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1 (* Title: NSComplexBin.ML |
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2 Author: Jacques D. Fleuriot |
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3 Copyright: 2001 University of Edinburgh |
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4 Descrition: Binary arithmetic for the nonstandard complex numbers |
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5 *) |
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6 |
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7 (** hcomplex_of_complex (coercion from complex to nonstandard complex) **) |
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8 |
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9 Goal "hcomplex_of_complex (number_of w) = number_of w"; |
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10 by (simp_tac (simpset() addsimps [hcomplex_number_of_def]) 1); |
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11 qed "hcomplex_number_of"; |
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12 Addsimps [hcomplex_number_of]; |
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13 |
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14 Goalw [hypreal_of_real_def] |
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15 "hcomplex_of_hypreal (hypreal_of_real x) = \ |
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16 \ hcomplex_of_complex(complex_of_real x)"; |
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17 by (simp_tac (simpset() addsimps [hcomplex_of_hypreal, |
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18 hcomplex_of_complex_def,complex_of_real_def]) 1); |
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19 qed "hcomplex_of_hypreal_eq_hcomplex_of_complex"; |
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20 |
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21 Goalw [complex_number_of_def,hypreal_number_of_def] |
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22 "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)"; |
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23 by (rtac (hcomplex_of_hypreal_eq_hcomplex_of_complex RS sym) 1); |
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24 qed "hcomplex_hypreal_number_of"; |
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25 |
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26 Goalw [hcomplex_number_of_def] "Numeral0 = (0::hcomplex)"; |
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27 by (simp_tac (simpset() addsimps [hcomplex_of_complex_zero RS sym]) 1); |
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28 qed "hcomplex_numeral_0_eq_0"; |
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29 |
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30 Goalw [hcomplex_number_of_def] "Numeral1 = (1::hcomplex)"; |
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31 by (simp_tac (simpset() addsimps [hcomplex_of_complex_one RS sym]) 1); |
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32 qed "hcomplex_numeral_1_eq_1"; |
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33 |
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34 (* |
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35 Goal "z + hcnj z = \ |
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36 \ hcomplex_of_hypreal (2 * hRe(z))"; |
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37 by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1); |
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38 by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add, |
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39 hypreal_mult,hcomplex_of_hypreal,complex_add_cnj])); |
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40 qed "hcomplex_add_hcnj"; |
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41 |
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42 Goal "z - hcnj z = \ |
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43 \ hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii"; |
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44 by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1); |
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45 by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff, |
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46 hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal, |
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47 complex_diff_cnj,iii_def,hcomplex_mult])); |
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48 qed "hcomplex_diff_hcnj"; |
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49 *) |
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50 |
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51 (** Addition **) |
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52 |
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53 Goal "(number_of v :: hcomplex) + number_of v' = number_of (bin_add v v')"; |
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54 by (simp_tac |
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55 (HOL_ss addsimps [hcomplex_number_of_def, |
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56 hcomplex_of_complex_add RS sym, add_complex_number_of]) 1); |
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57 qed "add_hcomplex_number_of"; |
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58 Addsimps [add_hcomplex_number_of]; |
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59 |
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60 |
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61 (** Subtraction **) |
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62 |
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63 Goalw [hcomplex_number_of_def] |
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64 "- (number_of w :: hcomplex) = number_of (bin_minus w)"; |
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65 by (simp_tac |
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66 (HOL_ss addsimps [minus_complex_number_of, hcomplex_of_complex_minus RS sym]) 1); |
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67 qed "minus_hcomplex_number_of"; |
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68 Addsimps [minus_hcomplex_number_of]; |
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69 |
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70 Goalw [hcomplex_number_of_def, hcomplex_diff_def] |
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71 "(number_of v :: hcomplex) - number_of w = \ |
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72 \ number_of (bin_add v (bin_minus w))"; |
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73 by (Simp_tac 1); |
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74 qed "diff_hcomplex_number_of"; |
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75 Addsimps [diff_hcomplex_number_of]; |
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76 |
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77 |
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78 (** Multiplication **) |
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79 |
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80 Goal "(number_of v :: hcomplex) * number_of v' = number_of (bin_mult v v')"; |
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81 by (simp_tac |
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82 (HOL_ss addsimps [hcomplex_number_of_def, |
