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1 (* Title: HOL/AxClasses/Tutorial/Group.ML |
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2 ID: $Id$ |
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3 Author: Markus Wenzel, TU Muenchen |
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4 |
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5 Some basic theorems of group theory. |
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6 *) |
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7 |
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8 open Group; |
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9 |
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10 fun sub r = standard (r RS subst); |
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11 fun ssub r = standard (r RS ssubst); |
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12 |
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13 |
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14 goal Group.thy "x <*> inv x = (1::'a::group)"; |
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15 br (sub left_unit) 1; |
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16 back(); |
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17 br (sub assoc) 1; |
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18 br (sub left_inv) 1; |
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19 back(); |
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20 back(); |
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21 br (ssub assoc) 1; |
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22 back(); |
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23 br (ssub left_inv) 1; |
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24 br (ssub assoc) 1; |
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25 br (ssub left_unit) 1; |
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26 br (ssub left_inv) 1; |
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27 br refl 1; |
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28 qed "right_inv"; |
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29 |
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30 |
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31 goal Group.thy "x <*> 1 = (x::'a::group)"; |
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32 br (sub left_inv) 1; |
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33 br (sub assoc) 1; |
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34 br (ssub right_inv) 1; |
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35 br (ssub left_unit) 1; |
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36 br refl 1; |
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37 qed "right_unit"; |
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38 |
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39 |
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40 goal Group.thy "e <*> x = x --> e = (1::'a::group)"; |
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41 br impI 1; |
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42 br (sub right_unit) 1; |
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43 back(); |
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44 by (res_inst_tac [("x", "x")] (sub right_inv) 1); |
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45 br (sub assoc) 1; |
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46 br arg_cong 1; |
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47 back(); |
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48 ba 1; |
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49 qed "strong_one_unit"; |
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50 |
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51 |
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52 goal Group.thy "EX! e. ALL x. e <*> x = (x::'a::group)"; |
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53 br ex1I 1; |
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54 br allI 1; |
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55 br left_unit 1; |
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56 br mp 1; |
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57 br strong_one_unit 1; |
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58 be allE 1; |
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59 ba 1; |
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60 qed "ex1_unit"; |
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61 |
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62 |
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63 goal Group.thy "ALL x. EX! e. e <*> x = (x::'a::group)"; |
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64 br allI 1; |
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65 br ex1I 1; |
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66 br left_unit 1; |
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67 br (strong_one_unit RS mp) 1; |
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68 ba 1; |
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69 qed "ex1_unit'"; |
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70 |
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71 |
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72 goal Group.thy "y <*> x = 1 --> y = inv (x::'a::group)"; |
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73 br impI 1; |
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74 br (sub right_unit) 1; |
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75 back(); |
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76 back(); |
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77 br (sub right_unit) 1; |
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78 by (res_inst_tac [("x", "x")] (sub right_inv) 1); |
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79 br (sub assoc) 1; |
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80 br (sub assoc) 1; |
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81 br arg_cong 1; |
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82 back(); |
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83 br (ssub left_inv) 1; |
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84 ba 1; |
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85 qed "one_inv"; |
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86 |
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87 |
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88 goal Group.thy "ALL x. EX! y. y <*> x = (1::'a::group)"; |
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89 br allI 1; |
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90 br ex1I 1; |
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91 br left_inv 1; |
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92 br mp 1; |
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93 br one_inv 1; |
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94 ba 1; |
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95 qed "ex1_inv"; |
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96 |
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97 |
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98 goal Group.thy "inv (x <*> y) = inv y <*> inv (x::'a::group)"; |
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99 br sym 1; |
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100 br mp 1; |
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101 br one_inv 1; |
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102 br (ssub assoc) 1; |
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103 br (sub assoc) 1; |
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104 back(); |
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105 br (ssub left_inv) 1; |
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106 br (ssub left_unit) 1; |
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107 br (ssub left_inv) 1; |
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108 br refl 1; |
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109 qed "inv_product"; |
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110 |
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111 |
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112 goal Group.thy "inv (inv x) = (x::'a::group)"; |
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113 br sym 1; |
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114 br (one_inv RS mp) 1; |
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115 br (ssub right_inv) 1; |
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116 br refl 1; |
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117 qed "inv_inv"; |
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118 |