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1 (* WN.020812: theorems in the Reals, |
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2 necessary for special rule sets, in addition to Isabelle2002. |
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3 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
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4 !!! THIS IS THE _least_ NUMBER OF ADDITIONAL THEOREMS !!! |
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5 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
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6 xxxI contain ^^^ instead of ^ in the respective theorem xxx in 2002 |
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7 changed by: Richard Lang 020912 |
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8 *) |
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9 |
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10 (* |
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11 use_thy"IsacKnowledge/Poly"; |
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12 use_thy"Poly"; |
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13 use_thy_only"IsacKnowledge/Poly"; |
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14 |
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15 remove_thy"Poly"; |
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16 use_thy"IsacKnowledge/Isac"; |
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17 |
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18 |
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19 use"ROOT.ML"; |
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20 cd"IsacKnowledge"; |
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21 *) |
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22 |
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23 Poly = Simplify + |
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24 |
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25 (*-------------------- consts-----------------------------------------------*) |
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26 consts |
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27 |
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28 is'_expanded'_in :: "[real, real] => bool" ("_ is'_expanded'_in _") |
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29 is'_poly'_in :: "[real, real] => bool" ("_ is'_poly'_in _") (*RL DA *) |
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30 has'_degree'_in :: "[real, real] => real" ("_ has'_degree'_in _")(*RL DA *) |
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31 is'_polyrat'_in :: "[real, real] => bool" ("_ is'_polyrat'_in _")(*RL030626*) |
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32 |
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33 is'_multUnordered :: "real => bool" ("_ is'_multUnordered") |
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34 is'_addUnordered :: "real => bool" ("_ is'_addUnordered") (*WN030618*) |
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35 is'_polyexp :: "real => bool" ("_ is'_polyexp") |
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36 |
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37 Expand'_binoms |
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38 :: "['y, \ |
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39 \ 'y] => 'y" |
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40 ("((Script Expand'_binoms (_ =))// \ |
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41 \ (_))" 9) |
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42 |
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43 (*-------------------- rules------------------------------------------------*) |
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44 rules (*.not contained in Isabelle2002, |
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45 stated as axioms, TODO: prove as theorems; |
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46 theorem-IDs 'xxxI' with ^^^ instead of ^ in 'xxx' in Isabelle2002.*) |
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47 |
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48 realpow_pow "(a ^^^ b) ^^^ c = a ^^^ (b * c)" |
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49 realpow_addI "r ^^^ (n + m) = r ^^^ n * r ^^^ m" |
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50 realpow_addI_assoc_l "r ^^^ n * (r ^^^ m * s) = r ^^^ (n + m) * s" |
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51 realpow_addI_assoc_r "s * r ^^^ n * r ^^^ m = s * r ^^^ (n + m)" |
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52 |
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53 realpow_oneI "r ^^^ 1 = r" |
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54 realpow_zeroI "r ^^^ 0 = 1" |
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55 realpow_eq_oneI "1 ^^^ n = 1" |
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56 realpow_multI "(r * s) ^^^ n = r ^^^ n * s ^^^ n" |
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57 realpow_multI_poly "[| r is_polyexp; s is_polyexp |] ==> \ |
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58 \(r * s) ^^^ n = r ^^^ n * s ^^^ n" |
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59 realpow_minus_oneI "-1 ^^^ (2 * n) = 1" |
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60 |
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61 realpow_twoI "r ^^^ 2 = r * r" |
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62 realpow_twoI_assoc_l "r * (r * s) = r ^^^ 2 * s" |
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63 realpow_twoI_assoc_r "s * r * r = s * r ^^^ 2" |
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64 realpow_two_atom "r is_atom ==> r * r = r ^^^ 2" |
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65 realpow_plus_1 "r * r ^^^ n = r ^^^ (n + 1)" |
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66 realpow_plus_1_assoc_l "r * (r ^^^ m * s) = r ^^^ (1 + m) * s" |
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67 realpow_plus_1_assoc_l2 "r ^^^ m * (r * s) = r ^^^ (1 + m) * s" |
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68 realpow_plus_1_assoc_r "s * r * r ^^^ m = s * r ^^^ (1 + m)" |
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69 realpow_plus_1_atom "r is_atom ==> r * r ^^^ n = r ^^^ (1 + n)" |
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70 realpow_def_atom "[| Not (r is_atom); 1 < n |] \ |
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71 \ ==> r ^^^ n = r * r ^^^ (n + -1)" |
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72 realpow_addI_atom "r is_atom ==> r ^^^ n * r ^^^ m = r ^^^ (n + m)" |
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73 |
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74 |
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75 realpow_minus_even "n is_even ==> (- r) ^^^ n = r ^^^ n" |
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76 realpow_minus_odd "Not (n is_even) ==> (- r) ^^^ n = -1 * r ^^^ n" |
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77 |
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78 |
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79 (* RL 020914 *) |
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80 real_pp_binom_times "(a + b)*(c + d) = a*c + a*d + b*c + b*d" |
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81 real_pm_binom_times "(a + b)*(c - d) = a*c - a*d + b*c - b*d" |
