1 (* Title : Real/RealDef.ML |
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2 ID : $Id$ |
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3 Author : Jacques D. Fleuriot |
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4 Copyright : 1998 University of Cambridge |
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5 Description : The reals |
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6 *) |
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7 |
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8 (*** Proving that realrel is an equivalence relation ***) |
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9 |
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10 Goal "[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] \ |
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11 \ ==> x1 + y3 = x3 + y1"; |
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12 by (res_inst_tac [("C","y2")] preal_add_right_cancel 1); |
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13 by (rotate_tac 1 1 THEN dtac sym 1); |
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14 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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15 by (rtac (preal_add_left_commute RS subst) 1); |
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16 by (res_inst_tac [("x1","x1")] (preal_add_assoc RS subst) 1); |
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17 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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18 qed "preal_trans_lemma"; |
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19 |
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20 (** Natural deduction for realrel **) |
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21 |
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22 Goalw [realrel_def] |
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23 "(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)"; |
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24 by (Blast_tac 1); |
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25 qed "realrel_iff"; |
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26 |
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27 Goalw [realrel_def] |
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28 "[| x1 + y2 = x2 + y1 |] ==> ((x1,y1),(x2,y2)): realrel"; |
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29 by (Blast_tac 1); |
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30 qed "realrelI"; |
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31 |
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32 Goalw [realrel_def] |
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33 "p: realrel --> (EX x1 y1 x2 y2. \ |
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34 \ p = ((x1,y1),(x2,y2)) & x1 + y2 = x2 + y1)"; |
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35 by (Blast_tac 1); |
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36 qed "realrelE_lemma"; |
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37 |
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38 val [major,minor] = Goal |
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39 "[| p: realrel; \ |
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40 \ !!x1 y1 x2 y2. [| p = ((x1,y1),(x2,y2)); x1+y2 = x2+y1 \ |
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41 \ |] ==> Q |] ==> Q"; |
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42 by (cut_facts_tac [major RS (realrelE_lemma RS mp)] 1); |
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43 by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); |
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44 qed "realrelE"; |
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45 |
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46 Goal "(x,x): realrel"; |
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47 by (case_tac "x" 1); |
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48 by (asm_simp_tac (simpset() addsimps [realrel_def]) 1); |
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49 qed "realrel_refl"; |
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50 |
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51 Goalw [equiv_def, refl_def, sym_def, trans_def] "equiv UNIV realrel"; |
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52 by (fast_tac (claset() addSIs [realrelI, realrel_refl] |
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53 addSEs [sym, realrelE, preal_trans_lemma]) 1); |
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54 qed "equiv_realrel"; |
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55 |
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56 (* (realrel `` {x} = realrel `` {y}) = ((x,y) : realrel) *) |
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57 bind_thm ("equiv_realrel_iff", |
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58 [equiv_realrel, UNIV_I, UNIV_I] MRS eq_equiv_class_iff); |
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59 |
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60 Goalw [REAL_def,realrel_def,quotient_def] "realrel``{(x,y)}: REAL"; |
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61 by (Blast_tac 1); |
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62 qed "realrel_in_real"; |
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63 |
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64 Goal "inj_on Abs_REAL REAL"; |
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65 by (rtac inj_on_inverseI 1); |
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66 by (etac Abs_REAL_inverse 1); |
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67 qed "inj_on_Abs_REAL"; |
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68 |
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69 Addsimps [equiv_realrel_iff,inj_on_Abs_REAL RS inj_on_iff, |
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70 realrel_iff, realrel_in_real, Abs_REAL_inverse]; |
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71 |
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72 Addsimps [equiv_realrel RS eq_equiv_class_iff]; |
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73 bind_thm ("eq_realrelD", equiv_realrel RSN (2,eq_equiv_class)); |
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74 |
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75 Goal "inj Rep_REAL"; |
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76 by (rtac inj_inverseI 1); |
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77 by (rtac Rep_REAL_inverse 1); |
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78 qed "inj_Rep_REAL"; |
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79 |
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80 (** real_of_preal: the injection from preal to real **) |
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81 Goal "inj(real_of_preal)"; |
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82 by (rtac injI 1); |
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83 by (rewtac real_of_preal_def); |
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84 by (dtac (inj_on_Abs_REAL RS inj_onD) 1); |
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85 by (REPEAT (rtac realrel_in_real 1)); |
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86 by (dtac eq_equiv_class 1); |
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87 by (rtac equiv_realrel 1); |
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88 by (Blast_tac 1); |
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89 by (asm_full_simp_tac (simpset() addsimps [realrel_def]) 1); |
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90 qed "inj_real_of_preal"; |
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91 |
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92 val [prem] = Goal |
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93 "(!!x y. z = Abs_REAL(realrel``{(x,y)}) ==> P) ==> P"; |
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94 by (res_inst_tac [("x1","z")] |
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95 (rewrite_rule [REAL_def] Rep_REAL RS quotientE) 1); |
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96 by (dres_inst_tac [("f","Abs_REAL")] arg_cong 1); |
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97 by (case_tac "x" 1); |
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98 by (rtac prem 1); |
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99 by (asm_full_simp_tac (simpset() addsimps [Rep_REAL_inverse]) 1); |
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100 qed "eq_Abs_REAL"; |
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101 |
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102 (**** real_minus: additive inverse on real ****) |
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103 |
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104 Goalw [congruent_def] |
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105 "congruent realrel (%p. (%(x,y). realrel``{(y,x)}) p)"; |
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106 by (Clarify_tac 1); |
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107 by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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108 qed "real_minus_congruent"; |
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109 |
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110 Goalw [real_minus_def] |
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111 "- (Abs_REAL(realrel``{(x,y)})) = Abs_REAL(realrel `` {(y,x)})"; |
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112 by (res_inst_tac [("f","Abs_REAL")] arg_cong 1); |
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113 by (simp_tac (simpset() addsimps [realrel_in_real RS Abs_REAL_inverse, |
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114 [equiv_realrel, real_minus_congruent] MRS UN_equiv_class]) 1); |
