(* Title: HOL/prod
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Ordered Pairs and the Cartesian product type
The unit type
The type definition admits the following unused axiom:
Abs_Unit_inverse "f: Unit ==> Rep_Unit(Abs_Unit(f)) = f"
*)
Prod = Set +
types
"*" 2 (infixr 20)
unit 0
arities
"*" :: (term,term)term
unit :: term
consts
Pair_Rep :: "['a,'b] => ['a,'b] => bool"
Prod :: "('a => 'b => bool)set"
Rep_Prod :: "'a * 'b => ('a => 'b => bool)"
Abs_Prod :: "('a => 'b => bool) => 'a * 'b"
fst :: "'a * 'b => 'a"
snd :: "'a * 'b => 'b"
split :: "['a * 'b, ['a,'b]=>'c] => 'c"
prod_fun :: "['a=>'b, 'c=>'d, 'a*'c] => 'b*'d"
Pair :: "['a,'b] => 'a * 'b"
"@Tuple" :: "args => 'a*'b" ("(1<_>)")
Sigma :: "['a set, 'a => 'b set] => ('a*'b)set"
Unit :: "bool set"
Rep_Unit :: "unit => bool"
Abs_Unit :: "bool => unit"
Unity :: "unit" ("<>")
translations
"<x,y,z>" == "<x,<y,z>>"
"<x,y>" == "Pair(x,y)"
"<x>" => "x"
rules
Pair_Rep_def "Pair_Rep == (%a b. %x y. x=a & y=b)"
Prod_def "Prod == {f. ? a b. f = Pair_Rep(a,b)}"
(*faking a type definition...*)
Rep_Prod "Rep_Prod(p): Prod"
Rep_Prod_inverse "Abs_Prod(Rep_Prod(p)) = p"
Abs_Prod_inverse "f: Prod ==> Rep_Prod(Abs_Prod(f)) = f"
(*defining the abstract constants*)
Pair_def "Pair(a,b) == Abs_Prod(Pair_Rep(a,b))"
fst_def "fst(p) == @a. ? b. p = <a,b>"
snd_def "snd(p) == @b. ? a. p = <a,b>"
split_def "split(p,c) == c(fst(p),snd(p))"
prod_fun_def "prod_fun(f,g) == (%z.split(z, %x y.<f(x), g(y)>))"
Sigma_def "Sigma(A,B) == UN x:A. UN y:B(x). {<x,y>}"
Unit_def "Unit == {p. p=True}"
(*faking a type definition...*)
Rep_Unit "Rep_Unit(u): Unit"
Rep_Unit_inverse "Abs_Unit(Rep_Unit(u)) = u"
(*defining the abstract constants*)
Unity_def "Unity == Abs_Unit(True)"
end