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1 (* Title: Relation.ML |
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2 ID: $Id$ |
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3 Authors: Riccardo Mattolini, Dip. Sistemi e Informatica |
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4 Lawrence C Paulson, Cambridge University Computer Laboratory |
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5 Copyright 1994 Universita' di Firenze |
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6 Copyright 1993 University of Cambridge |
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7 |
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8 Functions represented as relations in HOL Set Theory |
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9 *) |
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10 |
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11 val RSLIST = curry (op MRS); |
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12 |
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13 open Relation; |
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14 |
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15 goalw Relation.thy [converse_def] "!!a b r. <a,b>:r ==> <b,a>:converse(r)"; |
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16 by (simp_tac prod_ss 1); |
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17 by (fast_tac set_cs 1); |
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18 qed "converseI"; |
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19 |
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20 goalw Relation.thy [converse_def] "!!a b r. <a,b> : converse(r) ==> <b,a> : r"; |
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21 by (fast_tac comp_cs 1); |
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22 qed "converseD"; |
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23 |
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24 qed_goalw "converseE" Relation.thy [converse_def] |
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25 "[| yx : converse(r); \ |
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26 \ !!x y. [| yx=<y,x>; <x,y>:r |] ==> P \ |
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27 \ |] ==> P" |
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28 (fn [major,minor]=> |
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29 [ (rtac (major RS CollectE) 1), |
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30 (REPEAT (eresolve_tac [bexE,exE, conjE, minor] 1)), |
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31 (hyp_subst_tac 1), |
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32 (assume_tac 1) ]); |
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33 |
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34 val converse_cs = comp_cs addSIs [converseI] |
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35 addSEs [converseD,converseE]; |
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36 |
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37 qed_goalw "Domain_iff" Relation.thy [Domain_def] |
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38 "a: Domain(r) = (EX y. <a,y>: r)" |
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39 (fn _=> [ (fast_tac comp_cs 1) ]); |
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40 |
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41 qed_goal "DomainI" Relation.thy "!!a b r. <a,b>: r ==> a: Domain(r)" |
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42 (fn _ => [ (etac (exI RS (Domain_iff RS iffD2)) 1) ]); |
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43 |
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44 qed_goal "DomainE" Relation.thy |
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45 "[| a : Domain(r); !!y. <a,y>: r ==> P |] ==> P" |
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46 (fn prems=> |
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47 [ (rtac (Domain_iff RS iffD1 RS exE) 1), |
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48 (REPEAT (ares_tac prems 1)) ]); |
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49 |
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50 qed_goalw "RangeI" Relation.thy [Range_def] "!!a b r.<a,b>: r ==> b : Range(r)" |
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51 (fn _ => [ (etac (converseI RS DomainI) 1) ]); |
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52 |
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53 qed_goalw "RangeE" Relation.thy [Range_def] |
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54 "[| b : Range(r); !!x. <x,b>: r ==> P |] ==> P" |
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55 (fn major::prems=> |
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56 [ (rtac (major RS DomainE) 1), |
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57 (resolve_tac prems 1), |
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58 (etac converseD 1) ]); |
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59 |
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60 (*** Image of a set under a function/relation ***) |
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61 |
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62 qed_goalw "Image_iff" Relation.thy [Image_def] |
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63 "b : r^^A = (? x:A. <x,b>:r)" |
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64 (fn _ => [ fast_tac (comp_cs addIs [RangeI]) 1 ]); |
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65 |
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66 qed_goal "Image_singleton_iff" Relation.thy |
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67 "(b : r^^{a}) = (<a,b>:r)" |
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68 (fn _ => [ rtac (Image_iff RS trans) 1, |
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69 fast_tac comp_cs 1 ]); |
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70 |
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71 qed_goalw "ImageI" Relation.thy [Image_def] |
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72 "!!a b r. [| <a,b>: r; a:A |] ==> b : r^^A" |
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73 (fn _ => [ (REPEAT (ares_tac [CollectI,RangeI,bexI] 1)), |
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74 (resolve_tac [conjI ] 1), |
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75 (resolve_tac [RangeI] 1), |
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76 (REPEAT (fast_tac set_cs 1))]); |
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77 |
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78 qed_goalw "ImageE" Relation.thy [Image_def] |
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79 "[| b: r^^A; !!x.[| <x,b>: r; x:A |] ==> P |] ==> P" |
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80 (fn major::prems=> |
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81 [ (rtac (major RS CollectE) 1), |
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82 (safe_tac set_cs), |
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83 (etac RangeE 1), |
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84 (rtac (hd prems) 1), |
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85 (REPEAT (etac bexE 1 ORELSE ares_tac prems 1)) ]); |
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86 |
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87 qed_goal "Image_subset" Relation.thy |
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88 "!!A B r. r <= Sigma(A,%x.B) ==> r^^C <= B" |
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89 (fn _ => |
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90 [ (rtac subsetI 1), |
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91 (REPEAT (eresolve_tac [asm_rl, ImageE, subsetD RS SigmaD2] 1)) ]); |
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92 |
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93 val rel_cs = converse_cs addSIs [converseI] |
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94 addIs [ImageI, DomainI, RangeI] |
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95 addSEs [ImageE, DomainE, RangeE]; |
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96 |
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97 val rel_eq_cs = rel_cs addSIs [equalityI]; |
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98 |