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1 (* Title : Real/RealDef.ML |
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2 Author : Jacques D. Fleuriot |
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3 Copyright : 1998 University of Cambridge |
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4 Description : The reals |
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5 *) |
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6 |
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7 (*** Proving that realrel is an equivalence relation ***) |
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8 |
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9 Goal "[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] \ |
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10 \ ==> x1 + y3 = x3 + y1"; |
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11 by (res_inst_tac [("C","y2")] preal_add_right_cancel 1); |
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12 by (rotate_tac 1 1 THEN dtac sym 1); |
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13 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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14 by (rtac (preal_add_left_commute RS subst) 1); |
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15 by (res_inst_tac [("x1","x1")] (preal_add_assoc RS subst) 1); |
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16 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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17 qed "preal_trans_lemma"; |
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18 |
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19 (** Natural deduction for realrel **) |
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20 |
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21 Goalw [realrel_def] |
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22 "(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)"; |
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23 by (Blast_tac 1); |
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24 qed "realrel_iff"; |
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25 |
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26 Goalw [realrel_def] |
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27 "[| x1 + y2 = x2 + y1 |] ==> ((x1,y1),(x2,y2)): realrel"; |
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28 by (Blast_tac 1); |
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29 qed "realrelI"; |
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30 |
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31 Goalw [realrel_def] |
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32 "p: realrel --> (EX x1 y1 x2 y2. \ |
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33 \ p = ((x1,y1),(x2,y2)) & x1 + y2 = x2 + y1)"; |
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34 by (Blast_tac 1); |
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35 qed "realrelE_lemma"; |
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36 |
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37 val [major,minor] = goal thy |
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38 "[| p: realrel; \ |
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39 \ !!x1 y1 x2 y2. [| p = ((x1,y1),(x2,y2)); x1+y2 = x2+y1 \ |
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40 \ |] ==> Q |] ==> Q"; |
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41 by (cut_facts_tac [major RS (realrelE_lemma RS mp)] 1); |
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42 by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); |
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43 qed "realrelE"; |
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44 |
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45 AddSIs [realrelI]; |
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46 AddSEs [realrelE]; |
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47 |
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48 Goal "(x,x): realrel"; |
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49 by (stac surjective_pairing 1 THEN rtac (refl RS realrelI) 1); |
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50 qed "realrel_refl"; |
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51 |
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52 Goalw [equiv_def, refl_def, sym_def, trans_def] |
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53 "equiv {x::(preal*preal).True} realrel"; |
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54 by (fast_tac (claset() addSIs [realrel_refl] |
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55 addSEs [sym,preal_trans_lemma]) 1); |
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56 qed "equiv_realrel"; |
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57 |
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58 val equiv_realrel_iff = |
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59 [TrueI, TrueI] MRS |
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60 ([CollectI, CollectI] MRS |
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61 (equiv_realrel RS eq_equiv_class_iff)); |
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62 |
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63 Goalw [real_def,realrel_def,quotient_def] "realrel^^{(x,y)}:real"; |
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64 by (Blast_tac 1); |
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65 qed "realrel_in_real"; |
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66 |
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67 Goal "inj_on Abs_real real"; |
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68 by (rtac inj_on_inverseI 1); |
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69 by (etac Abs_real_inverse 1); |
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70 qed "inj_on_Abs_real"; |
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71 |
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72 Addsimps [equiv_realrel_iff,inj_on_Abs_real RS inj_on_iff, |
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73 realrel_iff, realrel_in_real, Abs_real_inverse]; |
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74 |
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75 Addsimps [equiv_realrel RS eq_equiv_class_iff]; |
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76 val eq_realrelD = equiv_realrel RSN (2,eq_equiv_class); |
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77 |
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78 Goal "inj(Rep_real)"; |
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79 by (rtac inj_inverseI 1); |
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80 by (rtac Rep_real_inverse 1); |
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81 qed "inj_Rep_real"; |
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82 |
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83 (** real_preal: the injection from preal to real **) |
