri.def(1)"∩/a B) a title="intersection" href="cic:/fakeuri.def(1)"∩/a C.
+#U #A #B #C #w % [ * #Aw * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ span class="autotactic"span class="autotrace"/span/span| * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ ]
+qed.
+
+(* We can also easily prove idempotency for union and intersection *)
+
+img class="anchor" src="icons/tick.png" id="union_idemp"lemma union_idemp: ∀U.∀A:U →Prop. 
+  A  a title="union" href="cic:/fakeuri.def(1)"∪/a A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #a % [* // | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a/span/span/] qed. 
+
+img class="anchor" src="icons/tick.png" id="cap_idemp"lemma cap_idemp: ∀U.∀A:U →Prop. 
+  A a title="intersection" href="cic:/fakeuri.def(1)"∩/a A a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A.
+#U #A #a % [* // | /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] qed. 
+
+(* We conclude our examples with a couple of distributivity theorems, and a 
+characterization of substraction in terms of interesection and complementation. *)
+
+img class="anchor" src="icons/tick.png" id="distribute_intersect"lemma distribute_intersect : ∀U.∀A,B,C:U→Prop. 
+  (A a title="union" href="cic:/fakeuri.def(1)"∪/a B) a title="intersection" href="cic:/fakeuri.def(1)"∩/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C) a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="intersection" href="cic:/fakeuri.def(1)"∩/a C).
+#U #A #B #C #w % [* * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ | * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] 
 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/] 
+img class="anchor" src="icons/tick.png" id="distribute_substract"lemma distribute_substract : ∀U.∀A,B,C:U→Prop. 
+  (A a title="union" href="cic:/fakeuri.def(1)"∪/a B) a title="substraction" href="cic:/fakeuri.def(1)"-/a C a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 (A a title="substraction" href="cic:/fakeuri.def(1)"-/a C) a title="union" href="cic:/fakeuri.def(1)"∪/a (B a title="substraction" href="cic:/fakeuri.def(1)"-/a C).
+#U #A #B #C #w % [* * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/ | * * /span class="autotactic"3span class="autotrace" trace a href="cic:/matita/basics/logic/Or.con(0,1,2)"or_introl/a, a href="cic:/matita/basics/logic/Or.con(0,2,2)"or_intror/a, a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/] 
 qed.
 
-lemma espilon_in_star: ∀S.∀A:word S → Prop.
-  A^* ϵ.
-#S #A @(ex_intro … [ ]) normalize /2/
+img class="anchor" src="icons/tick.png" id="substract_def"lemma substract_def:∀U.∀A,B:U→Prop. Aa title="substraction" href="cic:/fakeuri.def(1)"-/aB a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a a title="complement" href="cic:/fakeuri.def(1)"¬/aB.
+#U #A #B #w normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/
 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.
+(* 
+h2 class="section"Bool vs. Prop/h2
+In several situation it is important to assume to have a decidable equality 
+between elements of a set U, namely a boolean function eqb: U→U→bool such that
+for any pair of elements a and b in U, (eqb x y) is true if and only if x=y. 
+A set equipped with such an equality is called a DeqSet: *)
+
+img class="anchor" src="icons/tick.png" id="DeqSet"record DeqSet : Type[1] ≝ { carr :> Type[0];
+   eqb: carr → carr → a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a;
+   eqb_true: ∀x,y. (eqb x y a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) a title="iff" href="cic:/fakeuri.def(1)"↔/a (x a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a y)
+}.
+
+(* We use the notation == to denote the decidable equality, to distinguish it
+from the propositional equality. In particular, a term of the form a==b is a 
+boolean, while a=b is a proposition. *)
 
-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 
+notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
+interpretation "eqb" 'eqb a b = (eqb ? a b).
+
+(* 
+h2 class="section"Small Scale Reflection/h2
+It is convenient to have a simple way to reflect a proof of the fact 
+that (eqb a b) is true into a proof of the proposition (a = b); to this aim, 
+we introduce two operators "\P" and "\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)}. 
+  
+(* If H:eqb a b = true, then \P H: a = b, and conversely if h:a = b, then
+\b h: eqb a b = true. Let us see an example of their use: the following 
+statement asserts that we can reflect a proof that eqb a b is false into
+a proof of the proposition a ≠ b. *)
+
+img class="anchor" src="icons/tick.png" id="eqb_false"lemma eqb_false: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a,b:S. 
+  (a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a ? a b) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a a title="iff" href="cic:/fakeuri.def(1)"↔/a a a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a b.
