2 \ 5h1 class="section"
\ 6Naif Set Theory
\ 5/h1
\ 6
4 include "basics/types.ma".
5 include "basics/bool.ma".
7 In this Chapter we shall develop a naif theory of sets represented as
8 characteristic predicates over some universe
\ 5code
\ 6A
\ 5/code
\ 6, that is as objects of type
10 For instance the empty set is defined by the always false function: *)
12 definition empty_set ≝ λA:Type[0].λa:A.
\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"
\ 6False
\ 5/a
\ 6.
13 notation "\emptyv" non associative with precedence 90 for @{'empty_set}.
14 interpretation "empty set" 'empty_set = (empty_set ?).
16 (* Similarly, a singleton set contaning containing an element a, is defined
17 by by the characteristic function asserting equality with a *)
19 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.
20 (* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
21 interpretation "singleton" 'singl x = (singleton ? x).
23 (* The membership relation between an element of type A and a set S:A →Prop is
24 simply the predicate resulting from the application of S to a.
25 The operations of union, intersection, complement and substraction
26 are easily defined in terms of the propositional connectives of dijunction,
27 conjunction and negation *)
29 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.
30 interpretation "union" 'union a b = (union ? a b).
32 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.
33 interpretation "intersection" 'intersects a b = (intersection ? a b).
35 definition complement ≝ λU:Type[0].λA:U → Prop.λw.
\ 5a title="logical not" href="cic:/fakeuri.def(1)"
\ 6¬
\ 5/a
\ 6 A w.
36 interpretation "complement" 'not a = (complement ? a).
38 definition substraction := 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.
39 interpretation "substraction" 'minus a b = (substraction ? a b).
41 (* Finally, we use implication to define the inclusion relation between
44 definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
45 interpretation "subset" 'subseteq a b = (subset ? a b).
47 (* Two sets are equals if and only if they have the same elements, that is,
48 if the two characteristic functions are extensionally equivalent: *)
50 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.
51 notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
52 interpretation "extensional equality" 'eqP a b = (eqP ? a b).
54 (* This notion of equality is different from the intensional equality of
55 functions; the fact it defines an equivalence relation must be explicitly
58 lemma eqP_sym: ∀U.∀A,B:U →Prop.
59 A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 B → B
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 A.
60 #U #A #B #eqAB #a @
\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"
\ 6iff_sym
\ 5/a
\ 6 @eqAB qed.
62 lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
63 A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 B → B
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 C → A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 C.
64 #U #A #B #C #eqAB #eqBC #a @
\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"
\ 6iff_trans
\ 5/a
\ 6 // qed.
66 (* For the same reason, we must also prove that all the operations behave well
67 with respect to eqP: *)
69 lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
70 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.
71 #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.
73 lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
74 B
\ 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 A
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 C.
75 #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.
77 lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
78 A
\ 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 C
\ 5a title="intersection" href="cic:/fakeuri.def(1)"
\ 6∩
\ 5/a
\ 6 B.
79 #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.
81 lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
82 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.
83 #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.
85 lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
86 A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 C → A
\ 5a title="substraction" 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="substraction" href="cic:/fakeuri.def(1)"
\ 6-
\ 5/a
\ 6 B.
87 #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.
89 lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
90 B
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 C → A
\ 5a title="substraction" 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="substraction" href="cic:/fakeuri.def(1)"
\ 6-
\ 5/a
\ 6 C.
91 #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.
93 (* We can now prove several properties of the previous set-theoretic operations.
94 In particular, union is commutative and associative, and the empty set is an
97 lemma union_empty_r: ∀U.∀A:U→Prop.
98 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 \ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 A.
99 #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/]
102 lemma union_comm : ∀U.∀A,B:U →Prop.
103 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 B
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 A.
104 #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.
106 lemma union_assoc: ∀U.∀A,B,C:U → Prop.
107 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).
108 #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/]
111 (* In the same way we prove commutativity and associativity for set
114 lemma cap_comm : ∀U.∀A,B:U →Prop.
115 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 B
\ 5a title="intersection" href="cic:/fakeuri.def(1)"
\ 6∩
\ 5/a
\ 6 A.
116 #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.
118 lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
119 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)
\ 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 B)
\ 5a title="intersection" href="cic:/fakeuri.def(1)"
\ 6∩
\ 5/a
\ 6 C.
120 #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/ ]
123 (* We can also easily prove idempotency for union and intersection *)
125 lemma union_idemp: ∀U.∀A:U →Prop.
126 A
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 A.
127 #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.
129 lemma cap_idemp: ∀U.∀A:U →Prop.
130 A
\ 5a title="intersection" href="cic:/fakeuri.def(1)"
\ 6∩
\ 5/a
\ 6 A
\ 5a title="extensional equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 61 A.
131 #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.
133 (* We conclude our examples with a couple of distributivity theorems, and a
134 characterization of substraction in terms of interesection and complementation. *)
136 lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
137 (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
\ 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)
\ 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).