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83 hcomplex_of_complex_mult RS sym, mult_complex_number_of]) 1); |
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84 qed "mult_hcomplex_number_of"; |
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85 Addsimps [mult_hcomplex_number_of]; |
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86 |
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87 Goal "(2::hcomplex) = 1 + 1"; |
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88 by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1); |
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89 val lemma = result(); |
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90 |
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91 (*For specialist use: NOT as default simprules*) |
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92 Goal "2 * z = (z+z::hcomplex)"; |
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93 by (simp_tac (simpset() addsimps [lemma, hcomplex_add_mult_distrib]) 1); |
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94 qed "hcomplex_mult_2"; |
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95 |
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96 Goal "z * 2 = (z+z::hcomplex)"; |
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97 by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_2 1); |
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98 qed "hcomplex_mult_2_right"; |
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99 |
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100 (** Equals (=) **) |
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101 |
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102 Goal "((number_of v :: hcomplex) = number_of v') = \ |
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103 \ iszero (number_of (bin_add v (bin_minus v')))"; |
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104 by (simp_tac |
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105 (HOL_ss addsimps [hcomplex_number_of_def, |
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106 hcomplex_of_complex_eq_iff, eq_complex_number_of]) 1); |
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107 qed "eq_hcomplex_number_of"; |
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108 Addsimps [eq_hcomplex_number_of]; |
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109 |
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110 (*** New versions of existing theorems involving 0, 1hc ***) |
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111 |
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112 Goal "- 1 = (-1::hcomplex)"; |
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113 by (simp_tac (simpset() addsimps [hcomplex_numeral_1_eq_1 RS sym]) 1); |
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114 qed "hcomplex_minus_1_eq_m1"; |
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115 |
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116 Goal "-1 * z = -(z::hcomplex)"; |
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117 by (simp_tac (simpset() addsimps [hcomplex_minus_1_eq_m1 RS sym]) 1); |
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118 qed "hcomplex_mult_minus1"; |
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119 |
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120 Goal "z * -1 = -(z::hcomplex)"; |
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121 by (stac hcomplex_mult_commute 1 THEN rtac hcomplex_mult_minus1 1); |
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122 qed "hcomplex_mult_minus1_right"; |
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123 |
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124 Addsimps [hcomplex_mult_minus1,hcomplex_mult_minus1_right]; |
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125 |
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126 (*Maps 0 to Numeral0 and 1 to Numeral1 and -Numeral1 to -1*) |
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127 val hcomplex_numeral_ss = |
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128 complex_numeral_ss addsimps [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym, |
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129 hcomplex_minus_1_eq_m1]; |
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130 |
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131 fun rename_numerals th = |
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132 asm_full_simplify hcomplex_numeral_ss (Thm.transfer (the_context ()) th); |
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133 |
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134 |
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135 (*Now insert some identities previously stated for 0 and 1hc*) |
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136 |
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137 |
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138 Addsimps [hcomplex_numeral_0_eq_0,hcomplex_numeral_1_eq_1]; |
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139 |
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140 Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::hcomplex)"; |
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141 by (auto_tac (claset(),simpset() addsimps [hcomplex_add_assoc RS sym])); |
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142 qed "hcomplex_add_number_of_left"; |
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143 |
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144 Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::hcomplex)"; |
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145 by (simp_tac (simpset() addsimps [hcomplex_mult_assoc RS sym]) 1); |
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146 qed "hcomplex_mult_number_of_left"; |
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147 |
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148 Goalw [hcomplex_diff_def] |
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149 "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::hcomplex)"; |
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150 by (rtac hcomplex_add_number_of_left 1); |
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151 qed "hcomplex_add_number_of_diff1"; |
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152 |
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153 Goal "number_of v + (c - number_of w) = \ |
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154 \ number_of (bin_add v (bin_minus w)) + (c::hcomplex)"; |
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155 by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_def]@ hcomplex_add_ac)); |
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156 qed "hcomplex_add_number_of_diff2"; |
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157 |
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158 Addsimps [hcomplex_add_number_of_left, hcomplex_mult_number_of_left, |
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159 hcomplex_add_number_of_diff1, hcomplex_add_number_of_diff2]; |
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160 |
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161 |
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162 (**** Simprocs for numeric literals ****) |
