1 (* In this Chapter we shall develop a naif theory of sets represented as characteristic
2 predicates over some universe
\ 5code
\ 6A
\ 5/code
\ 6, that is as objects of type A→Prop. *)
4 include "basics/types.ma".
5 include "basics/bool.ma".
7 (**** For instance the empty set is defined by the always function predicate *)
9 definition empty_set ≝ λA:Type[0].λa:A.
\ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"
\ 6False
\ 5/a
\ 6.
10 notation "\emptyv" non associative with precedence 90 for @{'empty_set}.
11 interpretation "empty set" 'empty_set = (empty_set ?).
13 (* Similarly, a singleton set contaning containing an element a, is defined
14 by by the characteristic function asserting equality with a *)
16 definition singleton ≝ λA.λx,a:A.x
\ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"
\ 6=
\ 5/a
\ 6a.
17 (* notation "{x}" non associative with precedence 90 for @{'sing_lang $x}. *)
18 interpretation "singleton" 'singl x = (singleton ? x).
20 (* The membership relation between an element of type A and a set S:A →Prop is
21 simply the predicate resulting from the application of S to a.
22 The operations of union, intersection, complement and substraction
23 are easily defined in terms of the propositional connectives of dijunction,
24 conjunction and negation *)
26 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.
27 interpretation "union" 'union a b = (union ? a b).
29 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.
30 interpretation "intersection" 'intersects a b = (intersection ? a b).
32 definition complement ≝ λU:Type[0].λA:U → Prop.λw.
\ 5a title="logical not" href="cic:/fakeuri.def(1)"
\ 6¬
\ 5/a
\ 6 A w.
33 interpretation "complement" 'not a = (complement ? a).
35 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.
36 interpretation "substraction" 'minus a b = (substraction ? a b).
38 (* Finally, we use implication to define the inclusion relation between
41 definition subset: ∀A:Type[0].∀P,Q:A→Prop.Prop ≝ λA,P,Q.∀a:A.(P a → Q a).
42 interpretation "subset" 'subseteq a b = (subset ? a b).
44 (* Two sets are equals if and only if they have the same elements, that is,
45 if the two characteristic functions are extensionally equivalent: *)
47 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.
48 notation "A =1 B" non associative with precedence 45 for @{'eqP $A $B}.
49 interpretation "extensional equality" 'eqP a b = (eqP ? a b).
51 (* This notion of equality is different from the intensional equality of
52 functions; the fact it defines an equivalence relation must be explicitly
55 lemma eqP_sym: ∀U.∀A,B:U →Prop.
56 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.
57 #U #A #B #eqAB #a @
\ 5a href="cic:/matita/basics/logic/iff_sym.def(2)"
\ 6iff_sym
\ 5/a
\ 6 @eqAB qed.
59 lemma eqP_trans: ∀U.∀A,B,C:U →Prop.
60 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.
61 #U #A #B #C #eqAB #eqBC #a @
\ 5a href="cic:/matita/basics/logic/iff_trans.def(2)"
\ 6iff_trans
\ 5/a
\ 6 // qed.
63 (* For the same reason, we must also prove that all the operations behave well
64 with respect to eqP: *)
66 lemma eqP_union_r: ∀U.∀A,B,C:U →Prop.
67 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.
68 #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.
70 lemma eqP_union_l: ∀U.∀A,B,C:U →Prop.
71 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.
72 #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.
74 lemma eqP_intersect_r: ∀U.∀A,B,C:U →Prop.
75 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.
76 #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.
78 lemma eqP_intersect_l: ∀U.∀A,B,C:U →Prop.
79 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.
80 #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.
82 lemma eqP_substract_r: ∀U.∀A,B,C:U →Prop.
83 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.
84 #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.
86 lemma eqP_substract_l: ∀U.∀A,B,C:U →Prop.
87 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.
88 #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.
90 (* We can now prove several properties of the previous set-theoretic operations.
91 In particular, union is commutative and associative, and the empty set is an
94 lemma union_empty_r: ∀U.∀A:U→Prop.
95 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.
96 #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/]
99 lemma union_comm : ∀U.∀A,B:U →Prop.
100 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.
101 #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.
103 lemma union_assoc: ∀U.∀A,B,C:U → Prop.
104 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).
105 #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/]
108 (* In the same way we prove commutativity and associativity for set
111 lemma cap_comm : ∀U.∀A,B:U →Prop.
112 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.
113 #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.
115 lemma cap_assoc: ∀U.∀A,B,C:U→Prop.
116 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.
117 #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/ ]
120 (* We can also easily prove idempotency for union and intersection *)
122 lemma union_idemp: ∀U.∀A:U →Prop.
123 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.
124 #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.
126 lemma cap_idemp: ∀U.∀A:U →Prop.
127 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.
128 #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.
130 (* We conclude our examples with a couple of distributivity theorems, and a
131 characterization of substraction in terms of interesection and complementation. *)
133 lemma distribute_intersect : ∀U.∀A,B,C:U→Prop.
134 (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).
135 #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/]
138 lemma distribute_substract : ∀U.∀A,B,C:U→Prop.
139 (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).
140 #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/]
143 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.
144 #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/
147 (* In several situation it is important to assume to have a decidable equality
148 between elements of a set U, namely a boolean function eqb: U→U→bool such that
149 for any pair of elements a and b in U, (eqb x y) is true if and only if x=y.
