--- /dev/null
+(*
+\ 5h1 class="section"\ 6Naif Set Theory\ 5/h1\ 6
+*)
+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 \ 5code\ 6A\ 5/code\ 6, that is as objects of type
+A→Prop.
+For instance the empty set is defined by the always false function: *)
+
+\ 5img class="anchor" src="icons/tick.png" id="empty_set"\ 6definition empty_set ≝ λA:Type[0].λa:A.\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
+notation "∅" non associative with precedence 90 for @{'empty_set}.
+interpretation "empty set" 'empty_set = (empty_set ?).
+
+(* Similarly, a singleton set containing an element a, is defined
+by the characteristic function asserting equality with a *)
+
+\ 5img class="anchor" src="icons/tick.png" id="singleton"\ 6definition singleton ≝ λA.λx,a:A.x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6a.
+(* notation "{x}" non associative with precedence 90 for @{'singl $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 *)
+
+\ 5img class="anchor" src="icons/tick.png" id="union"\ 6definition union : ∀A:Type[0].∀P,Q.A → Prop ≝ λA,P,Q,a.P a \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 Q a.
+interpretation "union" 'union a b = (union ? a b).
+
+\ 5img class="anchor" src="icons/tick.png" id="intersection"\ 6definition intersection : ∀A:Type[0].∀P,Q.A→Prop ≝ λA,P,Q,a.P a \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 Q a.
+interpretation "intersection" 'intersects a b = (intersection ? a b).
+
+\ 5img class="anchor" src="icons/tick.png" id="complement"\ 6definition complement ≝ λU:Type[0].λA:U → Prop.λw.\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A w.
+interpretation "complement" 'not a = (complement ? a).
+
+\ 5img class="anchor" src="icons/tick.png" id="difference"\ 6definition difference := λU:Type[0].λA,B:U → Prop.λw.A w \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 \ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 B w.
+interpretation "difference" 'minus a b = (difference ? a b).
+
+(* Finally, we use implication to define the inclusion relation between
+sets *)
+
+\ 5img class="anchor" src="icons/tick.png" id="subset"\ 6definition 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).
+
+(*
+\ 5h2 class="section"\ 6Set Equality\ 5/h2\ 6
+Two sets are equals if and only if they have the same elements, that is,
+if the two characteristic functions are extensionally equivalent: *)
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP"\ 6definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 Q a.
+notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
+interpretation "extensional equality" 'eqP a b = (eqP ? a b).
+
+(* the fact it defines an equivalence relation must be explicitly proved: *)
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_sym"\ 6lemma eqP_sym: ∀U.∀A,B:U →Prop.
+ A =1 B → B =1 A.
+#U #A #B #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"\ 6iff_sym\ 5/a\ 6 @eqAB qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_trans"\ 6lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
+ A =1 B → B =1 C → A =1 C.
+#U #A #B #C #eqAB #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"\ 6iff_trans\ 5/a\ 6 // qed.
+
+(* For the same reason, we must also prove that all the operations behave well
+with respect to eqP: *)
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_union_r"\ 6lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B =1 C \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B.
+#U #A #B #C #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_or_r.def(2)"\ 6iff_or_r\ 5/a\ 6 @eqAB qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_union_l"\ 6lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B =1 A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C.
+#U #A #B #C #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_or_l.def(2)"\ 6iff_or_l\ 5/a\ 6 @eqBC qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_intersect_r"\ 6lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B =1 C \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B.
+#U #A #B #C #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_and_r.def(2)"\ 6iff_and_r\ 5/a\ 6 @eqAB qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_intersect_l"\ 6lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B =1 A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C.
+#U #A #B #C #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_and_l.def(2)"\ 6iff_and_l\ 5/a\ 6 @eqBC qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_substract_r"\ 6lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
+ A =1 C → A \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 B =1 C \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 B.
+#U #A #B #C #eqAB #a @\ 5a href="cic:/matita/basics/logic/iff_and_r.def(2)"\ 6iff_and_r\ 5/a\ 6 @eqAB qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="eqP_substract_l"\ 6lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
+ B =1 C → A \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 B =1 A \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C.
