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(* Title: HOL/Product_Type.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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Product_Type = 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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tuple_args patterns
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syntax
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"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))")
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"_tuple_arg" :: "'a => tuple_args" ("_")
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"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _")
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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)" == "Pair x y"
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"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
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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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global
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typedef unit = "{True}"
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consts
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"()" :: unit ("'(')")
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local
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defs
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Unity_def "() == Abs_unit True"
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end
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ML
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val print_translation = [("Sigma", dependent_tr' ("@Sigma", "@Times"))];
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