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163 |
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164 (** Combining of literal coefficients in sums of products **) |
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165 |
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166 Goal "(x = y) = (x-y = (0::hcomplex))"; |
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167 by (simp_tac (simpset() addsimps [hcomplex_diff_eq_eq]) 1); |
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168 qed "hcomplex_eq_iff_diff_eq_0"; |
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169 |
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170 (** For combine_numerals **) |
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171 |
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172 Goal "i*u + (j*u + k) = (i+j)*u + (k::hcomplex)"; |
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173 by (asm_simp_tac (simpset() addsimps [hcomplex_add_mult_distrib] |
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174 @ hcomplex_add_ac) 1); |
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175 qed "left_hcomplex_add_mult_distrib"; |
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176 |
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177 (** For cancel_numerals **) |
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178 |
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179 Goal "((x::hcomplex) = u + v) = (x - (u + v) = 0)"; |
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180 by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq])); |
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181 qed "hcomplex_eq_add_diff_eq_0"; |
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182 |
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183 Goal "((x::hcomplex) = n) = (x - n = 0)"; |
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184 by (auto_tac (claset(),simpset() addsimps [hcomplex_diff_eq_eq])); |
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185 qed "hcomplex_eq_diff_eq_0"; |
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186 |
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187 val hcomplex_rel_iff_rel_0_rls = [hcomplex_eq_diff_eq_0,hcomplex_eq_add_diff_eq_0]; |
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188 |
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189 Goal "!!i::hcomplex. (i*u + m = j*u + n) = ((i-j)*u + m = n)"; |
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190 by (auto_tac (claset(), simpset() addsimps [hcomplex_add_mult_distrib, |
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191 hcomplex_diff_def] @ hcomplex_add_ac)); |
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192 by (asm_simp_tac (simpset() addsimps [hcomplex_add_assoc RS sym]) 1); |
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193 by (simp_tac (simpset() addsimps [hcomplex_add_assoc]) 1); |
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194 qed "hcomplex_eq_add_iff1"; |
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195 |
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196 Goal "!!i::hcomplex. (i*u + m = j*u + n) = (m = (j-i)*u + n)"; |
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197 by (res_inst_tac [("z","i")] eq_Abs_hcomplex 1); |
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198 by (res_inst_tac [("z","j")] eq_Abs_hcomplex 1); |
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199 by (res_inst_tac [("z","u")] eq_Abs_hcomplex 1); |
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200 by (res_inst_tac [("z","m")] eq_Abs_hcomplex 1); |
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201 by (res_inst_tac [("z","n")] eq_Abs_hcomplex 1); |
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202 by (auto_tac (claset(), simpset() addsimps [hcomplex_diff,hcomplex_add, |
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203 hcomplex_mult,complex_eq_add_iff2])); |
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204 qed "hcomplex_eq_add_iff2"; |
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205 |
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206 |
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207 structure HComplex_Numeral_Simprocs = |
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208 struct |
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209 |
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210 (*Utilities*) |
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211 |
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212 val hcomplexT = Type("NSComplex.hcomplex",[]); |
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213 |
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214 fun mk_numeral n = HOLogic.number_of_const hcomplexT $ HOLogic.mk_bin n; |
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215 |
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216 val dest_numeral = Complex_Numeral_Simprocs.dest_numeral; |
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217 |
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218 val find_first_numeral = Complex_Numeral_Simprocs.find_first_numeral; |
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219 |
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220 val zero = mk_numeral 0; |
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221 val mk_plus = HOLogic.mk_binop "op +"; |
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222 |
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223 val uminus_const = Const ("uminus", hcomplexT --> hcomplexT); |
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224 |
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225 (*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*) |
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226 fun mk_sum [] = zero |
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227 | mk_sum [t,u] = mk_plus (t, u) |
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228 | mk_sum (t :: ts) = mk_plus (t, mk_sum ts); |
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229 |
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230 (*this version ALWAYS includes a trailing zero*) |
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231 fun long_mk_sum [] = zero |
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232 | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts); |
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233 |
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234 val dest_plus = HOLogic.dest_bin "op +" hcomplexT; |
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235 |
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236 (*decompose additions AND subtractions as a sum*) |
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237 fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) = |
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238 dest_summing (pos, t, dest_summing (pos, u, ts)) |
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239 | dest_summing (pos, Const ("op -", _) $ t $ u, ts) = |
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240 dest_summing (pos, t, dest_summing (not pos, u, ts)) |
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241 | dest_summing (pos, t, ts) = |
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242 if pos then t::ts else uminus_const$t :: ts; |
