(* Title: HOL/Product_Type.thy
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.
*)
Product_Type = Fun + equalities +
(** products **)
(* type definition *)
constdefs
Pair_Rep :: ['a, 'b] => ['a, 'b] => bool
"Pair_Rep == (%a b. %x y. x=a & y=b)"
global
typedef (Prod)
('a, 'b) "*" (infixr 20)
= "{f. ? a b. f = Pair_Rep (a::'a) (b::'b)}"
syntax (symbols)
"*" :: [type, type] => type ("(_ \\<times>/ _)" [21, 20] 20)
syntax (HTML output)
"*" :: [type, type] => type ("(_ \\<times>/ _)" [21, 20] 20)
(* abstract constants and syntax *)
consts
fst :: "'a * 'b => 'a"
snd :: "'a * 'b => 'b"
split :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
Pair :: "['a, 'b] => 'a * 'b"
Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set"
(* patterns -- extends pre-defined type "pttrn" used in abstractions *)
nonterminals
tuple_args patterns
syntax
"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))")
"_tuple_arg" :: "'a => tuple_args" ("_")
"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _")
"_pattern" :: [pttrn, patterns] => pttrn ("'(_,/ _')")
"" :: pttrn => patterns ("_")
"_patterns" :: [pttrn, patterns] => patterns ("_,/ _")
"@Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)
translations
"(x, y)" == "Pair x y"
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
"%(x,y,zs).b" == "split(%x (y,zs).b)"
"%(x,y).b" == "split(%x y. b)"
"_abs (Pair x y) t" => "%(x,y).t"
(* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
"SIGMA x:A. B" => "Sigma A (%x. B)"
"A <*> B" => "Sigma A (_K B)"
syntax (symbols)
"@Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3\\<Sigma> _\\<in>_./ _)" 10)
"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \\<times> _" [81, 80] 80)
(* definitions *)
local
defs
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 == (%c p. c (fst p) (snd p))"
prod_fun_def "prod_fun f g == split(%x y.(f(x), g(y)))"
Sigma_def "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
(** unit **)
global
typedef unit = "{True}"
consts
"()" :: unit ("'(')")
local
defs
Unity_def "() == Abs_unit True"
end
ML
val print_translation = [("Sigma", dependent_tr' ("@Sigma", "@Times"))];