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84 Goal "inj(real_preal)"; |
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85 by (rtac injI 1); |
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86 by (rewtac real_preal_def); |
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87 by (dtac (inj_on_Abs_real RS inj_onD) 1); |
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88 by (REPEAT (rtac realrel_in_real 1)); |
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89 by (dtac eq_equiv_class 1); |
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90 by (rtac equiv_realrel 1); |
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91 by (Blast_tac 1); |
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92 by Safe_tac; |
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93 by (Asm_full_simp_tac 1); |
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94 qed "inj_real_preal"; |
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95 |
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96 val [prem] = goal thy |
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97 "(!!x y. z = Abs_real(realrel^^{(x,y)}) ==> P) ==> P"; |
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98 by (res_inst_tac [("x1","z")] |
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99 (rewrite_rule [real_def] Rep_real RS quotientE) 1); |
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100 by (dres_inst_tac [("f","Abs_real")] arg_cong 1); |
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101 by (res_inst_tac [("p","x")] PairE 1); |
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102 by (rtac prem 1); |
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103 by (asm_full_simp_tac (simpset() addsimps [Rep_real_inverse]) 1); |
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104 qed "eq_Abs_real"; |
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105 |
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106 (**** real_minus: additive inverse on real ****) |
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107 |
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108 Goalw [congruent_def] |
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109 "congruent realrel (%p. split (%x y. realrel^^{(y,x)}) p)"; |
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110 by Safe_tac; |
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111 by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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112 qed "real_minus_congruent"; |
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113 |
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114 (*Resolve th against the corresponding facts for real_minus*) |
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115 val real_minus_ize = RSLIST [equiv_realrel, real_minus_congruent]; |
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116 |
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117 Goalw [real_minus_def] |
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118 "- (Abs_real(realrel^^{(x,y)})) = Abs_real(realrel ^^ {(y,x)})"; |
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119 by (res_inst_tac [("f","Abs_real")] arg_cong 1); |
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120 by (simp_tac (simpset() addsimps |
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121 [realrel_in_real RS Abs_real_inverse,real_minus_ize UN_equiv_class]) 1); |
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122 qed "real_minus"; |
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123 |
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124 Goal "- (- z) = (z::real)"; |
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125 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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126 by (asm_simp_tac (simpset() addsimps [real_minus]) 1); |
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127 qed "real_minus_minus"; |
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128 |
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129 Addsimps [real_minus_minus]; |
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130 |
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131 Goal "inj(%r::real. -r)"; |
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132 by (rtac injI 1); |
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133 by (dres_inst_tac [("f","uminus")] arg_cong 1); |
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134 by (asm_full_simp_tac (simpset() addsimps [real_minus_minus]) 1); |
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135 qed "inj_real_minus"; |
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136 |
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137 Goalw [real_zero_def] "-0r = 0r"; |
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138 by (simp_tac (simpset() addsimps [real_minus]) 1); |
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139 qed "real_minus_zero"; |
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140 |
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141 Addsimps [real_minus_zero]; |
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142 |
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143 Goal "(-x = 0r) = (x = 0r)"; |
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144 by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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145 by (auto_tac (claset(), |
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146 simpset() addsimps [real_zero_def, real_minus] @ preal_add_ac)); |
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147 qed "real_minus_zero_iff"; |
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148 |
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149 Addsimps [real_minus_zero_iff]; |
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150 |
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151 Goal "(-x ~= 0r) = (x ~= 0r)"; |
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152 by Auto_tac; |
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153 qed "real_minus_not_zero_iff"; |
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154 |
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155 (*** Congruence property for addition ***) |
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156 Goalw [congruent2_def] |
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157 "congruent2 realrel (%p1 p2. \ |
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158 \ split (%x1 y1. split (%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"; |
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159 by Safe_tac; |
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160 by (asm_simp_tac (simpset() addsimps [preal_add_assoc]) 1); |
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161 by (res_inst_tac [("z1.1","x1a")] (preal_add_left_commute RS ssubst) 1); |