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243 |
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244 fun dest_sum t = dest_summing (true, t, []); |
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245 |
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246 val mk_diff = HOLogic.mk_binop "op -"; |
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247 val dest_diff = HOLogic.dest_bin "op -" hcomplexT; |
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248 |
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249 val one = mk_numeral 1; |
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250 val mk_times = HOLogic.mk_binop "op *"; |
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251 |
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252 fun mk_prod [] = one |
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253 | mk_prod [t] = t |
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254 | mk_prod (t :: ts) = if t = one then mk_prod ts |
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255 else mk_times (t, mk_prod ts); |
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256 |
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257 val dest_times = HOLogic.dest_bin "op *" hcomplexT; |
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258 |
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259 fun dest_prod t = |
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260 let val (t,u) = dest_times t |
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261 in dest_prod t @ dest_prod u end |
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262 handle TERM _ => [t]; |
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263 |
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264 (*DON'T do the obvious simplifications; that would create special cases*) |
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265 fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts); |
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266 |
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267 (*Express t as a product of (possibly) a numeral with other sorted terms*) |
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268 fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t |
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269 | dest_coeff sign t = |
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270 let val ts = sort Term.term_ord (dest_prod t) |
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271 val (n, ts') = find_first_numeral [] ts |
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272 handle TERM _ => (1, ts) |
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273 in (sign*n, mk_prod ts') end; |
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274 |
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275 (*Find first coefficient-term THAT MATCHES u*) |
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276 fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) |
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277 | find_first_coeff past u (t::terms) = |
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278 let val (n,u') = dest_coeff 1 t |
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279 in if u aconv u' then (n, rev past @ terms) |
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280 else find_first_coeff (t::past) u terms |
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281 end |
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282 handle TERM _ => find_first_coeff (t::past) u terms; |
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283 |
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284 |
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285 |
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286 (*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*) |
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287 val add_0s = map rename_numerals [hcomplex_add_zero_left, hcomplex_add_zero_right]; |
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288 val mult_plus_1s = map rename_numerals [hcomplex_mult_one_left, hcomplex_mult_one_right]; |
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289 val mult_minus_1s = map rename_numerals [hcomplex_mult_minus1, hcomplex_mult_minus1_right]; |
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290 val mult_1s = mult_plus_1s @ mult_minus_1s; |
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291 |
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292 (*To perform binary arithmetic*) |
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293 val bin_simps = |
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294 [hcomplex_numeral_0_eq_0 RS sym, hcomplex_numeral_1_eq_1 RS sym, |
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295 add_hcomplex_number_of, hcomplex_add_number_of_left, |
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296 minus_hcomplex_number_of, diff_hcomplex_number_of, mult_hcomplex_number_of, |
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297 hcomplex_mult_number_of_left] @ bin_arith_simps @ bin_rel_simps; |
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298 |
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299 (*To evaluate binary negations of coefficients*) |
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300 val hcomplex_minus_simps = NCons_simps @ |
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301 [hcomplex_minus_1_eq_m1,minus_hcomplex_number_of, |
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302 bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min, |
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303 bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min]; |
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304 |
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305 |
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306 (*To let us treat subtraction as addition*) |
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307 val diff_simps = [hcomplex_diff_def, hcomplex_minus_add_distrib, |
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308 hcomplex_minus_minus]; |
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309 |
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310 (*push the unary minus down: - x * y = x * - y *) |
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311 val hcomplex_minus_mult_eq_1_to_2 = |
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312 [hcomplex_minus_mult_eq1 RS sym, hcomplex_minus_mult_eq2] MRS trans |
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313 |> standard; |
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314 |
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315 (*to extract again any uncancelled minuses*) |
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316 val hcomplex_minus_from_mult_simps = |
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317 [hcomplex_minus_minus, hcomplex_minus_mult_eq1 RS sym, |
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318 hcomplex_minus_mult_eq2 RS sym]; |
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319 |
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320 (*combine unary minus with numeric literals, however nested within a product*) |
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321 val hcomplex_mult_minus_simps = |
