src/ZF/ex/Equiv.thy
author clasohm
Thu Sep 16 12:20:38 1993 +0200 (1993-09-16)
changeset 0 a5a9c433f639
child 434 89d45187f04d
permissions -rw-r--r--
Initial revision
     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