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115 qed "real_minus"; |
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116 |
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117 Goal "- (- z) = (z::real)"; |
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118 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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119 by (asm_simp_tac (simpset() addsimps [real_minus]) 1); |
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120 qed "real_minus_minus"; |
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121 |
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122 Addsimps [real_minus_minus]; |
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123 |
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124 Goal "inj(%r::real. -r)"; |
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125 by (rtac injI 1); |
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126 by (dres_inst_tac [("f","uminus")] arg_cong 1); |
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127 by (asm_full_simp_tac (simpset() addsimps [real_minus_minus]) 1); |
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128 qed "inj_real_minus"; |
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129 |
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130 Goalw [real_zero_def] "- 0 = (0::real)"; |
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131 by (simp_tac (simpset() addsimps [real_minus]) 1); |
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132 qed "real_minus_zero"; |
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133 |
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134 Addsimps [real_minus_zero]; |
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135 |
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136 Goal "(-x = 0) = (x = (0::real))"; |
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137 by (res_inst_tac [("z","x")] eq_Abs_REAL 1); |
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138 by (auto_tac (claset(), |
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139 simpset() addsimps [real_zero_def, real_minus] @ preal_add_ac)); |
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140 qed "real_minus_zero_iff"; |
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141 |
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142 Addsimps [real_minus_zero_iff]; |
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143 |
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144 (*** Congruence property for addition ***) |
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145 Goalw [congruent2_def] |
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146 "congruent2 realrel (%p1 p2. \ |
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147 \ (%(x1,y1). (%(x2,y2). realrel``{(x1+x2, y1+y2)}) p2) p1)"; |
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148 by (clarify_tac (claset() addSEs [realrelE]) 1); |
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149 by (asm_simp_tac (simpset() addsimps [preal_add_assoc]) 1); |
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150 by (res_inst_tac [("z1.1","x1a")] (preal_add_left_commute RS ssubst) 1); |
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151 by (asm_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
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152 by (asm_simp_tac (simpset() addsimps preal_add_ac) 1); |
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153 qed "real_add_congruent2"; |
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154 |
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155 Goalw [real_add_def] |
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156 "Abs_REAL(realrel``{(x1,y1)}) + Abs_REAL(realrel``{(x2,y2)}) = \ |
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157 \ Abs_REAL(realrel``{(x1+x2, y1+y2)})"; |
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158 by (simp_tac (simpset() addsimps |
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159 [[equiv_realrel, real_add_congruent2] MRS UN_equiv_class2]) 1); |
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160 qed "real_add"; |
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161 |
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162 Goal "(z::real) + w = w + z"; |
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163 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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164 by (res_inst_tac [("z","w")] eq_Abs_REAL 1); |
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165 by (asm_simp_tac (simpset() addsimps preal_add_ac @ [real_add]) 1); |
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166 qed "real_add_commute"; |
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167 |
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168 Goal "((z1::real) + z2) + z3 = z1 + (z2 + z3)"; |
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169 by (res_inst_tac [("z","z1")] eq_Abs_REAL 1); |
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170 by (res_inst_tac [("z","z2")] eq_Abs_REAL 1); |
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171 by (res_inst_tac [("z","z3")] eq_Abs_REAL 1); |
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172 by (asm_simp_tac (simpset() addsimps [real_add, preal_add_assoc]) 1); |
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173 qed "real_add_assoc"; |
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174 |
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175 (*For AC rewriting*) |
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176 Goal "(x::real)+(y+z)=y+(x+z)"; |
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177 by(rtac ([real_add_assoc,real_add_commute] MRS |
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178 read_instantiate[("f","op +")](thm"mk_left_commute")) 1); |
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179 qed "real_add_left_commute"; |
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180 |
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181 (* real addition is an AC operator *) |
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182 bind_thms ("real_add_ac", [real_add_assoc,real_add_commute,real_add_left_commute]); |
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183 |
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184 Goalw [real_of_preal_def,real_zero_def] "(0::real) + z = z"; |
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185 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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186 by (asm_full_simp_tac (simpset() addsimps [real_add] @ preal_add_ac) 1); |
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187 qed "real_add_zero_left"; |
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188 Addsimps [real_add_zero_left]; |
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189 |
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190 Goal "z + (0::real) = z"; |
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191 by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
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192 qed "real_add_zero_right"; |
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193 Addsimps [real_add_zero_right]; |
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194 |
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195 Goalw [real_zero_def] "z + (-z) = (0::real)"; |
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196 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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197 by (asm_full_simp_tac (simpset() addsimps [real_minus, |
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198 real_add, preal_add_commute]) 1); |
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199 qed "real_add_minus"; |
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200 Addsimps [real_add_minus]; |
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201 |
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202 Goal "(-z) + z = (0::real)"; |
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203 by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
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204 qed "real_add_minus_left"; |
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205 Addsimps [real_add_minus_left]; |
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206 |
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207 |
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208 Goal "z + ((- z) + w) = (w::real)"; |
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209 by (simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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210 qed "real_add_minus_cancel"; |
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211 |
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212 Goal "(-z) + (z + w) = (w::real)"; |
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213 by (simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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214 qed "real_minus_add_cancel"; |
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215 |
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216 Addsimps [real_add_minus_cancel, real_minus_add_cancel]; |
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217 |
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218 Goal "EX y. (x::real) + y = 0"; |
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219 by (blast_tac (claset() addIs [real_add_minus]) 1); |
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220 qed "real_minus_ex"; |
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221 |
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222 Goal "EX! y. (x::real) + y = 0"; |
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223 by (auto_tac (claset() addIs [real_add_minus],simpset())); |