138 #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/]
141 lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
142 (A
\ 5a title="union" href="cic:/fakeuri.def(1)"
\ 6∪
\ 5/a
\ 6 B)
\ 5a title="substraction" 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="substraction" 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="substraction" href="cic:/fakeuri.def(1)"
\ 6-
\ 5/a
\ 6 C).
143 #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/]
146 lemma substract_def:∀U.∀A,B:U→Prop. A
\ 5a title="substraction" href="cic:/fakeuri.def(1)"
\ 6-
\ 5/a
\ 6B
\ 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 \ 5a title="complement" href="cic:/fakeuri.def(1)"
\ 6¬
\ 5/a
\ 6B.
147 #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/
150 (* In several situation it is important to assume to have a decidable equality
151 between elements of a set U, namely a boolean function eqb: U→U→bool such that
152 for any pair of elements a and b in U, (eqb x y) is true if and only if x=y.
153 A set equipped with such an equality is called a DeqSet: *)
155 record DeqSet : Type[1] ≝ { carr :> Type[0];
156 eqb: carr → carr →
\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"
\ 6bool
\ 5/a
\ 6;
157 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)
160 (* We use the notation == to denote the decidable equality, to distinguish it
161 from the propositional equality. In particular, a term of the form a==b is a
162 boolean, while a=b is a proposition. *)
164 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
165 interpretation "eqb" 'eqb a b = (eqb ? a b).
167 (* It is convenient to have a simple way to reflect a proof of the fact
168 that (eqb a b) is true into a proof of the proposition (a = b); to this aim,
169 we introduce two operators "\P" and "\b". *)
171 notation "\P H" non associative with precedence 90
172 for @{(proj1 … (eqb_true ???) $H)}.
174 notation "\b H" non associative with precedence 90
175 for @{(proj2 … (eqb_true ???) $H)}.
177 (* If H:eqb a b = true, then \P H: a = b, and conversely if h:a = b, then
178 \b h: eqb a b = true. Let us see an example of their use: the following
179 statement asserts that we can reflect a proof that eqb a b is false into
180 a proof of the proposition a ≠ b. *)
182 lemma eqb_false: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀a,b:S.
183 (
\ 5a href="cic:/matita/tutorial/chapter4/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.
185 (* We start the proof introducing the hypothesis, and then split the "if" and
190 (* The latter is easily reduced to prove the goal true=false under the assumption
192 [@(
\ 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
194 (* since by assumption H false is equal to (a==b), by rewriting we obtain the goal
195 true=(a==b) that is just the boolean version of H1 *)
197 <H @
\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"
\ 6sym_eq
\ 5/a
\ 6 @(\b H1)
199 (* In the "if" case, we proceed by cases over the boolean equality (a==b); if
200 (a==b) is false, the goal is trivial; the other case is absurd, since if (a==b) is
201 true, then by reflection a=b, while by hypothesis a≠b *)
203 |cases (
\ 5a href="cic:/matita/basics/bool/true_or_false.def(1)"
\ 6true_or_false
\ 5/a
\ 6 (
\ 5a href="cic:/matita/tutorial/chapter4/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)
207 (* We also introduce two operators "\Pf" and "\bf" to reflect a proof
208 of (a==b)=false into a proof of a≠b, and vice-versa *)
210 notation "\Pf H" non associative with precedence 90
211 for @{(proj1 … (eqb_false ???) $H)}.
213 notation "\bf H" non associative with precedence 90
214 for @{(proj2 … (eqb_false ???) $H)}.
216 (* The following statement proves that propositional equality in a
217 DeqSet is decidable in the traditional sense, namely either a=b or a≠b *)
219 lemma dec_eq: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/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.
220 #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/tutorial/chapter4/eqb.fix(0,0,3)"
\ 6eqb
\ 5/a
\ 6 ? a b)) #H
221 [%1 @(\P H) | %2 @(\Pf H)]
224 (* A simple example of a set with a decidable equality is bool. We first define
225 the boolean equality beqb, that is just the xand function, then prove that
226 beqb b1 b2 is true if and only if b1=b2, and finally build the type DeqBool by
227 instantiating the DeqSet record with the previous information *)
229 definition beqb ≝ λb1,b2.
230 match b1 with [ true ⇒ b2 | false ⇒
\ 5a href="cic:/matita/basics/bool/notb.def(1)"
\ 6notb
\ 5/a
\ 6 b2].
232 notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
234 lemma beqb_true: ∀b1,b2.
\ 5a href="cic:/matita/basics/logic/iff.def(1)"
\ 6iff
\ 5/a
\ 6 (
\ 5a href="cic:/matita/tutorial/chapter4/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) (b1
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6 b2).
235 #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/
238 definition DeqBool ≝
\ 5a href="cic:/matita/tutorial/chapter4/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/tutorial/chapter4/beqb.def(2)"
\ 6beqb
\ 5/a
\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"
\ 6beqb_true
\ 5/a
\ 6.
240 (* At this point, we would expect to be able to prove things like the
241 following: for any boolean b, if b==false is true then b=false.
242 Unfortunately, this would not work, unless we declare b of type
243 DeqBool (change the type in the following statement and see what
246 example exhint: ∀b:
\ 5a href="cic:/matita/tutorial/chapter4/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.