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162 by (asm_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
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163 by (asm_simp_tac (simpset() addsimps preal_add_ac) 1); |
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164 qed "real_add_congruent2"; |
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165 |
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166 (*Resolve th against the corresponding facts for real_add*) |
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167 val real_add_ize = RSLIST [equiv_realrel, real_add_congruent2]; |
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168 |
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169 Goalw [real_add_def] |
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170 "Abs_real(realrel^^{(x1,y1)}) + Abs_real(realrel^^{(x2,y2)}) = \ |
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171 \ Abs_real(realrel^^{(x1+x2, y1+y2)})"; |
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172 by (asm_simp_tac (simpset() addsimps [real_add_ize UN_equiv_class2]) 1); |
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173 qed "real_add"; |
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174 |
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175 Goal "(z::real) + w = w + z"; |
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176 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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177 by (res_inst_tac [("z","w")] eq_Abs_real 1); |
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178 by (asm_simp_tac (simpset() addsimps preal_add_ac @ [real_add]) 1); |
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179 qed "real_add_commute"; |
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180 |
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181 Goal "((z1::real) + z2) + z3 = z1 + (z2 + z3)"; |
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182 by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
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183 by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
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184 by (res_inst_tac [("z","z3")] eq_Abs_real 1); |
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185 by (asm_simp_tac (simpset() addsimps [real_add, preal_add_assoc]) 1); |
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186 qed "real_add_assoc"; |
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187 |
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188 (*For AC rewriting*) |
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189 Goal "(x::real)+(y+z)=y+(x+z)"; |
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190 by (rtac (real_add_commute RS trans) 1); |
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191 by (rtac (real_add_assoc RS trans) 1); |
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192 by (rtac (real_add_commute RS arg_cong) 1); |
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193 qed "real_add_left_commute"; |
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194 |
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195 (* real addition is an AC operator *) |
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196 val real_add_ac = [real_add_assoc,real_add_commute,real_add_left_commute]; |
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197 |
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198 Goalw [real_preal_def,real_zero_def] "0r + z = z"; |
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199 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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200 by (asm_full_simp_tac (simpset() addsimps [real_add] @ preal_add_ac) 1); |
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201 qed "real_add_zero_left"; |
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202 Addsimps [real_add_zero_left]; |
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203 |
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204 Goal "z + 0r = z"; |
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205 by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
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206 qed "real_add_zero_right"; |
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207 Addsimps [real_add_zero_right]; |
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208 |
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209 Goalw [real_zero_def] "z + -z = 0r"; |
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210 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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211 by (asm_full_simp_tac (simpset() addsimps [real_minus, |
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212 real_add, preal_add_commute]) 1); |
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213 qed "real_add_minus"; |
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214 Addsimps [real_add_minus]; |
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215 |
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216 Goal "-z + z = 0r"; |
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217 by (simp_tac (simpset() addsimps [real_add_commute]) 1); |
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218 qed "real_add_minus_left"; |
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219 Addsimps [real_add_minus_left]; |
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220 |
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221 |
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222 Goal "z + (- z + w) = (w::real)"; |
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223 by (simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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224 qed "real_add_minus_cancel"; |
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225 |
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226 Goal "(-z) + (z + w) = (w::real)"; |
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227 by (simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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228 qed "real_minus_add_cancel"; |
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229 |
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230 Addsimps [real_add_minus_cancel, real_minus_add_cancel]; |
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231 |
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232 Goal "? y. (x::real) + y = 0r"; |
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233 by (blast_tac (claset() addIs [real_add_minus]) 1); |
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234 qed "real_minus_ex"; |
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235 |
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236 Goal "?! y. (x::real) + y = 0r"; |
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237 by (auto_tac (claset() addIs [real_add_minus],simpset())); |
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238 by (dres_inst_tac [("f","%x. ya+x")] arg_cong 1); |