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224 by (dres_inst_tac [("f","%x. ya+x")] arg_cong 1); |
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225 by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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226 by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1); |
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227 qed "real_minus_ex1"; |
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228 |
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229 Goal "EX! y. y + (x::real) = 0"; |
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230 by (auto_tac (claset() addIs [real_add_minus_left],simpset())); |
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231 by (dres_inst_tac [("f","%x. x+ya")] arg_cong 1); |
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232 by (asm_full_simp_tac (simpset() addsimps [real_add_assoc]) 1); |
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233 by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1); |
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234 qed "real_minus_left_ex1"; |
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235 |
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236 Goal "x + y = (0::real) ==> x = -y"; |
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237 by (cut_inst_tac [("z","y")] real_add_minus_left 1); |
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238 by (res_inst_tac [("x1","y")] (real_minus_left_ex1 RS ex1E) 1); |
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239 by (Blast_tac 1); |
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240 qed "real_add_minus_eq_minus"; |
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241 |
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242 Goal "EX (y::real). x = -y"; |
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243 by (cut_inst_tac [("x","x")] real_minus_ex 1); |
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244 by (etac exE 1 THEN dtac real_add_minus_eq_minus 1); |
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245 by (Fast_tac 1); |
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246 qed "real_as_add_inverse_ex"; |
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247 |
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248 Goal "-(x + y) = (-x) + (- y :: real)"; |
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249 by (res_inst_tac [("z","x")] eq_Abs_REAL 1); |
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250 by (res_inst_tac [("z","y")] eq_Abs_REAL 1); |
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251 by (auto_tac (claset(),simpset() addsimps [real_minus,real_add])); |
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252 qed "real_minus_add_distrib"; |
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253 |
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254 Addsimps [real_minus_add_distrib]; |
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255 |
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256 Goal "((x::real) + y = x + z) = (y = z)"; |
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257 by (Step_tac 1); |
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258 by (dres_inst_tac [("f","%t. (-x) + t")] arg_cong 1); |
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259 by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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260 qed "real_add_left_cancel"; |
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261 |
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262 Goal "(y + (x::real)= z + x) = (y = z)"; |
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263 by (simp_tac (simpset() addsimps [real_add_commute,real_add_left_cancel]) 1); |
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264 qed "real_add_right_cancel"; |
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265 |
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266 Goal "(0::real) - x = -x"; |
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267 by (simp_tac (simpset() addsimps [real_diff_def]) 1); |
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268 qed "real_diff_0"; |
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269 |
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270 Goal "x - (0::real) = x"; |
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271 by (simp_tac (simpset() addsimps [real_diff_def]) 1); |
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272 qed "real_diff_0_right"; |
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273 |
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274 Goal "x - x = (0::real)"; |
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275 by (simp_tac (simpset() addsimps [real_diff_def]) 1); |
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276 qed "real_diff_self"; |
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277 |
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278 Addsimps [real_diff_0, real_diff_0_right, real_diff_self]; |
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279 |
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280 |
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281 (*** Congruence property for multiplication ***) |
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282 |
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283 Goal "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> \ |
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284 \ x * x1 + y * y1 + (x * y2 + x2 * y) = \ |
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285 \ x * x2 + y * y2 + (x * y1 + x1 * y)"; |
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286 by (asm_full_simp_tac (simpset() addsimps [preal_add_left_commute, |
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287 preal_add_assoc RS sym,preal_add_mult_distrib2 RS sym]) 1); |
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288 by (rtac (preal_mult_commute RS subst) 1); |
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289 by (res_inst_tac [("y1","x2")] (preal_mult_commute RS subst) 1); |
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290 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc, |
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291 preal_add_mult_distrib2 RS sym]) 1); |
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292 by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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293 qed "real_mult_congruent2_lemma"; |
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294 |
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295 Goal |
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296 "congruent2 realrel (%p1 p2. \ |
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297 \ (%(x1,y1). (%(x2,y2). realrel``{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"; |
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298 by (rtac (equiv_realrel RS congruent2_commuteI) 1); |
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299 by (Clarify_tac 1); |
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300 by (rewtac split_def); |
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301 by (asm_simp_tac (simpset() addsimps [preal_mult_commute,preal_add_commute]) 1); |
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302 by (auto_tac (claset(),simpset() addsimps [real_mult_congruent2_lemma])); |
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303 qed "real_mult_congruent2"; |
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304 |
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305 Goalw [real_mult_def] |
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306 "Abs_REAL((realrel``{(x1,y1)})) * Abs_REAL((realrel``{(x2,y2)})) = \ |
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307 \ Abs_REAL(realrel `` {(x1*x2+y1*y2,x1*y2+x2*y1)})"; |
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308 by (simp_tac (simpset() addsimps |
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309 [[equiv_realrel, real_mult_congruent2] MRS UN_equiv_class2]) 1); |
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310 qed "real_mult"; |
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311 |
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312 Goal "(z::real) * w = w * z"; |
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313 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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314 by (res_inst_tac [("z","w")] eq_Abs_REAL 1); |
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315 by (asm_simp_tac |
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316 (simpset() addsimps [real_mult] @ preal_add_ac @ preal_mult_ac) 1); |
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317 qed "real_mult_commute"; |
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318 |
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319 Goal "((z1::real) * z2) * z3 = z1 * (z2 * z3)"; |
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320 by (res_inst_tac [("z","z1")] eq_Abs_REAL 1); |
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321 by (res_inst_tac [("z","z2")] eq_Abs_REAL 1); |
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322 by (res_inst_tac [("z","z3")] eq_Abs_REAL 1); |
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323 by (asm_simp_tac (simpset() addsimps [preal_add_mult_distrib2,real_mult] @ |
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324 preal_add_ac @ preal_mult_ac) 1); |
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325 qed "real_mult_assoc"; |
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326 |
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327 Goal "(z1::real) * (z2 * z3) = z2 * (z1 * z3)"; |
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328 by(rtac ([real_mult_assoc,real_mult_commute] MRS |
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329 read_instantiate[("f","op *")](thm"mk_left_commute")) 1); |