+
+(* We start the proof introducing the hypothesis, and then split the "if" and
+"only if" cases *)
+ 
+#S #a #b % #H 
+
+(* The latter is easily reduced to prove the goal true=false under the assumption
+H1: a = b *)
+  [@(a href="cic:/matita/basics/logic/not_to_not.def(3)"not_to_not/a … a href="cic:/matita/basics/bool/not_eq_true_false.def(3)"not_eq_true_false/a) #H1 
+  
+(* since by assumption H false is equal to (a==b), by rewriting we obtain the goal 
+true=(a==b) that is just the boolean version of H1 *) 
+
+  <H @a href="cic:/matita/basics/logic/sym_eq.def(2)"sym_eq/a @(\b H1)
+
+(* In the "if" case, we proceed by cases over the boolean equality (a==b); if 
+(a==b) is false, the goal is trivial; the other case is absurd, since if (a==b) is
+true, then by reflection a=b, while by hypothesis a≠b *)
+  
+ |cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a ? a b)) // #H1 @a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"False_ind/a @(a href="cic:/matita/basics/logic/absurd.def(2)"absurd/a … (\P H1) H)
   ]
 qed.
+ 
+(* We also introduce two operators "\Pf" and "\bf" to reflect a proof
+of (a==b)=false into a proof of a≠b, and vice-versa *) 
 
-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 "\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)}. 
+
+(* The following statement proves that propositional equality in a 
+DeqSet is decidable in the traditional sense, namely either a=b or a≠b *)
+
+ img class="anchor" src="icons/tick.png" id="dec_eq"lemma dec_eq: ∀S:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀a,b:S. a a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b a title="logical or" href="cic:/fakeuri.def(1)"∨/a a a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"≠/a b.
+#S #a #b cases (a href="cic:/matita/basics/bool/true_or_false.def(1)"true_or_false/a (a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a ? a b)) #H
+  [%1 @(\P H) | %2 @(\Pf H)]
+qed.
+
+(* 
+h2 class="section"Unification Hints/h2
+A simple example of a set with a decidable equality is bool. We first define 
+the boolean equality beqb, that is just the xand function, then prove that 
+beqb b1 b2 is true if and only if b1=b2, and finally build the type DeqBool by 
+instantiating the DeqSet record with the previous information *)
+
+img class="anchor" src="icons/tick.png" id="beqb"definition beqb ≝ λb1,b2.
+  match b1 with [ true ⇒ b2 | false ⇒ a href="cic:/matita/basics/bool/notb.def(1)"notb/a b2].
+
+notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
+
+img class="anchor" src="icons/tick.png" id="beqb_true"lemma beqb_true: ∀b1,b2. a href="cic:/matita/basics/logic/iff.def(1)"iff/a (a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a b1 b2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a) (b1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a b2).
+#b1 #b2 cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace a href="cic:/matita/basics/logic/And.con(0,1,2)"conj/a/span/span/
 qed. 
-     
-lemma star_epsilon: ∀S:DeqSet.∀A:word S → Prop.
-  A^* ∪ {ϵ} =1 A^*.
-#S #A #w % /2/ * // 
+
+img class="anchor" src="icons/tick.png" id="DeqBool"definition DeqBool ≝ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"beqb_true/a.
+
+(* At this point, we would expect to be able to prove things like the
+following: for any boolean b, if b==false is true then b=false. 
+Unfortunately, this would not work, unless we declare b of type 
+DeqBool (change the type in the following statement and see what 
+happens). *)
+
+img class="anchor" src="icons/tick.png" id="exhint"example exhint: ∀b:a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"DeqBool/a. (ba title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → ba title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a.
+#b #H @(\P H) 
 qed.
-  
-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/
-  ]
+
+(* The point is that == expects in input a pair of objects whose type must be the 
+carrier of a DeqSet; bool is indeed the carrier of DeqBool, but the type inference 
+system has no knowledge of it (it is an information that has been supplied by the 
+user, and stored somewhere in the library). More explicitly, the type inference 
+inference system, would face an unification problem consisting to unify bool 
+against the carrier of something (a metavaribale) and it has no way to synthetize 
+the answer. To solve this kind of situations, matita provides a mechanism to hint 
+the system the expected solution. A unification hint is a kind of rule, whose rhd 
+is the unification problem, containing some metavariables X1, ..., Xn, and whose 
+left hand side is the solution suggested to the system, in the form of equations 
+Xi=Mi. The hint is accepted by the system if and only the solution is correct, that
+is, if it is a unifier for the given problem.
