src/HOL/Integ/Equiv.thy

Addition of Finite as parent allows cardinality theorems

(*  Title:      Equiv.thy
    ID:         $Id$
    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Equivalence relations in Higher-Order Set Theory 
*)

Equiv = Relation + Finite + 
consts
    refl,equiv  ::      "['a set,('a*'a) set]=>bool"
    sym         ::      "('a*'a) set=>bool"
    "'/"        ::      "['a set,('a*'a) set]=>'a set set"  (infixl 90) 
                        (*set of equiv classes*)
    congruent   ::      "[('a*'a) set,'a=>'b]=>bool"
    congruent2  ::      "[('a*'a) set,['a,'a]=>'b]=>bool"

defs
    refl_def      "refl A r == r <= A Times A & (ALL x: A. (x,x) : r)"
    sym_def       "sym(r)    == ALL x y. (x,y): r --> (y,x): r"
    equiv_def     "equiv A r == refl A r & sym(r) & trans(r)"
    quotient_def  "A/r == UN x:A. {r^^{x}}"  
    congruent_def   "congruent r b  == ALL y z. (y,z):r --> b(y)=b(z)"
    congruent2_def  "congruent2 r b == ALL y1 z1 y2 z2. 
           (y1,z1):r --> (y2,z2):r --> b y1 y2 = b z1 z2"
end