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330 qed "real_mult_left_commute"; |
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331 |
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332 (* real multiplication is an AC operator *) |
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333 bind_thms ("real_mult_ac", |
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334 [real_mult_assoc, real_mult_commute, real_mult_left_commute]); |
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335 |
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336 Goalw [real_one_def,pnat_one_def] "(1::real) * z = z"; |
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337 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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338 by (asm_full_simp_tac |
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339 (simpset() addsimps [real_mult, |
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340 preal_add_mult_distrib2,preal_mult_1_right] |
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341 @ preal_mult_ac @ preal_add_ac) 1); |
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342 qed "real_mult_1"; |
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343 |
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344 Addsimps [real_mult_1]; |
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345 |
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346 Goal "z * (1::real) = z"; |
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347 by (simp_tac (simpset() addsimps [real_mult_commute]) 1); |
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348 qed "real_mult_1_right"; |
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349 |
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350 Addsimps [real_mult_1_right]; |
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351 |
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352 Goalw [real_zero_def,pnat_one_def] "0 * z = (0::real)"; |
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353 by (res_inst_tac [("z","z")] eq_Abs_REAL 1); |
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354 by (asm_full_simp_tac (simpset() addsimps [real_mult, |
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355 preal_add_mult_distrib2,preal_mult_1_right] |
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356 @ preal_mult_ac @ preal_add_ac) 1); |
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357 qed "real_mult_0"; |
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358 |
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359 Goal "z * 0 = (0::real)"; |
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360 by (simp_tac (simpset() addsimps [real_mult_commute, real_mult_0]) 1); |
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361 qed "real_mult_0_right"; |
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362 |
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363 Addsimps [real_mult_0_right, real_mult_0]; |
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364 |
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365 Goal "(-x) * (y::real) = -(x * y)"; |
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366 by (res_inst_tac [("z","x")] eq_Abs_REAL 1); |
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367 by (res_inst_tac [("z","y")] eq_Abs_REAL 1); |
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368 by (auto_tac (claset(), |
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369 simpset() addsimps [real_minus,real_mult] |
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370 @ preal_mult_ac @ preal_add_ac)); |
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371 qed "real_mult_minus_eq1"; |
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372 Addsimps [real_mult_minus_eq1]; |
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373 |
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374 bind_thm("real_minus_mult_eq1", real_mult_minus_eq1 RS sym); |
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375 |
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376 Goal "x * (- y :: real) = -(x * y)"; |
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377 by (simp_tac (simpset() addsimps [inst "z" "x" real_mult_commute]) 1); |
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378 qed "real_mult_minus_eq2"; |
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379 Addsimps [real_mult_minus_eq2]; |
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380 |
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381 bind_thm("real_minus_mult_eq2", real_mult_minus_eq2 RS sym); |
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382 |
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383 Goal "(- (1::real)) * z = -z"; |
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384 by (Simp_tac 1); |
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385 qed "real_mult_minus_1"; |
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386 Addsimps [real_mult_minus_1]; |
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387 |
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388 Goal "z * (- (1::real)) = -z"; |
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389 by (stac real_mult_commute 1); |
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390 by (Simp_tac 1); |
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391 qed "real_mult_minus_1_right"; |
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392 Addsimps [real_mult_minus_1_right]; |
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393 |
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394 Goal "(-x) * (-y) = x * (y::real)"; |
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395 by (Simp_tac 1); |
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396 qed "real_minus_mult_cancel"; |
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397 |
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398 Addsimps [real_minus_mult_cancel]; |
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399 |
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400 Goal "(-x) * y = x * (- y :: real)"; |
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401 by (Simp_tac 1); |
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402 qed "real_minus_mult_commute"; |
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403 |
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404 (** Lemmas **) |
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405 |
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406 Goal "(z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"; |
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407 by (asm_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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408 qed "real_add_assoc_cong"; |
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409 |
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410 Goal "(z::real) + (v + w) = v + (z + w)"; |
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411 by (REPEAT (ares_tac [real_add_commute RS real_add_assoc_cong] 1)); |
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412 qed "real_add_assoc_swap"; |
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413 |
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414 Goal "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"; |
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415 by (res_inst_tac [("z","z1")] eq_Abs_REAL 1); |
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416 by (res_inst_tac [("z","z2")] eq_Abs_REAL 1); |
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417 by (res_inst_tac [("z","w")] eq_Abs_REAL 1); |
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418 by (asm_simp_tac |
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419 (simpset() addsimps [preal_add_mult_distrib2, real_add, real_mult] @ |
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420 preal_add_ac @ preal_mult_ac) 1); |
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421 qed "real_add_mult_distrib"; |
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422 |
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423 val real_mult_commute'= inst "z" "w" real_mult_commute; |
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424 |
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425 Goal "(w::real) * (z1 + z2) = (w * z1) + (w * z2)"; |
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426 by (simp_tac (simpset() addsimps [real_mult_commute', |
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427 real_add_mult_distrib]) 1); |
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428 qed "real_add_mult_distrib2"; |
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429 |
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430 Goalw [real_diff_def] "((z1::real) - z2) * w = (z1 * w) - (z2 * w)"; |
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431 by (simp_tac (simpset() addsimps [real_add_mult_distrib]) 1); |
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432 qed "real_diff_mult_distrib"; |
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433 |
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434 Goal "(w::real) * (z1 - z2) = (w * z1) - (w * z2)"; |
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435 by (simp_tac (simpset() addsimps [real_mult_commute', |
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436 real_diff_mult_distrib]) 1); |
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437 qed "real_diff_mult_distrib2"; |
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438 |
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439 (*** one and zero are distinct ***) |
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440 Goalw [real_zero_def,real_one_def] "0 ~= (1::real)"; |
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441 by (auto_tac (claset(), |
