1 (* Title: HOL/Real/real_arith0.ML |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow, TU Muenchen |
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4 Copyright 1999 TU Muenchen |
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5 |
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6 Instantiation of the generic linear arithmetic package for type real. |
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7 *) |
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8 |
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9 val add_zero_left = thm"Ring_and_Field.add_0"; |
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10 val add_zero_right = thm"Ring_and_Field.add_0_right"; |
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11 |
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12 val real_mult_left_mono = |
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13 read_instantiate_sg(sign_of RealBin.thy) [("a","?a::real")] mult_left_mono; |
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14 |
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15 |
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16 local |
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17 |
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18 (* reduce contradictory <= to False *) |
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19 val add_rules = |
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20 [order_less_irrefl, real_numeral_0_eq_0, real_numeral_1_eq_1, |
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21 real_minus_1_eq_m1, |
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22 add_real_number_of, minus_real_number_of, diff_real_number_of, |
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23 mult_real_number_of, eq_real_number_of, less_real_number_of, |
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24 le_real_number_of_eq_not_less, real_diff_def, |
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25 minus_add_distrib, minus_minus, mult_assoc, minus_zero, |
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26 add_zero_left, add_zero_right, left_minus, right_minus, |
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27 mult_left_zero, mult_right_zero, mult_1, mult_1_right, |
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28 minus_mult_left RS sym, minus_mult_right RS sym]; |
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29 |
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30 val simprocs = [Real_Times_Assoc.conv, Real_Numeral_Simprocs.combine_numerals]@ |
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31 Real_Numeral_Simprocs.cancel_numerals @ |
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32 Real_Numeral_Simprocs.eval_numerals; |
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33 |
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34 val mono_ss = simpset() addsimps |
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35 [add_mono,add_strict_mono, |
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36 real_add_less_le_mono,real_add_le_less_mono]; |
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37 |
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38 val add_mono_thms_real = |
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39 map (fn s => prove_goal (the_context ()) s |
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40 (fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 1])) |
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41 ["(i <= j) & (k <= l) ==> i + k <= j + (l::real)", |
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42 "(i = j) & (k <= l) ==> i + k <= j + (l::real)", |
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43 "(i <= j) & (k = l) ==> i + k <= j + (l::real)", |
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44 "(i = j) & (k = l) ==> i + k = j + (l::real)", |
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45 "(i < j) & (k = l) ==> i + k < j + (l::real)", |
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46 "(i = j) & (k < l) ==> i + k < j + (l::real)", |
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47 "(i < j) & (k <= l) ==> i + k < j + (l::real)", |
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48 "(i <= j) & (k < l) ==> i + k < j + (l::real)", |
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49 "(i < j) & (k < l) ==> i + k < j + (l::real)"]; |
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50 |
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51 fun cvar(th,_ $ (_ $ _ $ var)) = cterm_of (#sign(rep_thm th)) var; |
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52 |
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53 val real_mult_mono_thms = |
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54 [(rotate_prems 1 real_mult_less_mono2, |
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55 cvar(real_mult_less_mono2, hd(prems_of real_mult_less_mono2))), |
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56 (real_mult_left_mono, |
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57 cvar(real_mult_left_mono, hd(tl(prems_of real_mult_left_mono))))] |
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58 |
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59 in |
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60 |
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61 val fast_real_arith_simproc = Simplifier.simproc (Theory.sign_of (the_context ())) |
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62 "fast_real_arith" ["(m::real) < n","(m::real) <= n", "(m::real) = n"] |
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63 Fast_Arith.lin_arith_prover; |
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64 |
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65 val real_arith_setup = |
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66 [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} => |
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67 {add_mono_thms = add_mono_thms @ add_mono_thms_real, |
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68 mult_mono_thms = mult_mono_thms @ real_mult_mono_thms, |
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69 inj_thms = inj_thms, (*FIXME: add real*) |
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70 lessD = lessD, (*We don't change LA_Data_Ref.lessD because the real ordering is dense!*) |
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71 simpset = simpset addsimps add_rules |
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72 addsimprocs simprocs}), |
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73 arith_discrete ("RealDef.real",false), |
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74 Simplifier.change_simpset_of (op addsimprocs) [fast_real_arith_simproc]]; |
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75 |
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76 (* some thms for injection nat => real: |
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77 real_of_nat_zero |
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78 ?zero_eq_numeral_0 |
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79 real_of_nat_add |
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80 *) |
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81 |
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82 end; |
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83 |
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84 |
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85 (* Some test data [omitting examples that assume the ordering to be discrete!] |
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86 Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d"; |
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87 by (fast_arith_tac 1); |
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88 qed ""; |
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89 |
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90 Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c"; |
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91 by (fast_arith_tac 1); |
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92 qed ""; |
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93 |
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94 Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j"; |
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95 by (fast_arith_tac 1); |
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96 qed ""; |
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97 |
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98 Goal "!!a::real. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k"; |
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99 by (arith_tac 1); |
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100 qed ""; |
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101 |
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102 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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103 \ ==> a <= l"; |
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104 by (fast_arith_tac 1); |
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105 qed ""; |
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106 |
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107 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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108 \ ==> a+a+a+a <= l+l+l+l"; |
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109 by (fast_arith_tac 1); |
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110 qed ""; |
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111 |
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112 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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113 \ ==> a+a+a+a+a <= l+l+l+l+i"; |
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114 by (fast_arith_tac 1); |
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115 qed ""; |
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116 |
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117 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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118 \ ==> a+a+a+a+a+a <= l+l+l+l+i+l"; |
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119 by (fast_arith_tac 1); |
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120 qed ""; |
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121 |
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122 Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \ |
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123 \ ==> 6*a <= 5*l+i"; |
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124 by (fast_arith_tac 1); |
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125 qed ""; |
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126 |
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127 Goal "a<=b ==> a < b+(1::real)"; |
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128 by (fast_arith_tac 1); |
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129 qed ""; |
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130 |
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131 Goal "a<=b ==> a-(3::real) < b"; |
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132 by (fast_arith_tac 1); |
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133 qed ""; |
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134 |
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135 Goal "a<=b ==> a-(1::real) < b"; |
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136 by (fast_arith_tac 1); |
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137 qed ""; |
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138 |
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139 *) |
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