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322 [hcomplex_mult_assoc, hcomplex_minus_mult_eq1, hcomplex_minus_mult_eq_1_to_2]; |
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323 |
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324 (*Final simplification: cancel + and * *) |
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325 val simplify_meta_eq = |
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326 Int_Numeral_Simprocs.simplify_meta_eq |
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327 [hcomplex_add_zero_left, hcomplex_add_zero_right, |
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328 hcomplex_mult_zero_left, hcomplex_mult_zero_right, hcomplex_mult_one_left, |
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329 hcomplex_mult_one_right]; |
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330 |
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331 val prep_simproc = Complex_Numeral_Simprocs.prep_simproc; |
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332 |
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333 |
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334 structure CancelNumeralsCommon = |
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335 struct |
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336 val mk_sum = mk_sum |
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337 val dest_sum = dest_sum |
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338 val mk_coeff = mk_coeff |
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339 val dest_coeff = dest_coeff 1 |
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340 val find_first_coeff = find_first_coeff [] |
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341 val trans_tac = Real_Numeral_Simprocs.trans_tac |
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342 val norm_tac = |
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343 ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@ |
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344 hcomplex_minus_simps@hcomplex_add_ac)) |
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345 THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hcomplex_mult_minus_simps)) |
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346 THEN ALLGOALS |
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347 (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps@ |
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348 hcomplex_add_ac@hcomplex_mult_ac)) |
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349 val numeral_simp_tac = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps)) |
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350 val simplify_meta_eq = simplify_meta_eq |
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351 end; |
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352 |
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353 |
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354 structure EqCancelNumerals = CancelNumeralsFun |
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355 (open CancelNumeralsCommon |
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356 val prove_conv = Bin_Simprocs.prove_conv |
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357 val mk_bal = HOLogic.mk_eq |
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358 val dest_bal = HOLogic.dest_bin "op =" hcomplexT |
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359 val bal_add1 = hcomplex_eq_add_iff1 RS trans |
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360 val bal_add2 = hcomplex_eq_add_iff2 RS trans |
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361 ); |
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362 |
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363 |
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364 val cancel_numerals = |
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365 map prep_simproc |
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366 [("hcomplexeq_cancel_numerals", |
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367 ["(l::hcomplex) + m = n", "(l::hcomplex) = m + n", |
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368 "(l::hcomplex) - m = n", "(l::hcomplex) = m - n", |
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369 "(l::hcomplex) * m = n", "(l::hcomplex) = m * n"], |
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370 EqCancelNumerals.proc)]; |
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371 |
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372 structure CombineNumeralsData = |
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373 struct |
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374 val add = op + : int*int -> int |
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375 val mk_sum = long_mk_sum (*to work for e.g. #2*x + #3*x *) |
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376 val dest_sum = dest_sum |
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377 val mk_coeff = mk_coeff |
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378 val dest_coeff = dest_coeff 1 |
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379 val left_distrib = left_hcomplex_add_mult_distrib RS trans |
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380 val prove_conv = Bin_Simprocs.prove_conv_nohyps |
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381 val trans_tac = Real_Numeral_Simprocs.trans_tac |
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382 val norm_tac = |
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383 ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@ |
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384 hcomplex_minus_simps@hcomplex_add_ac)) |
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385 THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hcomplex_mult_minus_simps)) |
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386 THEN ALLGOALS (simp_tac (HOL_ss addsimps hcomplex_minus_from_mult_simps@ |
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387 hcomplex_add_ac@hcomplex_mult_ac)) |
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388 val numeral_simp_tac = ALLGOALS |
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389 (simp_tac (HOL_ss addsimps add_0s@bin_simps)) |
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390 val simplify_meta_eq = simplify_meta_eq |
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391 end; |
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392 |
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393 structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData); |
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394 |
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395 val combine_numerals = |
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396 prep_simproc ("hcomplex_combine_numerals", |
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397 ["(i::hcomplex) + j", "(i::hcomplex) - j"], |
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398 CombineNumerals.proc); |
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399 |
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400 (** Declarations for ExtractCommonTerm **) |
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401 |
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402 (*this version ALWAYS includes a trailing one*) |
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403 fun long_mk_prod [] = one |