+To make an example, in the previous case, the unification problem is bool = carr X,
+and the hint is to take X= mk_DeqSet bool beqb true. The hint is correct, since 
+bool is convertible with (carr (mk_DeqSet bool beb true)). *)
+
+unification hint  0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"≔/a ; 
+    X ≟ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"beqb_true/a
+(* ---------------------------------------- *) ⊢ 
+    a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a ≡ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a X.
+    
+unification hint  0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"≔/a b1,b2:a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a; 
+    X ≟ a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"beqb_true/a
+(* ---------------------------------------- *) ⊢ 
+    a href="cic:/matita/tutorial/chapter4/beqb.def(2)"beqb/a b1 b2 ≡ a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a X b1 b2.
+    
+(* After having provided the previous hints, we may rewrite example exhint 
+declaring b of type bool. *)
+ 
+img class="anchor" src="icons/tick.png" id="exhint1"example exhint1: ∀b:a href="cic:/matita/basics/bool/bool.ind(1,0,0)"bool/a. (b a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a) a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → b a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a. 
+#b #H @(\P 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/
+(* The cartesian product of two DeqSets is still a DeqSet. To prove
+this, we must as usual define the boolen equality function, and prove
+it correctly reflects propositional equality. *)
+
+img class="anchor" src="icons/tick.png" id="eq_pairs"definition eq_pairs ≝
+  λA,B:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.λp1,p2:Aa title="Product" href="cic:/fakeuri.def(1)"×/aB.(a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p1 a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a title="pair pi1" href="cic:/fakeuri.def(1)"\fst/a p2) a title="boolean and" href="cic:/fakeuri.def(1)"∧/a (a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p1 a title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/a a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p2).
+
+img class="anchor" src="icons/tick.png" id="eq_pairs_true"lemma eq_pairs_true: ∀A,B:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.∀p1,p2:Aa title="Product" href="cic:/fakeuri.def(1)"×/aB.
+  a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"eq_pairs/a A B p1 p2 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a a title="iff" href="cic:/fakeuri.def(1)"↔/a p1 a title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/a p2.
+#A #B * #a1 #b1 * #a2 #b2 %
+  [#H cases (a href="cic:/matita/basics/bool/andb_true.def(5)"andb_true/a …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) //
+  |#H destruct normalize >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … a2)) >(\b (a href="cic:/matita/basics/logic/eq.con(0,1,2)"refl/a … b2)) //
   ]
 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]
-qed.
+img class="anchor" src="icons/tick.png" id="DeqProd"definition DeqProd ≝ λA,B:a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"DeqSet/a.
+  a href="cic:/matita/tutorial/chapter4/DeqSet.con(0,1,0)"mk_DeqSet/a (Aa title="Product" href="cic:/fakeuri.def(1)"×/aB) (a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"eq_pairs/a A B) (a href="cic:/matita/tutorial/chapter4/eq_pairs_true.def(6)"eq_pairs_true/a A B).
+
+(* Having an unification problem of the kind T1×T2 = carr X, what kind 
+of hint can we give to the system? We expect T1 to be the carrier of a
+DeqSet C1, T2 to be the carrier of a DeqSet C2, and X to be DeqProd C1 C2.
+This is expressed by the following hint: *)
+
+unification hint  0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"≔/a C1,C2; 
+    T1 ≟ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a C1,
+    T2 ≟ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a C2,
+    X ≟ a href="cic:/matita/tutorial/chapter4/DeqProd.def(7)"DeqProd/a C1 C2
+(* ---------------------------------------- *) ⊢ 
+    T1a title="Product" href="cic:/fakeuri.def(1)"×/aT2 ≡ a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"carr/a X.
+
+unification hint  0 a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"≔/a T1,T2,p1,p2; 
+    X ≟ a href="cic:/matita/tutorial/chapter4/DeqProd.def(7)"DeqProd/a T1 T2
+(* ---------------------------------------- *) ⊢ 
+    a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"eq_pairs/a T1 T2 p1 p2 ≡ a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"eqb/a X p1 p2.
 
+img class="anchor" src="icons/tick.png" id="hint2"example hint2: ∀b1,b2. 
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ab1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="eqb" href="cic:/fakeuri.def(1)"=/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,b2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a → a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ab1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/aa title="Pair construction" href="cic:/fakeuri.def(1)"〉/aa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,b2a title="Pair construction" href="cic:/fakeuri.def(1)"〉/a.
+#b1 #b2 #H @(\P H).
\ No newline at end of file