clasohm@1475: (*  Title:      Relation.thy
nipkow@1128:     ID:         $Id$
clasohm@1475:     Author:     Riccardo Mattolini, Dip. Sistemi e Informatica
clasohm@1475:         and     Lawrence C Paulson, Cambridge University Computer Laboratory
nipkow@1128:     Copyright   1994 Universita' di Firenze
nipkow@1128:     Copyright   1993  University of Cambridge
nipkow@1128: *)
nipkow@1128: 
nipkow@1128: Relation = Prod +
nipkow@1128: consts
clasohm@1475:     id          :: "('a * 'a)set"               (*the identity relation*)
clasohm@1475:     O           :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
clasohm@1475:     trans       :: "('a * 'a)set => bool"       (*transitivity predicate*)
nipkow@1454:     converse    :: "('a * 'b)set => ('b * 'a)set"
nipkow@1454:     "^^"        :: "[('a * 'b) set, 'a set] => 'b set" (infixl 90)
nipkow@1454:     Domain      :: "('a * 'b) set => 'a set"
nipkow@1454:     Range       :: "('a * 'b) set => 'b set"
nipkow@1128: defs
clasohm@1475:     id_def      "id == {p. ? x. p = (x,x)}"
clasohm@1475:     comp_def    "r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
clasohm@1475:     trans_def     "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
nipkow@1454:     converse_def  "converse(r) == {(y,x). (x,y):r}"
nipkow@1454:     Domain_def    "Domain(r) == {x. ? y. (x,y):r}"
nipkow@1128:     Range_def     "Range(r) == Domain(converse(r))"
nipkow@1128:     Image_def     "r ^^ s == {y. y:Range(r) &  (? x:s. (x,y):r)}"
nipkow@1128: end