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239 by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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240 by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1); |
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241 qed "real_minus_ex1"; |
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242 |
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243 Goal "?! y. y + (x::real) = 0r"; |
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244 by (auto_tac (claset() addIs [real_add_minus_left],simpset())); |
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245 by (dres_inst_tac [("f","%x. x+ya")] arg_cong 1); |
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246 by (asm_full_simp_tac (simpset() addsimps [real_add_assoc]) 1); |
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247 by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1); |
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248 qed "real_minus_left_ex1"; |
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249 |
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250 Goal "x + y = 0r ==> x = -y"; |
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251 by (cut_inst_tac [("z","y")] real_add_minus_left 1); |
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252 by (res_inst_tac [("x1","y")] (real_minus_left_ex1 RS ex1E) 1); |
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253 by (Blast_tac 1); |
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254 qed "real_add_minus_eq_minus"; |
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255 |
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256 Goal "-(x + y) = -x + -(y::real)"; |
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257 by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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258 by (res_inst_tac [("z","y")] eq_Abs_real 1); |
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259 by (auto_tac (claset(),simpset() addsimps [real_minus,real_add])); |
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260 qed "real_minus_add_distrib"; |
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261 |
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262 Addsimps [real_minus_add_distrib]; |
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263 |
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264 Goal "((x::real) + y = x + z) = (y = z)"; |
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265 by (Step_tac 1); |
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266 by (dres_inst_tac [("f","%t.-x + t")] arg_cong 1); |
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267 by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1); |
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268 qed "real_add_left_cancel"; |
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269 |
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270 Goal "(y + (x::real)= z + x) = (y = z)"; |
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271 by (simp_tac (simpset() addsimps [real_add_commute,real_add_left_cancel]) 1); |
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272 qed "real_add_right_cancel"; |
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273 |
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274 Goal "0r - x = -x"; |
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275 by (simp_tac (simpset() addsimps [real_diff_def]) 1); |
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276 qed "real_diff_0"; |
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277 |
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278 Goal "x - 0r = x"; |
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279 by (simp_tac (simpset() addsimps [real_diff_def]) 1); |
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280 qed "real_diff_0_right"; |
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281 |
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282 Goal "x - x = 0r"; |
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283 by (simp_tac (simpset() addsimps [real_diff_def]) 1); |
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284 qed "real_diff_self"; |
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285 |
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286 Addsimps [real_diff_0, real_diff_0_right, real_diff_self]; |
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287 |
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288 |
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289 (*** Congruence property for multiplication ***) |
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290 |
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291 Goal "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> \ |
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292 \ x * x1 + y * y1 + (x * y2 + x2 * y) = \ |
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293 \ x * x2 + y * y2 + (x * y1 + x1 * y)"; |
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294 by (asm_full_simp_tac (simpset() addsimps [preal_add_left_commute, |
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295 preal_add_assoc RS sym,preal_add_mult_distrib2 RS sym]) 1); |
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296 by (rtac (preal_mult_commute RS subst) 1); |
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297 by (res_inst_tac [("y1","x2")] (preal_mult_commute RS subst) 1); |
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298 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc, |
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299 preal_add_mult_distrib2 RS sym]) 1); |
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300 by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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301 qed "real_mult_congruent2_lemma"; |
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302 |
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303 Goal |
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304 "congruent2 realrel (%p1 p2. \ |
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305 \ split (%x1 y1. split (%x2 y2. realrel^^{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)"; |
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306 by (rtac (equiv_realrel RS congruent2_commuteI) 1); |
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307 by Safe_tac; |
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308 by (rewtac split_def); |
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309 by (asm_simp_tac (simpset() addsimps [preal_mult_commute,preal_add_commute]) 1); |
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310 by (auto_tac (claset(),simpset() addsimps [real_mult_congruent2_lemma])); |
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311 qed "real_mult_congruent2"; |
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312 |
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313 (*Resolve th against the corresponding facts for real_mult*) |
