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1 (* Title: FOL/ex/Quantifiers_Int.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1991 University of Cambridge |
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5 *) |
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6 |
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7 header {* First-Order Logic: quantifier examples (classical version) *} |
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8 |
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9 theory Quantifiers_Cla |
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10 imports FOL |
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11 begin |
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12 |
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13 lemma "(ALL x y. P(x,y)) --> (ALL y x. P(x,y))" |
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14 by fast |
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15 |
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16 lemma "(EX x y. P(x,y)) --> (EX y x. P(x,y))" |
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17 by fast |
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18 |
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19 |
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20 -- {* Converse is false *} |
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21 lemma "(ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x) | Q(x))" |
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22 by fast |
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23 |
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24 lemma "(ALL x. P-->Q(x)) <-> (P--> (ALL x. Q(x)))" |
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25 by fast |
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26 |
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27 |
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28 lemma "(ALL x. P(x)-->Q) <-> ((EX x. P(x)) --> Q)" |
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29 by fast |
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30 |
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31 |
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32 text {* Some harder ones *} |
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33 |
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34 lemma "(EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))" |
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35 by fast |
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36 |
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37 -- {* Converse is false *} |
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38 lemma "(EX x. P(x)&Q(x)) --> (EX x. P(x)) & (EX x. Q(x))" |
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39 by fast |
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40 |
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41 |
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42 text {* Basic test of quantifier reasoning *} |
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43 |
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44 -- {* TRUE *} |
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45 lemma "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))" |
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46 by fast |
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47 |
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48 lemma "(ALL x. Q(x)) --> (EX x. Q(x))" |
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49 by fast |
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50 |
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51 |
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52 text {* The following should fail, as they are false! *} |
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53 |
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54 lemma "(ALL x. EX y. Q(x,y)) --> (EX y. ALL x. Q(x,y))" |
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55 apply fast? |
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56 oops |
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57 |
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58 lemma "(EX x. Q(x)) --> (ALL x. Q(x))" |
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59 apply fast? |
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60 oops |
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61 |
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62 lemma "P(?a) --> (ALL x. P(x))" |
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63 apply fast? |
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64 oops |
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65 |
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66 lemma "(P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))" |
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67 apply fast? |
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68 oops |
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69 |
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70 |
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71 text {* Back to things that are provable \dots *} |
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72 |
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73 lemma "(ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))" |
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74 by fast |
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75 |
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76 -- {* An example of why exI should be delayed as long as possible *} |
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77 lemma "(P --> (EX x. Q(x))) & P --> (EX x. Q(x))" |
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78 by fast |
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79 |
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80 lemma "(ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)" |
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81 by fast |
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82 |
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83 lemma "(ALL x. Q(x)) --> (EX x. Q(x))" |
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84 by fast |
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85 |
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86 |
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87 text {* Some slow ones *} |
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88 |
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89 -- {* Principia Mathematica *11.53 *} |
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90 lemma "(ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))" |
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91 by fast |
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92 |
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93 (*Principia Mathematica *11.55 *) |
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94 lemma "(EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))" |
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95 by fast |
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96 |
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97 (*Principia Mathematica *11.61 *) |
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98 lemma "(EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))" |
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99 by fast |
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100 |
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101 end |