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1 (* Title: HOLCF/porder.thy |
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2 ID: $Id$ |
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3 Author: Franz Regensburger |
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4 Copyright 1993 Technische Universitaet Muenchen |
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5 |
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6 Definition of class porder (partial order) |
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7 |
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8 The prototype theory for this class is void.thy |
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9 |
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10 *) |
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11 |
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12 Porder = Void + |
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13 |
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14 (* Introduction of new class. The witness is type void. *) |
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15 |
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16 classes po < term |
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17 |
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18 (* default type is still term ! *) |
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19 (* void is the prototype in po *) |
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20 |
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21 arities void :: po |
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22 |
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23 consts "<<" :: "['a,'a::po] => bool" (infixl 55) |
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24 |
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25 "<|" :: "['a set,'a::po] => bool" (infixl 55) |
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26 "<<|" :: "['a set,'a::po] => bool" (infixl 55) |
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27 lub :: "'a set => 'a::po" |
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28 is_tord :: "'a::po set => bool" |
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29 is_chain :: "(nat=>'a::po) => bool" |
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30 max_in_chain :: "[nat,nat=>'a::po]=>bool" |
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31 finite_chain :: "(nat=>'a::po)=>bool" |
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32 |
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33 rules |
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34 |
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35 (* class axioms: justification is theory Void *) |
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36 |
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37 refl_less "x << x" |
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38 (* witness refl_less_void *) |
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39 |
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40 antisym_less "[|x<<y ; y<<x |] ==> x = y" |
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41 (* witness antisym_less_void *) |
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42 |
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43 trans_less "[|x<<y ; y<<z |] ==> x<<z" |
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44 (* witness trans_less_void *) |
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45 |
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46 (* instance of << for the prototype void *) |
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47 |
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48 inst_void_po "(op <<)::[void,void]=>bool = less_void" |
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49 |
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50 (* class definitions *) |
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51 |
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52 is_ub "S <| x == ! y.y:S --> y<<x" |
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53 is_lub "S <<| x == S <| x & (! u. S <| u --> x << u)" |
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54 |
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55 lub "lub(S) = (@x. S <<| x)" |
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56 |
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57 (* Arbitrary chains are total orders *) |
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58 is_tord "is_tord(S) == ! x y. x:S & y:S --> (x<<y | y<<x)" |
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59 |
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60 |
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61 (* Here we use countable chains and I prefer to code them as functions! *) |
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62 is_chain "is_chain(F) == (! i.F(i) << F(Suc(i)))" |
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63 |
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64 |
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65 (* finite chains, needed for monotony of continouous functions *) |
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66 |
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67 max_in_chain_def "max_in_chain(i,C) == ! j. i <= j --> C(i) = C(j)" |
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68 |
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69 finite_chain_def "finite_chain(C) == is_chain(C) & (? i. max_in_chain(i,C))" |
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70 |
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71 end |