src/HOL/Prod.thy
author berghofe
Tue, 30 May 2000 18:02:49 +0200
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the is now defined using primrec, avoiding explicit use of arbitrary.
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(*  Title:      HOL/Prod.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1992  University of Cambridge
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Ordered Pairs and the Cartesian product type.
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The unit type.
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*)
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Prod = Fun + equalities +
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(** products **)
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(* type definition *)
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constdefs
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  Pair_Rep      :: ['a, 'b] => ['a, 'b] => bool
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  "Pair_Rep == (%a b. %x y. x=a & y=b)"
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global
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typedef (Prod)
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  ('a, 'b) "*"          (infixr 20)
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    = "{f. ? a b. f = Pair_Rep (a::'a) (b::'b)}"
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syntax (symbols)
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  "*"           :: [type, type] => type         ("(_ \\<times>/ _)" [21, 20] 20)
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syntax (HTML output)
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  "*"           :: [type, type] => type         ("(_ \\<times>/ _)" [21, 20] 20)
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(* abstract constants and syntax *)
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consts
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  fst           :: "'a * 'b => 'a"
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  snd           :: "'a * 'b => 'b"
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  split         :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
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  prod_fun      :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
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  Pair          :: "['a, 'b] => 'a * 'b"
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  Sigma         :: "['a set, 'a => 'b set] => ('a * 'b) set"
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(* patterns -- extends pre-defined type "pttrn" used in abstractions *)
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nonterminals
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  patterns
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syntax
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  "@Tuple"      :: "['a, args] => 'a * 'b"       ("(1'(_,/ _'))")
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  "_pattern"    :: [pttrn, patterns] => pttrn    ("'(_,/_')")
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  ""            :: pttrn => patterns             ("_")
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  "_patterns"   :: [pttrn, patterns] => patterns ("_,/_")
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  "@Sigma"      :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"   ("(3SIGMA _:_./ _)" 10)
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  "@Times"      :: "['a set, 'a => 'b set] => ('a * 'b) set"    (infixr "<*>" 80)
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translations
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  "(x, y, z)"    == "(x, (y, z))"
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  "(x, y)"       == "Pair x y"
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  "%(x,y,zs).b"  == "split(%x (y,zs).b)"
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  "%(x,y).b"     == "split(%x y. b)"
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  "_abs (Pair x y) t" => "%(x,y).t"
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  (* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
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     The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)
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  "SIGMA x:A. B" => "Sigma A (%x. B)"
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  "A <*> B"      => "Sigma A (_K B)"
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syntax (symbols)
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  "@Sigma"      :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"   ("(3\\<Sigma> _\\<in>_./ _)" 10)
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  "@Times"      :: "['a set, 'a => 'b set] => ('a * 'b) set"    ("_ \\<times> _" [81, 80] 80)
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(* definitions *)
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local
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defs
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  Pair_def      "Pair a b == Abs_Prod(Pair_Rep a b)"
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  fst_def       "fst p == @a. ? b. p = (a, b)"
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  snd_def       "snd p == @b. ? a. p = (a, b)"
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  split_def     "split == (%c p. c (fst p) (snd p))"
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  prod_fun_def  "prod_fun f g == split(%x y.(f(x), g(y)))"
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  Sigma_def     "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
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(** unit **)
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typedef  unit = "{True}"
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consts
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  "()"          :: unit                           ("'(')")
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defs
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  Unity_def     "() == Abs_unit True"
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end
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val print_translation = [("Sigma", dependent_tr' ("@Sigma", "@Times"))];