-(* 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. *)
-
+(*
+\ 5h1 class="section"\ 6Naif Set Theory\ 5/h1\ 6
+*)
include "basics/types.ma".
include "basics/bool.ma".
-
-(**** For instance the empty set is defined by the always function predicate *)
+(*
+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: *)
definition empty_set ≝ λA:Type[0].λa:A.\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
notation "\emptyv" non associative with precedence 90 for @{'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.x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6a.
+definition singleton ≝ λA.λx,a:A.x\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 6a.
(* 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 \ 5a title="logical or" href="cic:/fakeuri.def(1)"\ 6∨\ 5/a\ 6 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 \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 Q a.
+definition 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\ 5span class="error" title="Parse error: [term] expected after [sym∧] (in [term])"\ 6\ 5/span\ 6 Q a.
interpretation "intersection" 'intersects a b = (intersection ? a b).
definition complement ≝ λU:Type[0].λA:U → Prop.λw.\ 5a title="logical not" href="cic:/fakeuri.def(1)"\ 6¬\ 5/a\ 6 A w.
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).
-(* Two sets are equals if and only if they have the same elements, that is,
+(* \ 5h2 class="section"\ 6Set Quality\ 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: *)
-definition 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.
+definition eqP ≝ λA:Type[0].λP,Q:A → Prop.∀a:A.P a \ 5a title="iff" href="cic:/fakeuri.def(1)"\ 6↔\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym↔] (in [term])"\ 6\ 5/span\ 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).
-(* This notion of equality is different from the intensional equality of
+(*
+This notion of equality is different from the intensional equality of
functions; the fact it defines an equivalence relation must be explicitly
proved: *)
with respect to eqP: *)
lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
- A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B.
+ A \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5span class="error" title="Parse error: NUMBER '1' or [term] expected after [sym=] (in [term])"\ 6\ 5/span\ 61 C → A \ 5a title="union" href="cic:/fakeuri.def(1)"\ 6∪\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 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.
lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
#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.
lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
- B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6 C.
+ B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 C → A \ 5a title="intersection" href="cic:/fakeuri.def(1)"\ 6∩\ 5/a\ 6\ 5span class="error" title="Parse error: [term] expected after [sym∩] (in [term])"\ 6\ 5/span\ 6 B \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 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.
lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
lemma 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 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 61 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(0,2,2)"\ 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/]
+#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
#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.
-(* In several situation it is important to assume to have a decidable equality
+(*
+\ 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: *)
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
+(*
+\ 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". *)
[%1 @(\P H) | %2 @(\Pf H)]
qed.
-(* A simple example of a set with a decidable equality is bool. We first define
+(*
+\ 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, 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 *)
\ 5a href="cic:/matita/tutorial/chapter4/eq_pairs.def(4)"\ 6eq_pairs\ 5/a\ 6 T1 T2 p1 p2 ≡ \ 5a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"\ 6eqb\ 5/a\ 6 X p1 p2.
example hint2: ∀b1,b2.
- \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=\ 5a title="Pair construction" 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="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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" 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\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="eqb" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6=\ 5a title="Pair construction" 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="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="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6b1,\ 5a href="cic:/matita/basics/bool/bool.con(0,1,0)"\ 6true\ 5/a\ 6〉\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6\ 5a title="Pair construction" 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〉.
#b1 #b2 #H @(\P H).
\ No newline at end of file
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