+#U #A #B #C #eqBC #a @\ 5a href="cic:/matita/basics/logic/iff_and_l.def(2)"\ 6iff_and_l\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/iff_not.def(4)"\ 6iff_not\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+
+(*
+\ 5h2 class="section"\ 6Simple properties of sets\ 5/h2\ 6
+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: *)
+
+\ 5img class="anchor" src="icons/tick.png" id="union_empty_r"\ 6lemma union_empty_r: ∀U.∀A:U→Prop.
+ A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 \ 5a title="empty set" href="cic:/fakeuri.def(1)"\ 6∅\ 5/a\ 6 =1 A.
+#U #A #w % [* // normalize #abs @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="union_comm"\ 6lemma union_comm : ∀U.∀A,B:U →Prop.
+ A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B =1 B \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 A.
+#U #A #B #a % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="union_assoc"\ 6lemma union_assoc: ∀U.∀A,B,C:U → Prop.
+ A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C =1 A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (B \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 C).
+#S #A #B #C #w % [* [* /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ] | * [/\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
+qed.
+
+(* In the same way we prove commutativity and associativity for set
+interesection *)
+
+\ 5img class="anchor" src="icons/tick.png" id="cap_comm"\ 6lemma cap_comm : ∀U.∀A,B:U →Prop.
+ A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B =1 B \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 A.
+#U #A #B #a % * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="cap_assoc"\ 6lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
+ A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 (B \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C) =1 (A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B) \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C.
+#U #A #B #C #w % [ * #Aw * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ \ 5span class="autotactic"\ 6\ 5span class="autotrace"\ 6\ 5/span\ 6\ 5/span\ 6| * * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ ]
+qed.
+
+(* We can also easily prove idempotency for union and intersection *)
+
+\ 5img class="anchor" src="icons/tick.png" id="union_idemp"\ 6lemma union_idemp: ∀U.∀A:U →Prop.
+ A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 A =1 A.
+#U #A #a % [* // | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/] qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="cap_idemp"\ 6lemma cap_idemp: ∀U.∀A:U →Prop.
+ A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 A =1 A.
+#U #A #a % [* // | /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/] qed.
+
+(* We conclude our examples with a couple of distributivity theorems, and a
+characterization of substraction in terms of interesection and complementation. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="distribute_intersect"\ 6lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
+ (A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B) \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C =1 (A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (B \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C).
+#U #A #B #C #w % [* * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="distribute_substract"\ 6lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
+ (A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B) \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C =1 (A \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C) \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 (B \ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6 C).
+#U #A #B #C #w % [* * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/ | * * /\ 5span class="autotactic"\ 63\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/Or.con(0,1,2)"\ 6or_introl\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/Or.con(0,2,2)"\ 6or_intror\ 5/a\ 6, \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/]
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="substract_def"\ 6lemma substract_def:∀U.∀A,B:U→Prop. A\ 5a title="difference" href="cic:/fakeuri.def(1)"\ 6-\ 5/a\ 6B =1 A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 \ 5a title="complement" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6B.
+#U #A #B #w normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+qed.
+
+(*
+\ 5h2 class="section"\ 6Bool vs. Prop\ 5/h2\ 6
+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: *)
+
+\ 5img class="anchor" src="icons/tick.png" id="DeqSet"\ 6record DeqSet : Type[1] ≝ { carr :> Type[0];
+ eqb: carr → carr → \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6;
+ eqb_true: ∀x,y. (eqb x y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6) \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 (x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 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).
+
+(*
+\ 5h2 class="section"\ 6Small Scale Reflection\ 5/h2\ 6
+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. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="eqb_false"\ 6lemma eqb_false: ∀S:\ 5a href="cic:/matita/cicm2012/part2/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a,b:S.
+ (\ 5a href="cic:/matita/cicm2012/part2/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 ? a b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 a \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b.