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442 simpset() addsimps [preal_self_less_add_left RS preal_not_refl2])); |
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443 qed "real_zero_not_eq_one"; |
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444 |
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445 (*** existence of inverse ***) |
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446 (** lemma -- alternative definition of 0 **) |
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447 Goalw [real_zero_def] "0 = Abs_REAL (realrel `` {(x, x)})"; |
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448 by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
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449 qed "real_zero_iff"; |
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450 |
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451 Goalw [real_zero_def,real_one_def] |
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452 "!!(x::real). x ~= 0 ==> EX y. x*y = (1::real)"; |
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453 by (res_inst_tac [("z","x")] eq_Abs_REAL 1); |
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454 by (cut_inst_tac [("r1.0","xa"),("r2.0","y")] preal_linear 1); |
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455 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
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456 simpset() addsimps [real_zero_iff RS sym])); |
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457 by (res_inst_tac [("x", |
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458 "Abs_REAL (realrel `` \ |
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459 \ {(preal_of_prat(prat_of_pnat 1), \ |
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460 \ pinv(D) + preal_of_prat(prat_of_pnat 1))})")] exI 1); |
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461 by (res_inst_tac [("x", |
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462 "Abs_REAL (realrel `` \ |
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463 \ {(pinv(D) + preal_of_prat(prat_of_pnat 1),\ |
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464 \ preal_of_prat(prat_of_pnat 1))})")] exI 2); |
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465 by (auto_tac (claset(), |
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466 simpset() addsimps [real_mult, |
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467 pnat_one_def,preal_mult_1_right,preal_add_mult_distrib2, |
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468 preal_add_mult_distrib,preal_mult_1,preal_mult_inv_right] |
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469 @ preal_add_ac @ preal_mult_ac)); |
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470 qed "real_mult_inv_right_ex"; |
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471 |
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472 Goal "x ~= 0 ==> EX y. y*x = (1::real)"; |
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473 by (dtac real_mult_inv_right_ex 1); |
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474 by (auto_tac (claset(), simpset() addsimps [real_mult_commute])); |
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475 qed "real_mult_inv_left_ex"; |
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476 |
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477 Goalw [real_inverse_def] "x ~= 0 ==> inverse(x)*x = (1::real)"; |
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478 by (ftac real_mult_inv_left_ex 1); |
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479 by (Step_tac 1); |
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480 by (rtac someI2 1); |
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481 by Auto_tac; |
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482 qed "real_mult_inv_left"; |
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483 Addsimps [real_mult_inv_left]; |
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484 |
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485 Goal "x ~= 0 ==> x*inverse(x) = (1::real)"; |
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486 by (stac real_mult_commute 1); |
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487 by (auto_tac (claset(), simpset() addsimps [real_mult_inv_left])); |
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488 qed "real_mult_inv_right"; |
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489 Addsimps [real_mult_inv_right]; |
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490 |
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491 (** Inverse of zero! Useful to simplify certain equations **) |
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492 |
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493 Goalw [real_inverse_def] "inverse 0 = (0::real)"; |
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494 by (rtac someI2 1); |
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495 by (auto_tac (claset(), simpset() addsimps [real_zero_not_eq_one])); |
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496 qed "INVERSE_ZERO"; |
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497 |
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498 Goal "a / (0::real) = 0"; |
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499 by (simp_tac (simpset() addsimps [real_divide_def, INVERSE_ZERO]) 1); |
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500 qed "DIVISION_BY_ZERO"; |
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501 |
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502 fun real_div_undefined_case_tac s i = |
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503 case_tac s i THEN |
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504 asm_simp_tac (simpset() addsimps [DIVISION_BY_ZERO, INVERSE_ZERO]) i; |
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505 |
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506 |
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507 Goal "(c::real) ~= 0 ==> (c*a=c*b) = (a=b)"; |
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508 by Auto_tac; |
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509 by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1); |
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510 by (asm_full_simp_tac (simpset() addsimps real_mult_ac) 1); |
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511 qed "real_mult_left_cancel"; |
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512 |
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513 Goal "(c::real) ~= 0 ==> (a*c=b*c) = (a=b)"; |
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514 by (Step_tac 1); |
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515 by (dres_inst_tac [("f","%x. x*inverse c")] arg_cong 1); |
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516 by (asm_full_simp_tac (simpset() addsimps real_mult_ac) 1); |
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517 qed "real_mult_right_cancel"; |
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518 |
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519 Goal "c*a ~= c*b ==> a ~= b"; |
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520 by Auto_tac; |
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521 qed "real_mult_left_cancel_ccontr"; |
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522 |
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523 Goal "a*c ~= b*c ==> a ~= b"; |
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524 by Auto_tac; |
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525 qed "real_mult_right_cancel_ccontr"; |
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526 |
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527 Goalw [real_inverse_def] "x ~= 0 ==> inverse(x::real) ~= 0"; |
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528 by (ftac real_mult_inv_left_ex 1); |
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529 by (etac exE 1); |
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530 by (rtac someI2 1); |
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531 by (auto_tac (claset(), |
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532 simpset() addsimps [real_mult_0, real_zero_not_eq_one])); |
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533 qed "real_inverse_not_zero"; |
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534 |
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535 Goal "[| x ~= 0; y ~= 0 |] ==> x * y ~= (0::real)"; |
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536 by (Step_tac 1); |
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537 by (dres_inst_tac [("f","%z. inverse x*z")] arg_cong 1); |
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538 by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 1); |
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539 qed "real_mult_not_zero"; |
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540 |
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541 Goal "inverse(inverse (x::real)) = x"; |
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542 by (real_div_undefined_case_tac "x=0" 1); |
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543 by (res_inst_tac [("c1","inverse x")] (real_mult_right_cancel RS iffD1) 1); |
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544 by (etac real_inverse_not_zero 1); |
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545 by (auto_tac (claset() addDs [real_inverse_not_zero],simpset())); |
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546 qed "real_inverse_inverse"; |
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547 Addsimps [real_inverse_inverse]; |
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548 |
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549 Goalw [real_inverse_def] "inverse((1::real)) = (1::real)"; |
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550 by (cut_facts_tac [real_zero_not_eq_one RS |
