X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=weblib%2Ftutorial%2Fchapter4.ma;h=be8ddaa594212f9b0dacba4a7d7ddb8c81a23982;hb=a2ba04cd90e76937720b16e37cb12a39b46181e3;hp=1be877197f9cff9659a5e5573d508e9af1e2ef2d;hpb=f848433620419e71cd19ec0b4f58c717ac50f85e;p=helm.git

diff --git a/weblib/tutorial/chapter4.ma b/weblib/tutorial/chapter4.ma
index 1be877197..be8ddaa59 100644
--- a/weblib/tutorial/chapter4.ma
+++ b/weblib/tutorial/chapter4.ma
@@ -1,7 +1,9 @@
-(* 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. *)
+(* 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. *)
 
-include "basics/logic.ma".
+include "basics/types.ma".
+include "basics/bool.ma".
 
 (**** For instance the empty set is defined by the always function predicate *)
 
@@ -12,7 +14,7 @@ interpretation "empty set" 'empty_set = (empty_set ?).
 (* Similarly, a singleton set contaning containing an element a, is defined
 by by the characteristic function asserting equality with a *)
 
-definition singleton ≝ λA.λx,a:A.xa title="leibnitz's equality" href="cic:/fakeuri.def(1)"=/aa.
+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).
 
@@ -25,7 +27,7 @@ conjunction and negation *)
 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).
 
-definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a a title="logical and" href="cic:/fakeuri.def(1)"∧/a Q a.
+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).
 
 definition complement ≝ λU:Type[0].λA:U → Prop.λw.a title="logical not" href="cic:/fakeuri.def(1)"¬/a A w.
@@ -43,7 +45,7 @@ interpretation "subset" 'subseteq a b = (subset ? a b).
 (* Two sets are equals if and only if they have the same elements, that is,
 if the two characteristic functions are extensionally equivalent: *) 
 
-definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a a title="iff" href="cic:/fakeuri.def(1)"↔/a Q a.
+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).
 
@@ -63,7 +65,7 @@ lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
 with respect to eqP: *)
 
 lemma eqP_union_r: ∀U.∀A,B,C:U →Prop. 
-  A 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 C a title="union" href="cic:/fakeuri.def(1)"∪/a B.
+  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.
   
 lemma eqP_union_l: ∀U.∀A,B,C:U →Prop. 
@@ -75,7 +77,7 @@ lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
 #U #A #B #C #eqAB #a @a href="cic:/matita/basics/logic/iff_and_r.def(2)"iff_and_r/a @eqAB qed.
   
 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)"∩/a B a title="extensional equality" href="cic:/fakeuri.def(1)"=/a1 A a title="intersection" href="cic:/fakeuri.def(1)"∩/a C.
+  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.
 
 lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop. 
@@ -86,9 +88,9 @@ 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.
 
-(* 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: *)
+(* 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: *)
 
 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.
@@ -101,10 +103,10 @@ lemma union_comm : ∀U.∀A,B:U →Prop.
 
 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(0,2,2)"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/]
+#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
+(* In the same way we prove commutativity and associativity for set 
 interesection *)
 
 lemma cap_comm : ∀U.∀A,B:U →Prop. 
@@ -126,9 +128,8 @@ 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. *)
+(* We conclude our examples with a couple of distributivity theorems, and a 
+characterization of substraction in terms of interesection and complementation. *)
 
 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).
@@ -144,92 +145,174 @@ lemma substract_def:∀U.∀A,B:U→Prop. Aa title="substraction" href="cic:/fa
 #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.
 
-(****** DeqSet: a set with a decidbale equality ******)
+(* 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: *)
 
 record DeqSet : Type[1] ≝ { carr :> Type[0];
-   eqb: carr → carr → bool;
-   eqb_true: ∀x,y. (eqb x y = true) ↔ (x = y)
+   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. *)
+
 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
 interpretation "eqb" 'eqb a b = (eqb ? a b).
 
+(* 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)}. 
   
-lemma eqb_false: ∀S:DeqSet.∀a,b:S. 
-  (eqb ? a b) = false ↔ a ≠ b.
+(* 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. *)
+
+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 
-  [@(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)
+
+(* 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 *) 
+
 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 dec_eq: ∀S:DeqSet.∀a,b:S. a = b ∨ a ≠ b.
-#S #a #b cases (true_or_false (eqb ? a b)) #H
+
+(* The following statement proves that propositional equality in a 
+DeqSet is decidable in the traditional sense, namely either a=b or a≠b *)
+
+ 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.
 
+(* 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 *)
+
 definition beqb ≝ λb1,b2.
-  match b1 with [ true ⇒ b2 | false ⇒ notb 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 }.
-lemma beqb_true: ∀b1,b2. iff (beqb b1 b2 = true) (b1 = b2).
-#b1 #b2 cases b1 cases b2 normalize /span class="autotactic"2span class="autotrace" trace conj/span/span/
+
+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. 
 
-definition DeqBool ≝ mk_DeqSet bool beqb beqb_true.
+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). *)
+
+example exhint: ∀b:a href="cic:/matita/tutorial/chapter4/DeqBool.def(5)"DeqBool/a. (ba 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 → 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.
 
-unification hint  0 ≔ ; 
-    X ≟ mk_DeqSet bool beqb beqb_true
+(* 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
 (* ---------------------------------------- *) ⊢ 
-    bool ≡ carr X.
+    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 ≔ b1,b2:bool; 
-    X ≟ mk_DeqSet bool beqb beqb_true
+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
 (* ---------------------------------------- *) ⊢ 
-    beqb b1 b2 ≡ eqb X b1 b2.
+    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.
     
-example exhint: ∀b:bool. (b == false) = true → b = false. 
-#b #H @(\P H).
+(* After having provided the previous hints, we may rewrite example exhint 
+declaring b of type bool. *)
+ 
+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)"=/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.
 
-(* pairs *)
+(* 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. *)
+
 definition eq_pairs ≝
-  λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
+  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)"=/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)"=/a= a title="pair pi2" href="cic:/fakeuri.def(1)"\snd/a p2).
 
-lemma eq_pairs_true: ∀A,B:DeqSet.∀p1,p2:A×B.
-  eq_pairs A B p1 p2 = true ↔ p1 = p2.
+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 (andb_true …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) //
-  |#H destruct normalize >(\b (refl … a2)) >(\b (refl … 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.
 
-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
+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
 (* ---------------------------------------- *) ⊢ 
-    T1×T2 ≡ carr X.
+    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 ≔ T1,T2,p1,p2; 
-    X ≟ DeqProd T1 T2
+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
 (* ---------------------------------------- *) ⊢ 
-    eq_pairs T1 T2 p1 p2 ≡ eqb X p1 p2.
+    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.
 
 example hint2: ∀b1,b2. 
-  〈b1,true〉==〈false,b2〉=true → 〈b1,true〉=〈false,b2〉.
+  a title="Pair construction" href="cic:/fakeuri.def(1)"〈/ab1,a href="cic:/matita/basics/bool/bool.con(0,1,0)"true/a〉a title="eqb" href="cic:/fakeuri.def(1)"=/a=a title="Pair construction" href="cic:/fakeuri.def(1)"〈/aa href="cic:/matita/basics/bool/bool.con(0,2,0)"false/a,b2〉a 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/a〉a 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,b2〉.
 #b1 #b2 #H @(\P H).
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