150 A set equipped with such an equality is called a DeqSet: *)
152 record DeqSet : Type[1] ≝ { carr :> Type[0];
153 eqb: carr → carr →
\ 5a href="cic:/matita/basics/bool/bool.ind(1,0,0)"
\ 6bool
\ 5/a
\ 6;
154 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)
157 (* We use the notation == to denote the decidable equality, to distinguish it
158 from the propositional equality. In particular, a term of the form a==b is a
159 boolean, while a=b is a proposition. *)
161 notation "a == b" non associative with precedence 45 for @{ 'eqb $a $b }.
162 interpretation "eqb" 'eqb a b = (eqb ? a b).
164 (* It is convenient to have a simple way to reflect a proof of the fact
165 that (eqb a b) is true into a proof of the proposition (a = b); to this aim,
166 we introduce two operators "\P" and "\b". *)
168 notation "\P H" non associative with precedence 90
169 for @{(proj1 … (eqb_true ???) $H)}.
171 notation "\b H" non associative with precedence 90
172 for @{(proj2 … (eqb_true ???) $H)}.
174 (* If H:eqb a b = true, then \P H: a = b, and conversely if h:a = b, then
175 \b h: eqb a b = true. Let us see an example of their use: the following
176 statement asserts that we can reflect a proof that eqb a b is false into
177 a proof of the proposition a ≠ b. *)
179 lemma eqb_false: ∀S:
\ 5a href="cic:/matita/tutorial/chapter4/DeqSet.ind(1,0,0)"
\ 6DeqSet
\ 5/a
\ 6.∀a,b:S.
180 (
\ 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.
182 (* We start the proof introducing the hypothesis, and then split the "if" and
187 (* The latter is easily reduced to prove the goal true=false under the assumption
189 [@(
\ 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
191 (* since by assumption H false is equal to (a==b), by rewriting we obtain the goal
192 true=(a==b) that is just the boolean version of H1 *)
194 <H @
\ 5a href="cic:/matita/basics/logic/sym_eq.def(2)"
\ 6sym_eq
\ 5/a
\ 6 @(\b H1)
196 (* In the "if" case, we proceed by cases over the boolean equality (a==b); if
197 (a==b) is false, the goal is trivial; the other case is absurd, since if (a==b) is
198 true, then by reflection a=b, while by hypothesis a≠b *)
200 |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)
204 (* We also introduce two operators "\Pf" and "\bf" to reflect a proof
205 of (a==b)=false into a proof of a≠b, and vice-versa *)
207 notation "\Pf H" non associative with precedence 90
208 for @{(proj1 … (eqb_false ???) $H)}.
210 notation "\bf H" non associative with precedence 90
211 for @{(proj2 … (eqb_false ???) $H)}.
213 (* The following statement proves that propositional equality in a
214 DeqSet is decidable in the traditional sense, namely either a=b or a≠b *)
216 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.
217 #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
218 [%1 @(\P H) | %2 @(\Pf H)]
221 (* A simple example of a set with a decidable equality is bool. We first define
222 the boolean equality beqb, that is just the xand function, then prove that
223 beqb b1 b2 is true if and only if b1=b2, and finally build the type DeqBool by
224 instantiating the DeqSet record with the previous information *)
226 definition beqb ≝ λb1,b2.
227 match b1 with [ true ⇒ b2 | false ⇒
\ 5a href="cic:/matita/basics/bool/notb.def(1)"
\ 6notb
\ 5/a
\ 6 b2].
229 notation < "a == b" non associative with precedence 45 for @{beqb $a $b }.
231 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).
232 #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/
235 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.
237 (* At this point, we would expect to be able to prove things like the
238 following: for any boolean b, if b==false is true then b=false. *)
240 (* unification hint 0
\ 5a href="cic:/fakeuri.def(1)" title="hint_decl_Type1"
\ 6≔
\ 5/a
\ 6 ;
241 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
242 (* ---------------------------------------- *) ⊢
243 \ 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.
245 unification hint 0 ≔ b1,b2:bool;
246 X ≟ mk_DeqSet bool beqb beqb_true
247 (* ---------------------------------------- *) ⊢
248 beqb b1 b2 ≡ eqb X b1 b2. *)
250 example exhint: ∀b:bool. (b == false) = true → b = false.
255 definition eq_pairs ≝
256 λA,B:DeqSet.λp1,p2:A×B.(\fst p1 == \fst p2) ∧ (\snd p1 == \snd p2).
258 lemma eq_pairs_true: ∀A,B:DeqSet.∀p1,p2:A×B.
259 eq_pairs A B p1 p2 = true ↔ p1 = p2.
260 #A #B * #a1 #b1 * #a2 #b2 %
261 [#H cases (andb_true …H) normalize #eqa #eqb >(\P eqa) >(\P eqb) //
262 |#H destruct normalize >(\b (refl … a2)) >(\b (refl … b2)) //
266 definition DeqProd ≝ λA,B:DeqSet.
267 mk_DeqSet (A×B) (eq_pairs A B) (eq_pairs_true A B).
269 unification hint 0 ≔ C1,C2;
273 (* ---------------------------------------- *) ⊢
276 unification hint 0 ≔ T1,T2,p1,p2;
278 (* ---------------------------------------- *) ⊢
279 eq_pairs T1 T2 p1 p2 ≡ eqb X p1 p2.
281 example hint2: ∀b1,b2.
282 〈b1,true〉==〈false,b2〉=true → 〈b1,true〉=〈false,b2〉.