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314 val real_mult_ize = RSLIST [equiv_realrel, real_mult_congruent2]; |
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315 |
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316 Goalw [real_mult_def] |
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317 "Abs_real((realrel^^{(x1,y1)})) * Abs_real((realrel^^{(x2,y2)})) = \ |
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318 \ Abs_real(realrel ^^ {(x1*x2+y1*y2,x1*y2+x2*y1)})"; |
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319 by (simp_tac (simpset() addsimps [real_mult_ize UN_equiv_class2]) 1); |
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320 qed "real_mult"; |
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321 |
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322 Goal "(z::real) * w = w * z"; |
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323 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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324 by (res_inst_tac [("z","w")] eq_Abs_real 1); |
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325 by (asm_simp_tac |
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326 (simpset() addsimps [real_mult] @ preal_add_ac @ preal_mult_ac) 1); |
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327 qed "real_mult_commute"; |
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328 |
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329 Goal "((z1::real) * z2) * z3 = z1 * (z2 * z3)"; |
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330 by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
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331 by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
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332 by (res_inst_tac [("z","z3")] eq_Abs_real 1); |
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333 by (asm_simp_tac (simpset() addsimps [preal_add_mult_distrib2,real_mult] @ |
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334 preal_add_ac @ preal_mult_ac) 1); |
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335 qed "real_mult_assoc"; |
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336 |
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337 qed_goal "real_mult_left_commute" thy |
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338 "(z1::real) * (z2 * z3) = z2 * (z1 * z3)" |
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339 (fn _ => [rtac (real_mult_commute RS trans) 1, rtac (real_mult_assoc RS trans) 1, |
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340 rtac (real_mult_commute RS arg_cong) 1]); |
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341 |
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342 (* real multiplication is an AC operator *) |
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343 val real_mult_ac = [real_mult_assoc, real_mult_commute, real_mult_left_commute]; |
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344 |
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345 Goalw [real_one_def,pnat_one_def] "1r * z = z"; |
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346 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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347 by (asm_full_simp_tac |
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348 (simpset() addsimps [real_mult, |
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349 preal_add_mult_distrib2,preal_mult_1_right] |
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350 @ preal_mult_ac @ preal_add_ac) 1); |
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351 qed "real_mult_1"; |
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352 |
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353 Addsimps [real_mult_1]; |
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354 |
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355 Goal "z * 1r = z"; |
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356 by (simp_tac (simpset() addsimps [real_mult_commute]) 1); |
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357 qed "real_mult_1_right"; |
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358 |
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359 Addsimps [real_mult_1_right]; |
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360 |
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361 Goalw [real_zero_def,pnat_one_def] "0r * z = 0r"; |
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362 by (res_inst_tac [("z","z")] eq_Abs_real 1); |
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363 by (asm_full_simp_tac (simpset() addsimps [real_mult, |
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364 preal_add_mult_distrib2,preal_mult_1_right] |
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365 @ preal_mult_ac @ preal_add_ac) 1); |
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366 qed "real_mult_0"; |
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367 |
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368 Goal "z * 0r = 0r"; |
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369 by (simp_tac (simpset() addsimps [real_mult_commute, real_mult_0]) 1); |
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370 qed "real_mult_0_right"; |
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371 |
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372 Addsimps [real_mult_0_right, real_mult_0]; |
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373 |
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374 Goal "-(x * y) = -x * (y::real)"; |
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375 by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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376 by (res_inst_tac [("z","y")] eq_Abs_real 1); |
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377 by (auto_tac (claset(), |
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378 simpset() addsimps [real_minus,real_mult] |
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379 @ preal_mult_ac @ preal_add_ac)); |
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380 qed "real_minus_mult_eq1"; |
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381 |
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382 Goal "-(x * y) = x * -(y::real)"; |
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383 by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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384 by (res_inst_tac [("z","y")] eq_Abs_real 1); |
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385 by (auto_tac (claset(), |
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386 simpset() addsimps [real_minus,real_mult] |
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387 @ preal_mult_ac @ preal_add_ac)); |
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388 qed "real_minus_mult_eq2"; |
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389 |