+#S #a #b %
+(* same tactic on two goals *)
+#H
+ [@(\ 5a href="cic:/matita/basics/logic/not_to_not.def(3)"\ 6not_to_not\ 5/a\ 6 … \ 5a href="cic:/matita/basics/bool/not_eq_true_false.def(3)"\ 6not_eq_true_false\ 5/a\ 6) #H1
+ <H @\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"\ 6sym_eq\ 5/a\ 6 @(\b H1)
+ |cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/cicm2012/part2/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 ? a b)) // #H1 @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … (\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)}.
+
+(* The following statement proves that propositional equality in a
+DeqSet is decidable in the traditional sense, namely either a=b or a≠b *)
+
+ \ 5img class="anchor" src="icons/tick.png" id="dec_eq"\ 6lemma dec_eq: ∀S:\ 5a href="cic:/matita/cicm2012/part2/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀a,b:S. a \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 a \ 5a title="leibnitz's non-equality" href="cic:/fakeuri.def(1)"\ 6≠\ 5/a\ 6 b.
+#S #a #b cases (\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"\ 6true_or_false\ 5/a\ 6 (\ 5a href="cic:/matita/cicm2012/part2/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 ? a b)) #H
+ [%1 @(\P H) | %2 @(\Pf H)]
+qed.
+
+(*
+\ 5h2 class="section"\ 6Unification Hints\ 5/h2\ 6
+A simple example of a set with a decidable equality is bool. We first define
+the boolean equality beqb, 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 *)
+
+\ 5img class="anchor" src="icons/tick.png" id="beqb"\ 6definition beqb ≝ λb1,b2.
+ match b1 with [ true ⇒ b2 | false ⇒ \ 5a href="cic:/matita/basics/bool/notb.def(1)"\ 6notb\ 5/a\ 6 b2].
+
+notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
+\ 5img class="anchor" src="icons/tick.png" id="beqb_true"\ 6lemma beqb_true: ∀b1,b2. \ 5a href="cic:/matita/cicm2012/part2/beqb.def(2)"\ 6beqb\ 5/a\ 6 b1 b2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 b1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b2.
+#b1 #b2 cases b1 cases b2 normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5a href="cic:/matita/basics/logic/And.con(0,1,2)"\ 6conj\ 5/a\ 6\ 5/span\ 6\ 5/span\ 6/
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="DeqBool"\ 6definition DeqBool ≝ \ 5a href="cic:/matita/cicm2012/part2/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/cicm2012/part2/beqb.def(2)"\ 6beqb\ 5/a\ 6 \ 5a href="cic:/matita/cicm2012/part2/beqb_true.def(4)"\ 6beqb_true\ 5/a\ 6.
+
+(* 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). *)
+
+\ 5img class="anchor" src="icons/tick.png" id="exhint"\ 6example exhint: ∀b:\ 5a href="cic:/matita/cicm2012/part2/DeqBool.def(5)"\ 6DeqBool\ 5/a\ 6. (b=\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#b #H @(\P H)
+qed.
+
+(* 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 \ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"\ 6≔\ 5/a\ 6 ;
+ X ≟ \ 5a href="cic:/matita/cicm2012/part2/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/cicm2012/part2/beqb.def(2)"\ 6beqb\ 5/a\ 6 \ 5a href="cic:/matita/cicm2012/part2/beqb_true.def(4)"\ 6beqb_true\ 5/a\ 6
+(* ---------------------------------------- *) ⊢
+ \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 ≡ \ 5a href="cic:/matita/cicm2012/part2/carr.fix(0,0,2)"\ 6carr\ 5/a\ 6 X.
+
+unification hint 0 \ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"\ 6≔\ 5/a\ 6 b1,b2:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6;
+ X ≟ \ 5a href="cic:/matita/cicm2012/part2/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6 \ 5a href="cic:/matita/cicm2012/part2/beqb.def(2)"\ 6beqb\ 5/a\ 6 \ 5a href="cic:/matita/cicm2012/part2/beqb_true.def(4)"\ 6beqb_true\ 5/a\ 6
+(* ---------------------------------------- *) ⊢
+ \ 5a href="cic:/matita/cicm2012/part2/beqb.def(2)"\ 6beqb\ 5/a\ 6 b1 b2 ≡ \ 5a href="cic:/matita/cicm2012/part2/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 X b1 b2.