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551 not_sym RS real_mult_inv_left_ex] 1); |
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552 by (auto_tac (claset(), |
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553 simpset() addsimps [real_zero_not_eq_one RS not_sym])); |
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554 qed "real_inverse_1"; |
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555 Addsimps [real_inverse_1]; |
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556 |
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557 Goal "inverse(-x) = -inverse(x::real)"; |
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558 by (real_div_undefined_case_tac "x=0" 1); |
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559 by (res_inst_tac [("c1","-x")] (real_mult_right_cancel RS iffD1) 1); |
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560 by (stac real_mult_inv_left 2); |
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561 by Auto_tac; |
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562 qed "real_minus_inverse"; |
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563 |
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564 Goal "inverse(x*y) = inverse(x)*inverse(y::real)"; |
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565 by (real_div_undefined_case_tac "x=0" 1); |
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566 by (real_div_undefined_case_tac "y=0" 1); |
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567 by (forw_inst_tac [("y","y")] real_mult_not_zero 1 THEN assume_tac 1); |
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568 by (res_inst_tac [("c1","x")] (real_mult_left_cancel RS iffD1) 1); |
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569 by (auto_tac (claset(),simpset() addsimps [real_mult_assoc RS sym])); |
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570 by (res_inst_tac [("c1","y")] (real_mult_left_cancel RS iffD1) 1); |
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571 by (auto_tac (claset(),simpset() addsimps [real_mult_left_commute])); |
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572 by (asm_simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 1); |
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573 qed "real_inverse_distrib"; |
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574 |
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575 Goal "(x::real) * (y/z) = (x*y)/z"; |
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576 by (simp_tac (simpset() addsimps [real_divide_def, real_mult_assoc]) 1); |
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577 qed "real_times_divide1_eq"; |
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578 |
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579 Goal "(y/z) * (x::real) = (y*x)/z"; |
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580 by (simp_tac (simpset() addsimps [real_divide_def]@real_mult_ac) 1); |
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581 qed "real_times_divide2_eq"; |
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582 |
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583 Addsimps [real_times_divide1_eq, real_times_divide2_eq]; |
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584 |
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585 Goal "(x::real) / (y/z) = (x*z)/y"; |
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586 by (simp_tac (simpset() addsimps [real_divide_def, real_inverse_distrib]@ |
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587 real_mult_ac) 1); |
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588 qed "real_divide_divide1_eq"; |
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589 |
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590 Goal "((x::real) / y) / z = x/(y*z)"; |
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591 by (simp_tac (simpset() addsimps [real_divide_def, real_inverse_distrib, |
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592 real_mult_assoc]) 1); |
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593 qed "real_divide_divide2_eq"; |
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594 |
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595 Addsimps [real_divide_divide1_eq, real_divide_divide2_eq]; |
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596 |
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597 (** As with multiplication, pull minus signs OUT of the / operator **) |
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598 |
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599 Goal "(-x) / (y::real) = - (x/y)"; |
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600 by (simp_tac (simpset() addsimps [real_divide_def]) 1); |
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601 qed "real_minus_divide_eq"; |
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602 Addsimps [real_minus_divide_eq]; |
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603 |
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604 Goal "(x / -(y::real)) = - (x/y)"; |
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605 by (simp_tac (simpset() addsimps [real_divide_def, real_minus_inverse]) 1); |
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606 qed "real_divide_minus_eq"; |
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607 Addsimps [real_divide_minus_eq]; |
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608 |
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609 Goal "(x+y)/(z::real) = x/z + y/z"; |
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610 by (simp_tac (simpset() addsimps [real_divide_def, real_add_mult_distrib]) 1); |
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611 qed "real_add_divide_distrib"; |
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612 |
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613 (*The following would e.g. convert (x+y)/2 to x/2 + y/2. Sometimes that |
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614 leads to cancellations in x or y, but is also prevents "multiplying out" |
|
615 the 2 in e.g. (x+y)/2 = 5. |
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616 |
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617 Addsimps [inst "z" "number_of ?w" real_add_divide_distrib]; |
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618 **) |
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619 |
|
620 |
|
621 (*--------------------------------------------------------- |
|
622 Theorems for ordering |
|
623 --------------------------------------------------------*) |
|
624 (* prove introduction and elimination rules for real_less *) |
|
625 |
|
626 (* real_less is a strong order i.e. nonreflexive and transitive *) |
|
627 |
|
628 (*** lemmas ***) |
|
629 Goal "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"; |
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630 by (asm_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
|
631 qed "preal_lemma_eq_rev_sum"; |
|
632 |
|
633 Goal "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"; |
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634 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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635 qed "preal_add_left_commute_cancel"; |
|
636 |
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637 Goal "!!(x::preal). [| x + y2a = x2a + y; \ |
|
638 \ x + y2b = x2b + y |] \ |
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639 \ ==> x2a + y2b = x2b + y2a"; |
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640 by (dtac preal_lemma_eq_rev_sum 1); |
|
641 by (assume_tac 1); |
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642 by (thin_tac "x + y2b = x2b + y" 1); |
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643 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
644 by (dtac preal_add_left_commute_cancel 1); |
|
645 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
646 qed "preal_lemma_for_not_refl"; |
|
647 |
|
648 Goal "~ (R::real) < R"; |
|
649 by (res_inst_tac [("z","R")] eq_Abs_REAL 1); |
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650 by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
|
651 by (dtac preal_lemma_for_not_refl 1); |
|
652 by (assume_tac 1); |
|
653 by Auto_tac; |
|
654 qed "real_less_not_refl"; |
|
655 |
|
656 (*** y < y ==> P ***) |
|
657 bind_thm("real_less_irrefl", real_less_not_refl RS notE); |
|
658 AddSEs [real_less_irrefl]; |
|
659 |
|
660 Goal "!!(x::real). x < y ==> x ~= y"; |
|
661 by (auto_tac (claset(),simpset() addsimps [real_less_not_refl])); |
|
662 qed "real_not_refl2"; |
|
663 |
|
664 (* lemma re-arranging and eliminating terms *) |
|
665 Goal "!! (a::preal). [| a + b = c + d; \ |
|
666 \ x2b + d + (c + y2e) < a + y2b + (x2e + b) |] \ |
|
667 \ ==> x2b + y2e < x2e + y2b"; |
|
668 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
669 by (res_inst_tac [("C","c+d")] preal_add_left_less_cancel 1); |
|
670 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
671 qed "preal_lemma_trans"; |
|
672 |
|
673 (** heavy re-writing involved*) |
|
674 Goal "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; |
|
675 by (res_inst_tac [("z","R1")] eq_Abs_REAL 1); |
|
676 by (res_inst_tac [("z","R2")] eq_Abs_REAL 1); |
|
677 by (res_inst_tac [("z","R3")] eq_Abs_REAL 1); |
|
678 by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
|
679 by (REPEAT(rtac exI 1)); |
|
680 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
681 by (REPEAT(Blast_tac 2)); |
|
682 by (dtac preal_lemma_for_not_refl 1 THEN assume_tac 1); |
|
683 by (blast_tac (claset() addDs [preal_add_less_mono] |
|
684 addIs [preal_lemma_trans]) 1); |
|
685 qed "real_less_trans"; |
|
686 |
|
687 Goal "!! (R1::real). R1 < R2 ==> ~ (R2 < R1)"; |
|
688 by (rtac notI 1); |
|
689 by (dtac real_less_trans 1 THEN assume_tac 1); |
|
690 by (asm_full_simp_tac (simpset() addsimps [real_less_not_refl]) 1); |
