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1 (* Title: HOL/Prod.thy |
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2 ID: Prod.thy,v 1.5 1994/08/19 09:04:27 lcp Exp |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1992 University of Cambridge |
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5 |
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6 Ordered Pairs and the Cartesian product type. |
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7 The unit type. |
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8 *) |
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9 |
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10 Prod = Fun + |
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11 |
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12 (** Products **) |
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13 |
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14 (* type definition *) |
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15 |
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16 consts |
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17 Pair_Rep :: "['a, 'b] => ['a, 'b] => bool" |
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18 |
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19 defs |
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20 Pair_Rep_def "Pair_Rep == (%a b. %x y. x=a & y=b)" |
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21 |
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22 subtype (Prod) |
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23 ('a, 'b) "*" (infixr 20) |
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24 = "{f. ? a b. f = Pair_Rep (a::'a) (b::'b)}" |
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25 |
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26 |
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27 (* abstract constants and syntax *) |
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28 |
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29 consts |
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30 fst :: "'a * 'b => 'a" |
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31 snd :: "'a * 'b => 'b" |
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32 split :: "[['a, 'b] => 'c, 'a * 'b] => 'c" |
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33 prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd" |
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34 Pair :: "['a, 'b] => 'a * 'b" |
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35 Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set" |
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36 |
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37 syntax |
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38 "@Tuple" :: "args => 'a * 'b" ("(1<_>)") |
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39 |
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40 translations |
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41 "<x, y, z>" == "<x, <y, z>>" |
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42 "<x, y>" == "Pair x y" |
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43 "<x>" => "x" |
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44 |
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45 defs |
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46 Pair_def "Pair a b == Abs_Prod(Pair_Rep a b)" |
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47 fst_def "fst(p) == @a. ? b. p = <a, b>" |
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48 snd_def "snd(p) == @b. ? a. p = <a, b>" |
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49 split_def "split c p == c (fst p) (snd p)" |
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50 prod_fun_def "prod_fun f g == split(%x y.<f(x), g(y)>)" |
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51 Sigma_def "Sigma A B == UN x:A. UN y:B(x). {<x, y>}" |
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52 |
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53 |
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54 |
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55 (** Unit **) |
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56 |
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57 subtype (Unit) |
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58 unit = "{p. p = True}" |
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59 |
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60 consts |
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61 Unity :: "unit" ("<>") |
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62 |
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63 defs |
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64 Unity_def "Unity == Abs_Unit(True)" |
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65 |
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66 end |