(* Title: HOL/Relation.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
Relation = Product_Type +
constdefs
converse :: "('a * 'b) set => ('b * 'a) set" ("(_^-1)" [1000] 999)
"r^-1 == {(y, x). (x, y) : r}"
syntax (xsymbols)
converse :: "('a * 'b) set => ('b * 'a) set" ("(_\\<inverse>)" [1000] 999)
constdefs
comp :: "[('b * 'c) set, ('a * 'b) set] => ('a * 'c) set" (infixr "O" 60)
"r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
Image :: "[('a * 'b) set, 'a set] => 'b set" (infixl "``" 90)
"r `` s == {y. ? x:s. (x,y):r}"
Id :: "('a * 'a) set" (*the identity relation*)
"Id == {p. ? x. p = (x,x)}"
diag :: "'a set => ('a * 'a) set" (*diagonal: identity over a set*)
"diag(A) == UN x:A. {(x,x)}"
Domain :: "('a * 'b) set => 'a set"
"Domain(r) == {x. ? y. (x,y):r}"
Range :: "('a * 'b) set => 'b set"
"Range(r) == Domain(r^-1)"
Field :: "('a * 'a) set => 'a set"
"Field r == Domain r Un Range r"
refl :: "['a set, ('a * 'a) set] => bool" (*reflexivity over a set*)
"refl A r == r <= A <*> A & (ALL x: A. (x,x) : r)"
sym :: "('a * 'a) set => bool" (*symmetry predicate*)
"sym(r) == ALL x y. (x,y): r --> (y,x): r"
antisym:: "('a * 'a) set => bool" (*antisymmetry predicate*)
"antisym(r) == ALL x y. (x,y):r --> (y,x):r --> x=y"
trans :: "('a * 'a) set => bool" (*transitivity predicate*)
"trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
single_valued :: "('a * 'b) set => bool"
"single_valued r == !x y. (x,y):r --> (!z. (x,z):r --> y=z)"
fun_rel_comp :: "['a => 'b, ('b * 'c) set] => ('a => 'c) set"
"fun_rel_comp f R == {g. !x. (f x, g x) : R}"
inv_image :: "('b * 'b) set => ('a => 'b) => ('a * 'a) set"
"inv_image r f == {(x,y). (f(x), f(y)) : r}"
syntax
reflexive :: "('a * 'a) set => bool" (*reflexivity over a type*)
translations
"reflexive" == "refl UNIV"
end