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404 | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts); |
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405 |
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406 (*Find first term that matches u*) |
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407 fun find_first past u [] = raise TERM("find_first", []) |
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408 | find_first past u (t::terms) = |
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409 if u aconv t then (rev past @ terms) |
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410 else find_first (t::past) u terms |
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411 handle TERM _ => find_first (t::past) u terms; |
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412 |
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413 (*Final simplification: cancel + and * *) |
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414 fun cancel_simplify_meta_eq cancel_th th = |
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415 Int_Numeral_Simprocs.simplify_meta_eq |
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416 [hcomplex_mult_one_left, hcomplex_mult_one_right] |
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417 (([th, cancel_th]) MRS trans); |
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418 |
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419 (*** Making constant folding work for 0 and 1 too ***) |
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420 |
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421 structure HComplexAbstractNumeralsData = |
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422 struct |
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423 val dest_eq = HOLogic.dest_eq o HOLogic.dest_Trueprop o concl_of |
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424 val is_numeral = Bin_Simprocs.is_numeral |
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425 val numeral_0_eq_0 = hcomplex_numeral_0_eq_0 |
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426 val numeral_1_eq_1 = hcomplex_numeral_1_eq_1 |
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427 val prove_conv = Bin_Simprocs.prove_conv_nohyps_novars |
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428 fun norm_tac simps = ALLGOALS (simp_tac (HOL_ss addsimps simps)) |
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429 val simplify_meta_eq = Bin_Simprocs.simplify_meta_eq |
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430 end |
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431 |
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432 structure HComplexAbstractNumerals = AbstractNumeralsFun (HComplexAbstractNumeralsData) |
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433 |
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434 (*For addition, we already have rules for the operand 0. |
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435 Multiplication is omitted because there are already special rules for |
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436 both 0 and 1 as operands. Unary minus is trivial, just have - 1 = -1. |
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437 For the others, having three patterns is a compromise between just having |
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438 one (many spurious calls) and having nine (just too many!) *) |
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439 val eval_numerals = |
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440 map prep_simproc |
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441 [("hcomplex_add_eval_numerals", |
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442 ["(m::hcomplex) + 1", "(m::hcomplex) + number_of v"], |
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443 HComplexAbstractNumerals.proc add_hcomplex_number_of), |
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444 ("hcomplex_diff_eval_numerals", |
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445 ["(m::hcomplex) - 1", "(m::hcomplex) - number_of v"], |
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446 HComplexAbstractNumerals.proc diff_hcomplex_number_of), |
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447 ("hcomplex_eq_eval_numerals", |
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448 ["(m::hcomplex) = 0", "(m::hcomplex) = 1", "(m::hcomplex) = number_of v"], |
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449 HComplexAbstractNumerals.proc eq_hcomplex_number_of)] |
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450 |
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451 end; |
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452 |
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453 Addsimprocs HComplex_Numeral_Simprocs.eval_numerals; |
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454 Addsimprocs HComplex_Numeral_Simprocs.cancel_numerals; |
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455 Addsimprocs [HComplex_Numeral_Simprocs.combine_numerals]; |
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456 |
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457 |
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458 (*examples: |
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459 print_depth 22; |
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460 set timing; |
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461 set trace_simp; |
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462 fun test s = (Goal s, by (Simp_tac 1)); |
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463 |
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464 test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::hcomplex)"; |
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465 test " 2*u = (u::hcomplex)"; |
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466 test "(i + j + 12 + (k::hcomplex)) - 15 = y"; |
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467 test "(i + j + 12 + (k::hcomplex)) - 5 = y"; |
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468 |
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469 test "( 2*x - (u*v) + y) - v* 3*u = (w::hcomplex)"; |
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470 test "( 2*x*u*v + (u*v)* 4 + y) - v*u* 4 = (w::hcomplex)"; |
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471 test "( 2*x*u*v + (u*v)* 4 + y) - v*u = (w::hcomplex)"; |
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472 test "u*v - (x*u*v + (u*v)* 4 + y) = (w::hcomplex)"; |
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473 |
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474 test "(i + j + 12 + (k::hcomplex)) = u + 15 + y"; |
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475 test "(i + j* 2 + 12 + (k::hcomplex)) = j + 5 + y"; |
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476 |
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477 test " 2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::hcomplex)"; |
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478 |
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479 test "a + -(b+c) + b = (d::hcomplex)"; |
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480 test "a + -(b+c) - b = (d::hcomplex)"; |
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481 |