|
691 qed "real_less_not_sym"; |
|
692 |
|
693 (* [| x < y; ~P ==> y < x |] ==> P *) |
|
694 bind_thm ("real_less_asym", real_less_not_sym RS contrapos_np); |
|
695 |
|
696 Goalw [real_of_preal_def] |
|
697 "real_of_preal ((z1::preal) + z2) = \ |
|
698 \ real_of_preal z1 + real_of_preal z2"; |
|
699 by (asm_simp_tac (simpset() addsimps [real_add, |
|
700 preal_add_mult_distrib,preal_mult_1] addsimps preal_add_ac) 1); |
|
701 qed "real_of_preal_add"; |
|
702 |
|
703 Goalw [real_of_preal_def] |
|
704 "real_of_preal ((z1::preal) * z2) = \ |
|
705 \ real_of_preal z1* real_of_preal z2"; |
|
706 by (full_simp_tac (simpset() addsimps [real_mult, |
|
707 preal_add_mult_distrib2,preal_mult_1, |
|
708 preal_mult_1_right,pnat_one_def] |
|
709 @ preal_add_ac @ preal_mult_ac) 1); |
|
710 qed "real_of_preal_mult"; |
|
711 |
|
712 Goalw [real_of_preal_def] |
|
713 "!!(x::preal). y < x ==> \ |
|
714 \ EX m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m"; |
|
715 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
716 simpset() addsimps preal_add_ac)); |
|
717 qed "real_of_preal_ExI"; |
|
718 |
|
719 Goalw [real_of_preal_def] |
|
720 "!!(x::preal). EX m. Abs_REAL (realrel `` {(x,y)}) = \ |
|
721 \ real_of_preal m ==> y < x"; |
|
722 by (auto_tac (claset(), |
|
723 simpset() addsimps |
|
724 [preal_add_commute,preal_add_assoc])); |
|
725 by (asm_full_simp_tac (simpset() addsimps |
|
726 [preal_add_assoc RS sym,preal_self_less_add_left]) 1); |
|
727 qed "real_of_preal_ExD"; |
|
728 |
|
729 Goal "(EX m. Abs_REAL (realrel `` {(x,y)}) = real_of_preal m) = (y < x)"; |
|
730 by (blast_tac (claset() addSIs [real_of_preal_ExI,real_of_preal_ExD]) 1); |
|
731 qed "real_of_preal_iff"; |
|
732 |
|
733 (*** Gleason prop 9-4.4 p 127 ***) |
|
734 Goalw [real_of_preal_def,real_zero_def] |
|
735 "EX m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"; |
|
736 by (res_inst_tac [("z","x")] eq_Abs_REAL 1); |
|
737 by (auto_tac (claset(),simpset() addsimps [real_minus] @ preal_add_ac)); |
|
738 by (cut_inst_tac [("r1.0","x"),("r2.0","y")] preal_linear 1); |
|
739 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
740 simpset() addsimps [preal_add_assoc RS sym])); |
|
741 by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
|
742 qed "real_of_preal_trichotomy"; |
|
743 |
|
744 Goal "!!P. [| !!m. x = real_of_preal m ==> P; \ |
|
745 \ x = 0 ==> P; \ |
|
746 \ !!m. x = -(real_of_preal m) ==> P |] ==> P"; |
|
747 by (cut_inst_tac [("x","x")] real_of_preal_trichotomy 1); |
|
748 by Auto_tac; |
|
749 qed "real_of_preal_trichotomyE"; |
|
750 |
|
751 Goalw [real_of_preal_def] |
|
752 "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"; |
|
753 by (auto_tac (claset(),simpset() addsimps [real_less_def] @ preal_add_ac)); |
|
754 by (auto_tac (claset(),simpset() addsimps [preal_add_assoc RS sym])); |
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755 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
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756 qed "real_of_preal_lessD"; |
|
757 |
|
758 Goal "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"; |
|
759 by (dtac preal_less_add_left_Ex 1); |
|
760 by (auto_tac (claset(), |
|
761 simpset() addsimps [real_of_preal_add, |
|
762 real_of_preal_def,real_less_def])); |
|
763 by (REPEAT(rtac exI 1)); |
|
764 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
765 by (REPEAT(Blast_tac 2)); |
|
766 by (simp_tac (simpset() addsimps [preal_self_less_add_left] |
|
767 delsimps [preal_add_less_iff2]) 1); |
|
768 qed "real_of_preal_lessI"; |
|
769 |
|
770 Goal "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"; |
|
771 by (blast_tac (claset() addIs [real_of_preal_lessI,real_of_preal_lessD]) 1); |
|
772 qed "real_of_preal_less_iff1"; |
|
773 |
|
774 Addsimps [real_of_preal_less_iff1]; |
|
775 |
|
776 Goal "- real_of_preal m < real_of_preal m"; |
|
777 by (auto_tac (claset(), |
|
778 simpset() addsimps |
|
779 [real_of_preal_def,real_less_def,real_minus])); |
|
780 by (REPEAT(rtac exI 1)); |
|
781 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
782 by (REPEAT(Blast_tac 2)); |
|
783 by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
784 by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
785 preal_add_assoc RS sym]) 1); |
|
786 qed "real_of_preal_minus_less_self"; |
|
787 |
|
788 Goalw [real_zero_def] "- real_of_preal m < 0"; |
|
789 by (auto_tac (claset(), |
|
790 simpset() addsimps [real_of_preal_def, |
|
791 real_less_def,real_minus])); |
|
792 by (REPEAT(rtac exI 1)); |
|
793 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
794 by (REPEAT(Blast_tac 2)); |
|
795 by (full_simp_tac (simpset() addsimps |
|
796 [preal_self_less_add_right] @ preal_add_ac) 1); |
|
797 qed "real_of_preal_minus_less_zero"; |
|
798 |
|
799 Goal "~ 0 < - real_of_preal m"; |
|
800 by (cut_facts_tac [real_of_preal_minus_less_zero] 1); |
|
801 by (fast_tac (claset() addDs [real_less_trans] |
|
802 addEs [real_less_irrefl]) 1); |
|
803 qed "real_of_preal_not_minus_gt_zero"; |
|
804 |
|
805 Goalw [real_zero_def] "0 < real_of_preal m"; |
|
806 by (auto_tac (claset(),simpset() addsimps |
|
807 [real_of_preal_def,real_less_def,real_minus])); |
|
808 by (REPEAT(rtac exI 1)); |
|
809 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
810 by (REPEAT(Blast_tac 2)); |
|
811 by (full_simp_tac (simpset() addsimps |
|
812 [preal_self_less_add_right] @ preal_add_ac) 1); |
|
813 qed "real_of_preal_zero_less"; |
|
814 |
|
815 Goal "~ real_of_preal m < 0"; |
|
816 by (cut_facts_tac [real_of_preal_zero_less] 1); |
|
817 by (blast_tac (claset() addDs [real_less_trans] |
|
818 addEs [real_less_irrefl]) 1); |
|
819 qed "real_of_preal_not_less_zero"; |
|
820 |
|
821 Goal "0 < - (- real_of_preal m)"; |
|
822 by (simp_tac (simpset() addsimps |
|
823 [real_of_preal_zero_less]) 1); |
|
824 qed "real_minus_minus_zero_less"; |
|
825 |
|
826 (* another lemma *) |
|
827 Goalw [real_zero_def] |
|
828 "0 < real_of_preal m + real_of_preal m1"; |
|
829 by (auto_tac (claset(), |
|
830 simpset() addsimps [real_of_preal_def, |
|
831 real_less_def,real_add])); |
|
832 by (REPEAT(rtac exI 1)); |
|
833 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
834 by (REPEAT(Blast_tac 2)); |
|
835 by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
836 by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
837 preal_add_assoc RS sym]) 1); |
|
838 qed "real_of_preal_sum_zero_less"; |
|
839 |
|
840 Goal "- real_of_preal m < real_of_preal m1"; |
|
841 by (auto_tac (claset(), |
|
842 simpset() addsimps [real_of_preal_def, |
|
843 real_less_def,real_minus])); |
|
844 by (REPEAT(rtac exI 1)); |
|
845 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
846 by (REPEAT(Blast_tac 2)); |
|
847 by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
848 by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
849 preal_add_assoc RS sym]) 1); |
|
850 qed "real_of_preal_minus_less_all"; |
|
851 |
|
852 Goal "~ real_of_preal m < - real_of_preal m1"; |
|
853 by (cut_facts_tac [real_of_preal_minus_less_all] 1); |
|
854 by (blast_tac (claset() addDs [real_less_trans] |
|
855 addEs [real_less_irrefl]) 1); |
|
856 qed "real_of_preal_not_minus_gt_all"; |
|
857 |
|
858 Goal "- real_of_preal m1 < - real_of_preal m2 \ |
|
859 \ ==> real_of_preal m2 < real_of_preal m1"; |
|
860 by (auto_tac (claset(), |
|
861 simpset() addsimps [real_of_preal_def, |
|
862 real_less_def,real_minus])); |
|
863 by (REPEAT(rtac exI 1)); |
|
864 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
865 by (REPEAT(Blast_tac 2)); |
|
866 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
867 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
868 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
869 qed "real_of_preal_minus_less_rev1"; |
|
870 |
|
871 Goal "real_of_preal m1 < real_of_preal m2 \ |
|
872 \ ==> - real_of_preal m2 < - real_of_preal m1"; |
|
873 by (auto_tac (claset(), |
|
874 simpset() addsimps [real_of_preal_def, |
|
875 real_less_def,real_minus])); |
|
876 by (REPEAT(rtac exI 1)); |
|
877 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
878 by (REPEAT(Blast_tac 2)); |
|
879 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
880 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
|
881 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
882 qed "real_of_preal_minus_less_rev2"; |
|
883 |
|
884 Goal "(- real_of_preal m1 < - real_of_preal m2) = \ |
|
885 \ (real_of_preal m2 < real_of_preal m1)"; |
|
886 by (blast_tac (claset() addSIs [real_of_preal_minus_less_rev1, |
|
887 real_of_preal_minus_less_rev2]) 1); |
|
888 qed "real_of_preal_minus_less_rev_iff"; |
|
889 |
|
890 Addsimps [real_of_preal_minus_less_rev_iff]; |
|
891 |
|
892 (*** linearity ***) |
|
893 Goal "(R1::real) < R2 | R1 = R2 | R2 < R1"; |
|
894 by (res_inst_tac [("x","R1")] real_of_preal_trichotomyE 1); |
|
895 by (ALLGOALS(res_inst_tac [("x","R2")] real_of_preal_trichotomyE)); |
|
896 by (auto_tac (claset() addSDs [preal_le_anti_sym], |
|
897 simpset() addsimps [preal_less_le_iff,real_of_preal_minus_less_zero, |
|
898 real_of_preal_zero_less,real_of_preal_minus_less_all])); |
|
899 qed "real_linear"; |
|
900 |
|
901 Goal "!!w::real. (w ~= z) = (w<z | z<w)"; |
|
902 by (cut_facts_tac [real_linear] 1); |
|
903 by (Blast_tac 1); |
|
904 qed "real_neq_iff"; |
|
905 |
|
906 Goal "!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P; \ |
|
907 \ R2 < R1 ==> P |] ==> P"; |
|
908 by (cut_inst_tac [("R1.0","R1"),("R2.0","R2")] real_linear 1); |
|
909 by Auto_tac; |
|
910 qed "real_linear_less2"; |
|
911 |
|
912 (*** Properties of <= ***) |
|
913 |
|
914 Goalw [real_le_def] "~(w < z) ==> z <= (w::real)"; |
|
915 by (assume_tac 1); |
|
916 qed "real_leI"; |
|
917 |
|
918 Goalw [real_le_def] "z<=w ==> ~(w<(z::real))"; |
|
919 by (assume_tac 1); |
|
920 qed "real_leD"; |
|
921 |
|
922 bind_thm ("real_leE", make_elim real_leD); |
|
923 |
|
924 Goal "(~(w < z)) = (z <= (w::real))"; |
|
925 by (blast_tac (claset() addSIs [real_leI,real_leD]) 1); |
|
926 qed "real_less_le_iff"; |
|