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390 Goal "- 1r * z = -z"; |
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391 by (simp_tac (simpset() addsimps [real_minus_mult_eq1 RS sym]) 1); |
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392 qed "real_mult_minus_1"; |
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393 |
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394 Addsimps [real_mult_minus_1]; |
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395 |
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396 Goal "z * - 1r = -z"; |
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397 by (stac real_mult_commute 1); |
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398 by (Simp_tac 1); |
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399 qed "real_mult_minus_1_right"; |
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400 |
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401 Addsimps [real_mult_minus_1_right]; |
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402 |
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403 Goal "-x * -y = x * (y::real)"; |
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404 by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym, |
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405 real_minus_mult_eq1 RS sym]) 1); |
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406 qed "real_minus_mult_cancel"; |
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407 |
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408 Addsimps [real_minus_mult_cancel]; |
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409 |
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410 Goal "-x * y = x * -(y::real)"; |
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411 by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym, |
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412 real_minus_mult_eq1 RS sym]) 1); |
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413 qed "real_minus_mult_commute"; |
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414 |
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415 (*----------------------------------------------------------------------------- |
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416 |
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417 -----------------------------------------------------------------------------*) |
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418 |
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419 (** Lemmas **) |
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420 |
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421 qed_goal "real_add_assoc_cong" thy |
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422 "!!z. (z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)" |
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423 (fn _ => [(asm_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1)]); |
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424 |
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425 qed_goal "real_add_assoc_swap" thy "(z::real) + (v + w) = v + (z + w)" |
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426 (fn _ => [(REPEAT (ares_tac [real_add_commute RS real_add_assoc_cong] 1))]); |
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427 |
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428 Goal "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"; |
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429 by (res_inst_tac [("z","z1")] eq_Abs_real 1); |
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430 by (res_inst_tac [("z","z2")] eq_Abs_real 1); |
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431 by (res_inst_tac [("z","w")] eq_Abs_real 1); |
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432 by (asm_simp_tac |
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433 (simpset() addsimps [preal_add_mult_distrib2, real_add, real_mult] @ |
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434 preal_add_ac @ preal_mult_ac) 1); |
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435 qed "real_add_mult_distrib"; |
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436 |
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437 val real_mult_commute'= read_instantiate [("z","w")] real_mult_commute; |
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438 |
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439 Goal "(w::real) * (z1 + z2) = (w * z1) + (w * z2)"; |
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440 by (simp_tac (simpset() addsimps [real_mult_commute',real_add_mult_distrib]) 1); |
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441 qed "real_add_mult_distrib2"; |
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442 |
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443 (*** one and zero are distinct ***) |
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444 Goalw [real_zero_def,real_one_def] "0r ~= 1r"; |
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445 by (auto_tac (claset(), |
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446 simpset() addsimps [preal_self_less_add_left RS preal_not_refl2])); |
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447 qed "real_zero_not_eq_one"; |
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448 |
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449 (*** existence of inverse ***) |
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450 (** lemma -- alternative definition for 0r **) |
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451 Goalw [real_zero_def] "0r = Abs_real (realrel ^^ {(x, x)})"; |
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452 by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
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453 qed "real_zero_iff"; |
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454 |
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455 Goalw [real_zero_def,real_one_def] |
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456 "!!(x::real). x ~= 0r ==> ? y. x*y = 1r"; |
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457 by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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458 by (cut_inst_tac [("r1.0","xa"),("r2.0","y")] preal_linear 1); |
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459 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
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460 simpset() addsimps [real_zero_iff RS sym])); |
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461 by (res_inst_tac [("x","Abs_real (realrel ^^ {(@#$#1p,pinv(D)+@#$#1p)})")] exI 1); |
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462 by (res_inst_tac [("x","Abs_real (realrel ^^ {(pinv(D)+@#$#1p,@#$#1p)})")] exI 2); |
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463 by (auto_tac (claset(), |
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464 simpset() addsimps [real_mult, |