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482 (*negative numerals*) |
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483 test "(i + j + -2 + (k::hcomplex)) - (u + 5 + y) = zz"; |
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484 |
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485 test "(i + j + -12 + (k::hcomplex)) - 15 = y"; |
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486 test "(i + j + 12 + (k::hcomplex)) - -15 = y"; |
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487 test "(i + j + -12 + (k::hcomplex)) - -15 = y"; |
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488 *) |
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489 |
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490 (** Constant folding for hcomplex plus and times **) |
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491 |
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492 structure HComplex_Times_Assoc_Data : ASSOC_FOLD_DATA = |
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493 struct |
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494 val ss = HOL_ss |
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495 val eq_reflection = eq_reflection |
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496 val sg_ref = Sign.self_ref (Theory.sign_of (the_context ())) |
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497 val T = HComplex_Numeral_Simprocs.hcomplexT |
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498 val plus = Const ("op *", [T,T] ---> T) |
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499 val add_ac = hcomplex_mult_ac |
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500 end; |
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501 |
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502 structure HComplex_Times_Assoc = Assoc_Fold (HComplex_Times_Assoc_Data); |
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503 |
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504 Addsimprocs [HComplex_Times_Assoc.conv]; |
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505 |
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506 Addsimps [hcomplex_of_complex_zero_iff]; |
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507 |
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508 (*Simplification of x-y = 0 *) |
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509 |
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510 AddIffs [hcomplex_eq_iff_diff_eq_0 RS sym]; |
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511 |
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512 (** extra thms **) |
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513 |
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514 Goal "(hcnj z = 0) = (z = 0)"; |
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515 by (auto_tac (claset(),simpset() addsimps [hcomplex_hcnj_zero_iff])); |
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516 qed "hcomplex_hcnj_num_zero_iff"; |
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517 Addsimps [hcomplex_hcnj_num_zero_iff]; |
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518 |
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519 Goal "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})"; |
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520 by (simp_tac (simpset() addsimps [hcomplex_zero_def RS meta_eq_to_obj_eq RS sym]) 1); |
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521 qed "hcomplex_zero_num"; |
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522 |
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523 Goal "1 = Abs_hcomplex (hcomplexrel `` {%n. 1})"; |
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524 by (simp_tac (simpset() addsimps [hcomplex_one_def RS meta_eq_to_obj_eq RS sym]) 1); |
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525 qed "hcomplex_one_num"; |
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526 |
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527 (*** Real and imaginary stuff ***) |
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528 |
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529 Goalw [hcomplex_number_of_def] |
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530 "((number_of xa :: hcomplex) + iii * number_of ya = \ |
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531 \ number_of xb + iii * number_of yb) = \ |
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532 \ (((number_of xa :: hcomplex) = number_of xb) & \ |
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533 \ ((number_of ya :: hcomplex) = number_of yb))"; |
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534 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff, |
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535 hcomplex_hypreal_number_of])); |
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536 qed "hcomplex_number_of_eq_cancel_iff"; |
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537 Addsimps [hcomplex_number_of_eq_cancel_iff]; |
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538 |
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539 Goalw [hcomplex_number_of_def] |
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540 "((number_of xa :: hcomplex) + number_of ya * iii = \ |
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541 \ number_of xb + number_of yb * iii) = \ |
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542 \ (((number_of xa :: hcomplex) = number_of xb) & \ |
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543 \ ((number_of ya :: hcomplex) = number_of yb))"; |
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544 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA, |
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545 hcomplex_hypreal_number_of])); |
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546 qed "hcomplex_number_of_eq_cancel_iffA"; |
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547 Addsimps [hcomplex_number_of_eq_cancel_iffA]; |
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548 |
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549 Goalw [hcomplex_number_of_def] |
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550 "((number_of xa :: hcomplex) + number_of ya * iii = \ |
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551 \ number_of xb + iii * number_of yb) = \ |
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552 \ (((number_of xa :: hcomplex) = number_of xb) & \ |
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553 \ ((number_of ya :: hcomplex) = number_of yb))"; |
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554 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB, |
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555 hcomplex_hypreal_number_of])); |
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556 qed "hcomplex_number_of_eq_cancel_iffB"; |
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557 Addsimps [hcomplex_number_of_eq_cancel_iffB]; |
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558 |
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559 Goalw [hcomplex_number_of_def] |
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560 "((number_of xa :: hcomplex) + iii * number_of ya = \ |
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561 \ number_of xb + number_of yb * iii) = \ |
