equal
deleted
inserted
replaced
|
1 (* Title: ZF/ex/equiv.thy |
|
2 ID: $Id$ |
|
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
4 Copyright 1993 University of Cambridge |
|
5 |
|
6 Equivalence relations in Zermelo-Fraenkel Set Theory |
|
7 *) |
|
8 |
|
9 Equiv = Trancl + |
|
10 consts |
|
11 refl,equiv :: "[i,i]=>o" |
|
12 sym :: "i=>o" |
|
13 "'/" :: "[i,i]=>i" (infixl 90) (*set of equiv classes*) |
|
14 congruent :: "[i,i=>i]=>o" |
|
15 congruent2 :: "[i,[i,i]=>i]=>o" |
|
16 |
|
17 rules |
|
18 refl_def "refl(A,r) == r <= (A*A) & (ALL x: A. <x,x> : r)" |
|
19 sym_def "sym(r) == ALL x y. <x,y>: r --> <y,x>: r" |
|
20 equiv_def "equiv(A,r) == refl(A,r) & sym(r) & trans(r)" |
|
21 quotient_def "A/r == {r``{x} . x:A}" |
|
22 congruent_def "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)" |
|
23 |
|
24 congruent2_def |
|
25 "congruent2(r,b) == ALL y1 z1 y2 z2. \ |
|
26 \ <y1,z1>:r --> <y2,z2>:r --> b(y1,y2) = b(z1,z2)" |
|
27 |
|
28 end |