(* 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.
*)
theory Product_Type = Fun
files ("Product_Type_lemmas.ML") ("Tools/split_rule.ML"):
(** 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. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
proof
fix a b show "Pair_Rep a b : ?Prod"
by blast
qed
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)
print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *}
(* definitions *)
local
defs
Pair_def: "Pair a b == Abs_Prod(Pair_Rep a b)"
fst_def: "fst p == SOME a. EX b. p = (a, b)"
snd_def: "snd p == SOME b. EX 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}"
proof
show "True : ?unit"
by blast
qed
consts
"()" :: unit ("'(')")
local
defs
Unity_def: "() == Abs_unit True"
(** lemmas and tool setup **)
use "Product_Type_lemmas.ML"
constdefs
internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
"internal_split == split"
lemma internal_split_conv: "internal_split c (a, b) = c a b"
by (simp only: internal_split_def split_conv)
hide const internal_split
use "Tools/split_rule.ML"
setup SplitRule.setup
end