927 |
|
928 Goalw [real_le_def] "~ z <= w ==> w<(z::real)"; |
|
929 by (Blast_tac 1); |
|
930 qed "not_real_leE"; |
|
931 |
|
932 Goalw [real_le_def] "!!(x::real). x <= y ==> x < y | x = y"; |
|
933 by (cut_facts_tac [real_linear] 1); |
|
934 by (blast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
|
935 qed "real_le_imp_less_or_eq"; |
|
936 |
|
937 Goalw [real_le_def] "z<w | z=w ==> z <=(w::real)"; |
|
938 by (cut_facts_tac [real_linear] 1); |
|
939 by (fast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
|
940 qed "real_less_or_eq_imp_le"; |
|
941 |
|
942 Goal "(x <= (y::real)) = (x < y | x=y)"; |
|
943 by (REPEAT(ares_tac [iffI, real_less_or_eq_imp_le, real_le_imp_less_or_eq] 1)); |
|
944 qed "real_le_less"; |
|
945 |
|
946 Goal "w <= (w::real)"; |
|
947 by (simp_tac (simpset() addsimps [real_le_less]) 1); |
|
948 qed "real_le_refl"; |
|
949 |
|
950 (* Axiom 'linorder_linear' of class 'linorder': *) |
|
951 Goal "(z::real) <= w | w <= z"; |
|
952 by (simp_tac (simpset() addsimps [real_le_less]) 1); |
|
953 by (cut_facts_tac [real_linear] 1); |
|
954 by (Blast_tac 1); |
|
955 qed "real_le_linear"; |
|
956 |
|
957 Goal "[| i <= j; j <= k |] ==> i <= (k::real)"; |
|
958 by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
|
959 rtac real_less_or_eq_imp_le, |
|
960 blast_tac (claset() addIs [real_less_trans])]); |
|
961 qed "real_le_trans"; |
|
962 |
|
963 Goal "[| z <= w; w <= z |] ==> z = (w::real)"; |
|
964 by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
|
965 fast_tac (claset() addEs [real_less_irrefl,real_less_asym])]); |
|
966 qed "real_le_anti_sym"; |
|
967 |
|
968 Goal "[| ~ y < x; y ~= x |] ==> x < (y::real)"; |
|
969 by (rtac not_real_leE 1); |
|
970 by (blast_tac (claset() addDs [real_le_imp_less_or_eq]) 1); |
|
971 qed "not_less_not_eq_real_less"; |
|
972 |
|
973 (* Axiom 'order_less_le' of class 'order': *) |
|
974 Goal "((w::real) < z) = (w <= z & w ~= z)"; |
|
975 by (simp_tac (simpset() addsimps [real_le_def, real_neq_iff]) 1); |
|
976 by (blast_tac (claset() addSEs [real_less_asym]) 1); |
|
977 qed "real_less_le"; |
|
978 |
|
979 Goal "(0 < -R) = (R < (0::real))"; |
|
980 by (res_inst_tac [("x","R")] real_of_preal_trichotomyE 1); |
|
981 by (auto_tac (claset(), |
|
982 simpset() addsimps [real_of_preal_not_minus_gt_zero, |
|
983 real_of_preal_not_less_zero,real_of_preal_zero_less, |
|
984 real_of_preal_minus_less_zero])); |
|
985 qed "real_minus_zero_less_iff"; |
|
986 Addsimps [real_minus_zero_less_iff]; |
|
987 |
|
988 Goal "(-R < 0) = ((0::real) < R)"; |
|
989 by (res_inst_tac [("x","R")] real_of_preal_trichotomyE 1); |
|
990 by (auto_tac (claset(), |
|
991 simpset() addsimps [real_of_preal_not_minus_gt_zero, |
|
992 real_of_preal_not_less_zero,real_of_preal_zero_less, |
|
993 real_of_preal_minus_less_zero])); |
|
994 qed "real_minus_zero_less_iff2"; |
|
995 Addsimps [real_minus_zero_less_iff2]; |
|
996 |
|
997 (*Alternative definition for real_less*) |
|
998 Goal "R < S ==> EX T::real. 0 < T & R + T = S"; |
|
999 by (res_inst_tac [("x","R")] real_of_preal_trichotomyE 1); |
|
1000 by (ALLGOALS(res_inst_tac [("x","S")] real_of_preal_trichotomyE)); |
|
1001 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
|
1002 simpset() addsimps [real_of_preal_not_minus_gt_all, |
|
1003 real_of_preal_add, real_of_preal_not_less_zero, |
|
1004 real_less_not_refl, |
|
1005 real_of_preal_not_minus_gt_zero])); |
|
1006 by (res_inst_tac [("x","real_of_preal D")] exI 1); |
|
1007 by (res_inst_tac [("x","real_of_preal m+real_of_preal ma")] exI 2); |
|
1008 by (res_inst_tac [("x","real_of_preal D")] exI 3); |
|
1009 by (auto_tac (claset(), |
|
1010 simpset() addsimps [real_of_preal_zero_less, |
|
1011 real_of_preal_sum_zero_less,real_add_assoc])); |
|
1012 qed "real_less_add_positive_left_Ex"; |
|
1013 |
|
1014 (** change naff name(s)! **) |
|
1015 Goal "(W < S) ==> (0 < S + (-W::real))"; |
|
1016 by (dtac real_less_add_positive_left_Ex 1); |
|
1017 by (auto_tac (claset(), |
|
1018 simpset() addsimps [real_add_minus, |
|
1019 real_add_zero_right] @ real_add_ac)); |
|
1020 qed "real_less_sum_gt_zero"; |
|
1021 |
|
1022 Goal "!!S::real. T = S + W ==> S = T + (-W)"; |
|
1023 by (asm_simp_tac (simpset() addsimps real_add_ac) 1); |
|
1024 qed "real_lemma_change_eq_subj"; |
|
1025 |
|
1026 (* FIXME: long! *) |
|
1027 Goal "(0 < S + (-W::real)) ==> (W < S)"; |
|
1028 by (rtac ccontr 1); |
|
1029 by (dtac (real_leI RS real_le_imp_less_or_eq) 1); |
|
1030 by (auto_tac (claset(), |
|
1031 simpset() addsimps [real_less_not_refl])); |
|
1032 by (EVERY1[dtac real_less_add_positive_left_Ex, etac exE, etac conjE]); |
|
1033 by (Asm_full_simp_tac 1); |
|
1034 by (dtac real_lemma_change_eq_subj 1); |
|
1035 by Auto_tac; |
|
1036 by (dtac real_less_sum_gt_zero 1); |
|
1037 by (asm_full_simp_tac (simpset() addsimps real_add_ac) 1); |
|
1038 by (EVERY1[rotate_tac 1, dtac (real_add_left_commute RS ssubst)]); |
|
1039 by (auto_tac (claset() addEs [real_less_asym], simpset())); |
|
1040 qed "real_sum_gt_zero_less"; |
|
1041 |
|
1042 Goal "(0 < S + (-W::real)) = (W < S)"; |
|
1043 by (blast_tac (claset() addIs [real_less_sum_gt_zero, |
|
1044 real_sum_gt_zero_less]) 1); |
|
1045 qed "real_less_sum_gt_0_iff"; |
|
1046 |
|
1047 |
|
1048 Goalw [real_diff_def] "(x<y) = (x-y < (0::real))"; |
|
1049 by (stac (real_minus_zero_less_iff RS sym) 1); |
|
1050 by (simp_tac (simpset() addsimps [real_add_commute, |
|
1051 real_less_sum_gt_0_iff]) 1); |
|
1052 qed "real_less_eq_diff"; |
|
1053 |
|
1054 |
|
1055 (*** Subtraction laws ***) |
|
1056 |
|
1057 Goal "x + (y - z) = (x + y) - (z::real)"; |
|
1058 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
1059 qed "real_add_diff_eq"; |
|
1060 |
|
1061 Goal "(x - y) + z = (x + z) - (y::real)"; |
|
1062 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
1063 qed "real_diff_add_eq"; |
|
1064 |
|
1065 Goal "(x - y) - z = x - (y + (z::real))"; |
|
1066 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
1067 qed "real_diff_diff_eq"; |
|
1068 |
|
1069 Goal "x - (y - z) = (x + z) - (y::real)"; |
|
1070 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
1071 qed "real_diff_diff_eq2"; |
|
1072 |
|
1073 Goal "(x-y < z) = (x < z + (y::real))"; |
|
1074 by (stac real_less_eq_diff 1); |
|
1075 by (res_inst_tac [("y1", "z")] (real_less_eq_diff RS ssubst) 1); |
|
1076 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
1077 qed "real_diff_less_eq"; |
|
1078 |
|
1079 Goal "(x < z-y) = (x + (y::real) < z)"; |
|
1080 by (stac real_less_eq_diff 1); |
|
1081 by (res_inst_tac [("y1", "z-y")] (real_less_eq_diff RS ssubst) 1); |
|
1082 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
1083 qed "real_less_diff_eq"; |
|
1084 |
|
1085 Goalw [real_le_def] "(x-y <= z) = (x <= z + (y::real))"; |
|
1086 by (simp_tac (simpset() addsimps [real_less_diff_eq]) 1); |
|
1087 qed "real_diff_le_eq"; |
|
1088 |
|
1089 Goalw [real_le_def] "(x <= z-y) = (x + (y::real) <= z)"; |
|
1090 by (simp_tac (simpset() addsimps [real_diff_less_eq]) 1); |
|
1091 qed "real_le_diff_eq"; |
|
1092 |
|
1093 Goalw [real_diff_def] "(x-y = z) = (x = z + (y::real))"; |
|
1094 by (auto_tac (claset(), simpset() addsimps [real_add_assoc])); |
|
1095 qed "real_diff_eq_eq"; |
|
1096 |
|
1097 Goalw [real_diff_def] "(x = z-y) = (x + (y::real) = z)"; |
|
1098 by (auto_tac (claset(), simpset() addsimps [real_add_assoc])); |
|
1099 qed "real_eq_diff_eq"; |
|
1100 |
|
1101 (*This list of rewrites simplifies (in)equalities by bringing subtractions |
|
1102 to the top and then moving negative terms to the other side. |
|
1103 Use with real_add_ac*) |
|
1104 bind_thms ("real_compare_rls", |
|
1105 [symmetric real_diff_def, |
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1106 real_add_diff_eq, real_diff_add_eq, real_diff_diff_eq, real_diff_diff_eq2, |
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1107 real_diff_less_eq, real_less_diff_eq, real_diff_le_eq, real_le_diff_eq, |
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1108 real_diff_eq_eq, real_eq_diff_eq]); |
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1109 |
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1110 |
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1111 (** For the cancellation simproc. |
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1112 The idea is to cancel like terms on opposite sides by subtraction **) |
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1113 |
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1114 Goal "(x::real) - y = x' - y' ==> (x<y) = (x'<y')"; |
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1115 by (stac real_less_eq_diff 1); |
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1116 by (res_inst_tac [("y1", "y")] (real_less_eq_diff RS ssubst) 1); |
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1117 by (Asm_simp_tac 1); |
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1118 qed "real_less_eqI"; |
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1119 |
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1120 Goal "(x::real) - y = x' - y' ==> (y<=x) = (y'<=x')"; |
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1121 by (dtac real_less_eqI 1); |
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1122 by (asm_simp_tac (simpset() addsimps [real_le_def]) 1); |
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1123 qed "real_le_eqI"; |
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1124 |
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1125 Goal "(x::real) - y = x' - y' ==> (x=y) = (x'=y')"; |
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1126 by Safe_tac; |
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1127 by (ALLGOALS |
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1128 (asm_full_simp_tac |
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1129 (simpset() addsimps [real_eq_diff_eq, real_diff_eq_eq]))); |
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1130 qed "real_eq_eqI"; |
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