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562 \ (((number_of xa :: hcomplex) = number_of xb) & \ |
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563 \ ((number_of ya :: hcomplex) = number_of yb))"; |
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564 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC, |
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565 hcomplex_hypreal_number_of])); |
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566 qed "hcomplex_number_of_eq_cancel_iffC"; |
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567 Addsimps [hcomplex_number_of_eq_cancel_iffC]; |
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568 |
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569 Goalw [hcomplex_number_of_def] |
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570 "((number_of xa :: hcomplex) + iii * number_of ya = \ |
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571 \ number_of xb) = \ |
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572 \ (((number_of xa :: hcomplex) = number_of xb) & \ |
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573 \ ((number_of ya :: hcomplex) = 0))"; |
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574 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2, |
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575 hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff])); |
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576 qed "hcomplex_number_of_eq_cancel_iff2"; |
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577 Addsimps [hcomplex_number_of_eq_cancel_iff2]; |
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578 |
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579 Goalw [hcomplex_number_of_def] |
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580 "((number_of xa :: hcomplex) + number_of ya * iii = \ |
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581 \ number_of xb) = \ |
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582 \ (((number_of xa :: hcomplex) = number_of xb) & \ |
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583 \ ((number_of ya :: hcomplex) = 0))"; |
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584 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a, |
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585 hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff])); |
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586 qed "hcomplex_number_of_eq_cancel_iff2a"; |
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587 Addsimps [hcomplex_number_of_eq_cancel_iff2a]; |
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588 |
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589 Goalw [hcomplex_number_of_def] |
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590 "((number_of xa :: hcomplex) + iii * number_of ya = \ |
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591 \ iii * number_of yb) = \ |
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592 \ (((number_of xa :: hcomplex) = 0) & \ |
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593 \ ((number_of ya :: hcomplex) = number_of yb))"; |
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594 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3, |
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595 hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff])); |
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596 qed "hcomplex_number_of_eq_cancel_iff3"; |
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597 Addsimps [hcomplex_number_of_eq_cancel_iff3]; |
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598 |
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599 Goalw [hcomplex_number_of_def] |
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600 "((number_of xa :: hcomplex) + number_of ya * iii= \ |
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601 \ iii * number_of yb) = \ |
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602 \ (((number_of xa :: hcomplex) = 0) & \ |
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603 \ ((number_of ya :: hcomplex) = number_of yb))"; |
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604 by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a, |
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605 hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff])); |
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606 qed "hcomplex_number_of_eq_cancel_iff3a"; |
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607 Addsimps [hcomplex_number_of_eq_cancel_iff3a]; |
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608 |
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609 Goalw [hcomplex_number_of_def] "hcnj (number_of v :: hcomplex) = number_of v"; |
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610 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1); |
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611 by (rtac hcomplex_hcnj_hcomplex_of_hypreal 1); |
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612 qed "hcomplex_number_of_hcnj"; |
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613 Addsimps [hcomplex_number_of_hcnj]; |
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614 |
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615 Goalw [hcomplex_number_of_def] |
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616 "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)"; |
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617 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1); |
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618 by (auto_tac (claset(), HOL_ss addsimps [hcmod_hcomplex_of_hypreal])); |
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619 qed "hcomplex_number_of_hcmod"; |
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620 Addsimps [hcomplex_number_of_hcmod]; |
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621 |
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622 Goalw [hcomplex_number_of_def] |
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623 "hRe(number_of v :: hcomplex) = number_of v"; |
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624 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1); |
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625 by (auto_tac (claset(), HOL_ss addsimps [hRe_hcomplex_of_hypreal])); |
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626 qed "hcomplex_number_of_hRe"; |
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627 Addsimps [hcomplex_number_of_hRe]; |
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628 |
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629 Goalw [hcomplex_number_of_def] |
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630 "hIm(number_of v :: hcomplex) = 0"; |
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631 by (rtac (hcomplex_hypreal_number_of RS ssubst) 1); |
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632 by (auto_tac (claset(), HOL_ss addsimps [hIm_hcomplex_of_hypreal])); |
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633 qed "hcomplex_number_of_hIm"; |
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634 Addsimps [hcomplex_number_of_hIm]; |
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635 |
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636 |
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637 |