+
+(* After having provided the previous hints, we may rewrite example exhint
+declaring b of type bool. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="exhint1"\ 6example exhint1: ∀b:\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"\ 6bool\ 5/a\ 6. (b =\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6.
+#b #H @(\P H)
+qed.
+
+(* 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. *)
+
+\ 5img class="anchor" src="icons/tick.png" id="eq_pairs"\ 6definition eq_pairs ≝
+ λA,B:\ 5a href="cic:/matita/cicm2012/part2/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.λp1,p2:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B.(\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p1 =\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p2) \ 5a title="boolean and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p1 =\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p2).
+
+\ 5img class="anchor" src="icons/tick.png" id="eq_pairs_true"\ 6lemma eq_pairs_true: ∀A,B:\ 5a href="cic:/matita/cicm2012/part2/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.∀p1,p2:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B.
+ \ 5a href="cic:/matita/cicm2012/part2/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 A B p1 p2 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6 p1 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 p2.
+#A #B * #a1 #b1 * #a2 #b2 %
+ [#H cases (\ 5a href="cic:/matita/basics/bool/andb_true.def(5)"\ 6andb_true\ 5/a\ 6 …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) //
+ |#H destruct normalize >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … a2)) >(\b (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 … b2)) //
+ ]
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="DeqProd"\ 6definition DeqProd ≝ λA,B:\ 5a href="cic:/matita/cicm2012/part2/DeqSet.ind(1,0,0)"\ 6DeqSet\ 5/a\ 6.
+ \ 5a href="cic:/matita/cicm2012/part2/DeqSet.con(0,1,0)"\ 6mk_DeqSet\ 5/a\ 6 (A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B) (\ 5a href="cic:/matita/cicm2012/part2/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 A B) (\ 5a href="cic:/matita/cicm2012/part2/eq_pairs_true.def(6)"\ 6eq_pairs_true\ 5/a\ 6 A B).
+
+(* Having a 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 \ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"\ 6≔\ 5/a\ 6 C1,C2;
+ T1 ≟ \ 5a href="cic:/matita/cicm2012/part2/carr.fix(0,0,2)"\ 6carr\ 5/a\ 6 C1,
+ T2 ≟ \ 5a href="cic:/matita/cicm2012/part2/carr.fix(0,0,2)"\ 6carr\ 5/a\ 6 C2,
+ X ≟ \ 5a href="cic:/matita/cicm2012/part2/DeqProd.def(7)"\ 6DeqProd\ 5/a\ 6 C1 C2
+(* ---------------------------------------- *) ⊢
+ T1\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6T2 ≡ \ 5a href="cic:/matita/cicm2012/part2/carr.fix(0,0,2)"\ 6carr\ 5/a\ 6 X.
+
+unification hint 0 \ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"\ 6≔\ 5/a\ 6 T1,T2,p1,p2;
+ X ≟ \ 5a href="cic:/matita/cicm2012/part2/DeqProd.def(7)"\ 6DeqProd\ 5/a\ 6 T1 T2
+(* ---------------------------------------- *) ⊢
+ \ 5a href="cic:/matita/cicm2012/part2/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 T1 T2 p1 p2 ≡ \ 5a href="cic:/matita/cicm2012/part2/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 X p1 p2.
+
+\ 5img class="anchor" src="icons/tick.png" id="hint2"\ 6example hint2: ∀b1,b2.
+ 〈b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6=\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6〈\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,b2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6 → 〈b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6〈\ 5a href="cic:/matita/basics/bool/bool.con(0,2,0)"\ 6false\ 5/a\ 6,b2\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
+#b1 #b2 #H @(\P H)
+qed.
projects / helm.git / blobdiff