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465 pnat_one_def,preal_mult_1_right,preal_add_mult_distrib2, |
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466 preal_add_mult_distrib,preal_mult_1,preal_mult_inv_right] |
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467 @ preal_add_ac @ preal_mult_ac)); |
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468 qed "real_mult_inv_right_ex"; |
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469 |
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470 Goal "!!(x::real). x ~= 0r ==> ? y. y*x = 1r"; |
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471 by (asm_simp_tac (simpset() addsimps [real_mult_commute, |
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472 real_mult_inv_right_ex]) 1); |
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473 qed "real_mult_inv_left_ex"; |
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474 |
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475 Goalw [rinv_def] "!!(x::real). x ~= 0r ==> rinv(x)*x = 1r"; |
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476 by (forward_tac [real_mult_inv_left_ex] 1); |
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477 by (Step_tac 1); |
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478 by (rtac selectI2 1); |
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479 by Auto_tac; |
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480 qed "real_mult_inv_left"; |
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481 |
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482 Goal "!!(x::real). x ~= 0r ==> x*rinv(x) = 1r"; |
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483 by (auto_tac (claset() addIs [real_mult_commute RS subst], |
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484 simpset() addsimps [real_mult_inv_left])); |
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485 qed "real_mult_inv_right"; |
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486 |
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487 Goal "(c::real) ~= 0r ==> (c*a=c*b) = (a=b)"; |
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488 by Auto_tac; |
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489 by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1); |
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490 by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac) 1); |
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491 qed "real_mult_left_cancel"; |
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492 |
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493 Goal "(c::real) ~= 0r ==> (a*c=b*c) = (a=b)"; |
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494 by (Step_tac 1); |
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495 by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1); |
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496 by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac) 1); |
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497 qed "real_mult_right_cancel"; |
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498 |
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499 Goalw [rinv_def] "x ~= 0r ==> rinv(x) ~= 0r"; |
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500 by (forward_tac [real_mult_inv_left_ex] 1); |
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501 by (etac exE 1); |
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502 by (rtac selectI2 1); |
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503 by (auto_tac (claset(), |
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504 simpset() addsimps [real_mult_0, |
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505 real_zero_not_eq_one])); |
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506 qed "rinv_not_zero"; |
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507 |
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508 Addsimps [real_mult_inv_left,real_mult_inv_right]; |
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509 |
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510 Goal "x ~= 0r ==> rinv(rinv x) = x"; |
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511 by (res_inst_tac [("c1","rinv x")] (real_mult_right_cancel RS iffD1) 1); |
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512 by (etac rinv_not_zero 1); |
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513 by (auto_tac (claset() addDs [rinv_not_zero],simpset())); |
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514 qed "real_rinv_rinv"; |
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515 |
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516 Goalw [rinv_def] "rinv(1r) = 1r"; |
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517 by (cut_facts_tac [real_zero_not_eq_one RS |
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518 not_sym RS real_mult_inv_left_ex] 1); |
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519 by (etac exE 1); |
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520 by (rtac selectI2 1); |
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521 by (auto_tac (claset(), |
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522 simpset() addsimps |
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523 [real_zero_not_eq_one RS not_sym])); |
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524 qed "real_rinv_1"; |
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525 |
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526 Goal "x ~= 0r ==> rinv(-x) = -rinv(x)"; |
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527 by (res_inst_tac [("c1","-x")] (real_mult_right_cancel RS iffD1) 1); |
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528 by Auto_tac; |
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529 qed "real_minus_rinv"; |
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530 |
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531 (*** theorems for ordering ***) |
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532 (* prove introduction and elimination rules for real_less *) |
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533 |
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534 (* real_less is a strong order i.e nonreflexive and transitive *) |
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535 (*** lemmas ***) |
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536 Goal "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y"; |
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537 by (asm_simp_tac (simpset() addsimps [preal_add_commute]) 1); |
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538 qed "preal_lemma_eq_rev_sum"; |
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539 |
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540 Goal "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1"; |
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541 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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542 qed "preal_add_left_commute_cancel"; |