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82 real_mp_binom_times "(a - b)*(c + d) = a*c + a*d - b*c - b*d" |
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83 real_mm_binom_times "(a - b)*(c - d) = a*c - a*d - b*c + b*d" |
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84 real_plus_binom_pow3 "(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3" |
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85 real_plus_binom_pow3_poly "[| a is_polyexp; b is_polyexp |] ==> \ |
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86 \(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3" |
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87 real_minus_binom_pow3 "(a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3" |
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88 real_minus_binom_pow3_p "(a + -1 * b)^^^3 = a^^^3 + -3*a^^^2*b + 3*a*b^^^2 + -1*b^^^3" |
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89 (* real_plus_binom_pow "[| n is_const; 3 < n |] ==> \ |
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90 \(a + b)^^^n = (a + b) * (a + b)^^^(n - 1)" *) |
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91 real_plus_binom_pow4 "(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a + b)" |
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92 real_plus_binom_pow4_poly "[| a is_polyexp; b is_polyexp |] ==> \ |
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93 \(a + b)^^^4 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a + b)" |
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94 real_plus_binom_pow5 "(a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a^^^2 + 2*a*b + b^^^2)" |
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95 |
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96 real_plus_binom_pow5_poly "[| a is_polyexp; b is_polyexp |] ==> \ |
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97 \(a + b)^^^5 = (a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3)*(a^^^2 + 2*a*b + b^^^2)" |
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98 |
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99 real_diff_plus "a - b = a + -b" (*17.3.03: do_NOT_use*) |
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100 real_diff_minus "a - b = a + -1 * b" |
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101 real_plus_binom_times "(a + b)*(a + b) = a^^^2 + 2*a*b + b^^^2" |
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102 real_minus_binom_times "(a - b)*(a - b) = a^^^2 - 2*a*b + b^^^2" |
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103 (*WN071229 changed for Schaerding -----vvv*) |
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104 (*real_plus_binom_pow2 "(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"*) |
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105 real_plus_binom_pow2 "(a + b)^^^2 = (a + b) * (a + b)" |
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106 (*WN071229 changed for Schaerding -----^^^*) |
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107 real_plus_binom_pow2_poly "[| a is_polyexp; b is_polyexp |] ==> \ |
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108 \(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2" |
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109 real_minus_binom_pow2 "(a - b)^^^2 = a^^^2 - 2*a*b + b^^^2" |
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110 real_minus_binom_pow2_p "(a - b)^^^2 = a^^^2 + -2*a*b + b^^^2" |
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111 real_plus_minus_binom1 "(a + b)*(a - b) = a^^^2 - b^^^2" |
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112 real_plus_minus_binom1_p "(a + b)*(a - b) = a^^^2 + -1*b^^^2" |
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113 real_plus_minus_binom1_p_p "(a + b)*(a + -1 * b) = a^^^2 + -1*b^^^2" |
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114 real_plus_minus_binom2 "(a - b)*(a + b) = a^^^2 - b^^^2" |
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115 real_plus_minus_binom2_p "(a - b)*(a + b) = a^^^2 + -1*b^^^2" |
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116 real_plus_minus_binom2_p_p "(a + -1 * b)*(a + b) = a^^^2 + -1*b^^^2" |
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117 real_plus_binom_times1 "(a + 1*b)*(a + -1*b) = a^^^2 + -1*b^^^2" |
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118 real_plus_binom_times2 "(a + -1*b)*(a + 1*b) = a^^^2 + -1*b^^^2" |
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119 |
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120 real_num_collect "[| l is_const; m is_const |] ==> \ |
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121 \l * n + m * n = (l + m) * n" |
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122 (* FIXME.MG.0401: replace 'real_num_collect_assoc' |
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123 by 'real_num_collect_assoc_l' ... are equal, introduced by MG ! *) |
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124 real_num_collect_assoc "[| l is_const; m is_const |] ==> \ |
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125 \l * n + (m * n + k) = (l + m) * n + k" |
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126 real_num_collect_assoc_l "[| l is_const; m is_const |] ==> \ |
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127 \l * n + (m * n + k) = (l + m) |
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128 * n + k" |
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129 real_num_collect_assoc_r "[| l is_const; m is_const |] ==> \ |
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130 \(k + m * n) + l * n = k + (l + m) * n" |
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131 real_one_collect "m is_const ==> n + m * n = (1 + m) * n" |
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132 (* FIXME.MG.0401: replace 'real_one_collect_assoc' |
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133 by 'real_one_collect_assoc_l' ... are equal, introduced by MG ! *) |
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134 real_one_collect_assoc "m is_const ==> n + (m * n + k) = (1 + m)* n + k" |
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135 |
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136 real_one_collect_assoc_l "m is_const ==> n + (m * n + k) = (1 + m) * n + k" |
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137 real_one_collect_assoc_r "m is_const ==>(k + n) + m * n = k + (1 + m) * n" |
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138 |
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139 (* FIXME.MG.0401: replace 'real_mult_2_assoc' |
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140 by 'real_mult_2_assoc_l' ... are equal, introduced by MG ! *) |
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141 real_mult_2_assoc "z1 + (z1 + k) = 2 * z1 + k" |
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142 real_mult_2_assoc_l "z1 + (z1 + k) = 2 * z1 + k" |
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143 real_mult_2_assoc_r "(k + z1) + z1 = k + 2 * z1" |
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144 |
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145 real_add_mult_distrib_poly "w is_polyexp ==> (z1 + z2) * w = z1 * w + z2 * w" |
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146 real_add_mult_distrib2_poly "w is_polyexp ==> w * (z1 + z2) = w * z1 + w * z2" |
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147 end |