250 (* The point is that == expects in input a pair of objects whose type must be the
251 carrier of a DeqSet; bool is indeed the carrier of DeqBool, but the type inference
252 system has no knowledge of it (it is an information that has been supplied by the
253 user, and stored somewhere in the library). More explicitly, the type inference
254 inference system, would face an unification problem consisting to unify bool
255 against the carrier of something (a metavaribale) and it has no way to synthetize
256 the answer. To solve this kind of situations, matita provides a mechanism to hint
257 the system the expected solution. A unification hint is a kind of rule, whose rhd
258 is the unification problem, containing some metavariables X1, ..., Xn, and whose
259 left hand side is the solution suggested to the system, in the form of equations
260 Xi=Mi. The hint is accepted by the system if and only the solution is correct, that
261 is, if it is a unifier for the given problem.
262 To make an example, in the previous case, the unification problem is bool = carr X,
263 and the hint is to take X= mk_DeqSet bool beqb true. The hint is correct, since
264 bool is convertible with (carr (mk_DeqSet bool beb true)). *)
266 unification hint 0
\ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"
\ 6≔
\ 5/a
\ 6 ;
267 X ≟
\ 5a href="cic:/matita/tutorial/chapter4/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/tutorial/chapter4/beqb.def(2)"
\ 6beqb
\ 5/a
\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"
\ 6beqb_true
\ 5/a
\ 6
268 (* ---------------------------------------- *) ⊢
269 \ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"
\ 6bool
\ 5/a
\ 6 ≡
\ 5a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"
\ 6carr
\ 5/a
\ 6 X.
271 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;
272 X ≟
\ 5a href="cic:/matita/tutorial/chapter4/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/tutorial/chapter4/beqb.def(2)"
\ 6beqb
\ 5/a
\ 6 \ 5a href="cic:/matita/tutorial/chapter4/beqb_true.def(4)"
\ 6beqb_true
\ 5/a
\ 6
273 (* ---------------------------------------- *) ⊢
274 \ 5a href="cic:/matita/tutorial/chapter4/beqb.def(2)"
\ 6beqb
\ 5/a
\ 6 b1 b2 ≡
\ 5a href="cic:/matita/tutorial/chapter4/eqb.fix(0,0,3)"
\ 6eqb
\ 5/a
\ 6 X b1 b2.
276 (* After having provided the previous hints, we may rewrite example exhint
277 declaring b of type bool. *)
279 example 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.
283 (* The cartesian product of two DeqSets is still a DeqSet. To prove
284 this, we must as usual define the boolen equality function, and prove
285 it correctly reflects propositional equality. *)
287 definition eq_pairs ≝
288 λA,B:
\ 5a href="cic:/matita/tutorial/chapter4/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).
290 lemma eq_pairs_true: ∀A,B:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀p1,p2:A
\ 5a title="Product" href="cic:/fakeuri.def(1)"
\ 6×
\ 5/a
\ 6B.
291 \ 5a href="cic:/matita/tutorial/chapter4/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.
292 #A #B * #a1 #b1 * #a2 #b2 %
293 [#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) //
294 |#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)) //
298 definition DeqProd ≝ λA,B:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.
299 \ 5a href="cic:/matita/tutorial/chapter4/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/tutorial/chapter4/eq_pairs.def(4)"
\ 6eq_pairs
\ 5/a
\ 6 A B) (
\ 5a href="cic:/matita/tutorial/chapter4/eq_pairs_true.def(6)"
\ 6eq_pairs_true
\ 5/a
\ 6 A B).
301 (* Having an unification problem of the kind T1×T2 = carr X, what kind
302 of hint can we give to the system? We expect T1 to be the carrier of a
303 DeqSet C1, T2 to be the carrier of a DeqSet C2, and X to be DeqProd C1 C2.
304 This is expressed by the following hint: *)
306 unification hint 0
\ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"
\ 6≔
\ 5/a
\ 6 C1,C2;
307 T1 ≟
\ 5a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"
\ 6carr
\ 5/a
\ 6 C1,
308 T2 ≟
\ 5a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"
\ 6carr
\ 5/a
\ 6 C2,
309 X ≟
\ 5a href="cic:/matita/tutorial/chapter4/DeqProd.def(7)"
\ 6DeqProd
\ 5/a
\ 6 C1 C2
310 (* ---------------------------------------- *) ⊢
311 T1
\ 5a title="Product" href="cic:/fakeuri.def(1)"
\ 6×
\ 5/a
\ 6T2 ≡
\ 5a href="cic:/matita/tutorial/chapter4/carr.fix(0,0,2)"
\ 6carr
\ 5/a
\ 6 X.
313 unification hint 0
\ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type0"
\ 6≔
\ 5/a
\ 6 T1,T2,p1,p2;
314 X ≟
\ 5a href="cic:/matita/tutorial/chapter4/DeqProd.def(7)"
\ 6DeqProd
\ 5/a
\ 6 T1 T2
315 (* ---------------------------------------- *) ⊢
316 \ 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.
318 example hint2: ∀b1,b2.
319 \ 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〉.