src/ZF/ex/Equiv.thy
 changeset 0 a5a9c433f639 child 434 89d45187f04d
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/ZF/ex/Equiv.thy	Thu Sep 16 12:20:38 1993 +0200
1.3 @@ -0,0 +1,28 @@
1.4 +(*  Title: 	ZF/ex/equiv.thy
1.5 +    ID:         \$Id\$
1.6 +    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
1.7 +    Copyright   1993  University of Cambridge
1.8 +
1.9 +Equivalence relations in Zermelo-Fraenkel Set Theory
1.10 +*)
1.11 +
1.12 +Equiv = Trancl +
1.13 +consts
1.14 +    refl,equiv 	::      "[i,i]=>o"
1.15 +    sym         ::      "i=>o"
1.16 +    "'/"        ::      "[i,i]=>i"  (infixl 90)  (*set of equiv classes*)
1.17 +    congruent	::	"[i,i=>i]=>o"
1.18 +    congruent2  ::      "[i,[i,i]=>i]=>o"
1.19 +
1.20 +rules
1.21 +    refl_def      "refl(A,r) == r <= (A*A) & (ALL x: A. <x,x> : r)"
1.22 +    sym_def       "sym(r) == ALL x y. <x,y>: r --> <y,x>: r"
1.23 +    equiv_def     "equiv(A,r) == refl(A,r) & sym(r) & trans(r)"
1.24 +    quotient_def  "A/r == {r``{x} . x:A}"
1.25 +    congruent_def "congruent(r,b) == ALL y z. <y,z>:r --> b(y)=b(z)"
1.26 +
1.27 +    congruent2_def
1.28 +       "congruent2(r,b) == ALL y1 z1 y2 z2. \
1.29 +\           <y1,z1>:r --> <y2,z2>:r --> b(y1,y2) = b(z1,z2)"
1.30 +
1.31 +end
```