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543 |
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544 Goal "!!(x::preal). [| x + y2a = x2a + y; \ |
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545 \ x + y2b = x2b + y |] \ |
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546 \ ==> x2a + y2b = x2b + y2a"; |
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547 by (dtac preal_lemma_eq_rev_sum 1); |
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548 by (assume_tac 1); |
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549 by (thin_tac "x + y2b = x2b + y" 1); |
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550 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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551 by (dtac preal_add_left_commute_cancel 1); |
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552 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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553 qed "preal_lemma_for_not_refl"; |
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554 |
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555 Goal "~ (R::real) < R"; |
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556 by (res_inst_tac [("z","R")] eq_Abs_real 1); |
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557 by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
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558 by (dtac preal_lemma_for_not_refl 1); |
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559 by (assume_tac 1 THEN rotate_tac 2 1); |
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560 by (auto_tac (claset(),simpset() addsimps [preal_less_not_refl])); |
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561 qed "real_less_not_refl"; |
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562 |
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563 (*** y < y ==> P ***) |
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564 bind_thm("real_less_irrefl", real_less_not_refl RS notE); |
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565 AddSEs [real_less_irrefl]; |
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566 |
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567 Goal "!!(x::real). x < y ==> x ~= y"; |
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568 by (auto_tac (claset(),simpset() addsimps [real_less_not_refl])); |
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569 qed "real_not_refl2"; |
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570 |
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571 (* lemma re-arranging and eliminating terms *) |
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572 Goal "!! (a::preal). [| a + b = c + d; \ |
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573 \ x2b + d + (c + y2e) < a + y2b + (x2e + b) |] \ |
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574 \ ==> x2b + y2e < x2e + y2b"; |
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575 by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1); |
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576 by (res_inst_tac [("C","c+d")] preal_add_left_less_cancel 1); |
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577 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
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578 qed "preal_lemma_trans"; |
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579 |
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580 (** heavy re-writing involved*) |
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581 Goal "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; |
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582 by (res_inst_tac [("z","R1")] eq_Abs_real 1); |
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583 by (res_inst_tac [("z","R2")] eq_Abs_real 1); |
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584 by (res_inst_tac [("z","R3")] eq_Abs_real 1); |
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585 by (auto_tac (claset(),simpset() addsimps [real_less_def])); |
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586 by (REPEAT(rtac exI 1)); |
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587 by (EVERY[rtac conjI 1, rtac conjI 2]); |
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588 by (REPEAT(Blast_tac 2)); |
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589 by (dtac preal_lemma_for_not_refl 1 THEN assume_tac 1); |
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590 by (blast_tac (claset() addDs [preal_add_less_mono] |
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591 addIs [preal_lemma_trans]) 1); |
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592 qed "real_less_trans"; |
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593 |
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594 Goal "!! (R1::real). [| R1 < R2; R2 < R1 |] ==> P"; |
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595 by (dtac real_less_trans 1 THEN assume_tac 1); |
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596 by (asm_full_simp_tac (simpset() addsimps [real_less_not_refl]) 1); |
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597 qed "real_less_asym"; |
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598 |
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599 (****)(****)(****)(****)(****)(****)(****)(****)(****)(****) |
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600 (****** Map and more real_less ******) |
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601 (*** mapping from preal into real ***) |
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602 Goalw [real_preal_def] |
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603 "%#((z1::preal) + z2) = %#z1 + %#z2"; |
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604 by (asm_simp_tac (simpset() addsimps [real_add, |
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605 preal_add_mult_distrib,preal_mult_1] addsimps preal_add_ac) 1); |
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606 qed "real_preal_add"; |
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607 |
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608 Goalw [real_preal_def] |
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609 "%#((z1::preal) * z2) = %#z1* %#z2"; |
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610 by (full_simp_tac (simpset() addsimps [real_mult, |
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611 preal_add_mult_distrib2,preal_mult_1, |
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612 preal_mult_1_right,pnat_one_def] |
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613 @ preal_add_ac @ preal_mult_ac) 1); |
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614 qed "real_preal_mult"; |
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615 |
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616 Goalw [real_preal_def] |
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617 "!!(x::preal). y < x ==> ? m. