(* Title: HOL/Prod.thy
ID: Prod.thy,v 1.5 1994/08/19 09:04:27 lcp Exp
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
Ordered Pairs and the Cartesian product type.
The unit type.
*)
Prod = Fun +
(** Products **)
(* type definition *)
consts
Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
defs
Pair_Rep_def "Pair_Rep == (%a b. %x y. x=a & y=b)"
subtype (Prod)
('a, 'b) "*" (infixr 20)
= "{f. ? a b. f = Pair_Rep (a::'a) (b::'b)}"
(* 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 **)
types pttrns
syntax
"@Tuple" :: "['a, args] => 'a * 'b" ("(1'(_,/ _'))")
"@pttrn" :: "[pttrn,pttrns] => pttrn" ("'(_,/_')")
"" :: " pttrn => pttrns" ("_")
"@pttrns" :: "[pttrn,pttrns] => pttrns" ("_,/_")
translations
"(x, y, z)" == "(x, (y, z))"
"(x, y)" == "Pair x y"
"%(x,y,zs).b" == "split(%x (y,zs).b)"
"%(x,y).b" == "split(%x y.b)"
(* The <= direction fails if split has more than one argument because
ast-matching fails. Otherwise it would work fine *)
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 **)
subtype (Unit)
unit = "{p. p = True}"
consts
"()" :: "unit" ("'(')")
defs
Unity_def "() == Abs_Unit(True)"
(* start 8bit 1 *)
(* end 8bit 1 *)
end
(*
ML
local open Syntax
fun pttrn(_ $ s $ t) = const"@pttrn" $ s $ t;
fun pttrns s t = const"@pttrns" $ s $ t;
fun split2(Abs(x,T,t)) =
let val (pats,u) = split1 t
in (pttrns (Free(x,T)) pats, subst_bounds([free x],u)) end
| split2(Const("split",_) $ r) =
let val (pats,s) = split2(r)
val (pats2,t) = split1(s)
in (pttrns (pttrn pats) pats2, t) end
and split1(Abs(x,T,t)) = (Free(x,T), subst_bounds([free x],t))
| split1(Const("split",_)$t) = split2(t);
fun split_tr'(t::args) =
let val (pats,ft) = split2(t)
in list_comb(const"_lambda" $ pttrn pats $ ft, args) end;
in
val print_translation = [("split", split_tr')];
end;
*)