--- a/src/HOL/Relation.thy Fri Jan 26 13:43:36 1996 +0100
+++ b/src/HOL/Relation.thy Fri Jan 26 20:25:39 1996 +0100
@@ -11,17 +11,16 @@
id :: "('a * 'a)set" (*the identity relation*)
O :: "[('b * 'c)set, ('a * 'b)set] => ('a * 'c)set" (infixr 60)
trans :: "('a * 'a)set => bool" (*transitivity predicate*)
- converse :: "('a*'a) set => ('a*'a) set"
- "^^" :: "[('a*'a) set,'a set] => 'a set" (infixl 90)
- Domain :: "('a*'a) set => 'a set"
- Range :: "('a*'a) set => 'a set"
+ converse :: "('a * 'b)set => ('b * 'a)set"
+ "^^" :: "[('a * 'b) set, 'a set] => 'b set" (infixl 90)
+ Domain :: "('a * 'b) set => 'a set"
+ Range :: "('a * 'b) set => 'b set"
defs
id_def "id == {p. ? x. p = (x,x)}"
- comp_def (*composition of relations*)
- "r O s == {xz. ? x y z. xz = (x,z) & (x,y):s & (y,z):r}"
+ comp_def "r O s == {(x,z). ? y. (x,y):s & (y,z):r}"
trans_def "trans(r) == (!x y z. (x,y):r --> (y,z):r --> (x,z):r)"
- converse_def "converse(r) == {z. (? w:r. ? x y. w=(x,y) & z=(y,x))}"
- Domain_def "Domain(r) == {z. ! x. (z=x --> (? y. (x,y):r))}"
+ converse_def "converse(r) == {(y,x). (x,y):r}"
+ Domain_def "Domain(r) == {x. ? y. (x,y):r}"
Range_def "Range(r) == Domain(converse(r))"
Image_def "r ^^ s == {y. y:Range(r) & (? x:s. (x,y):r)}"
end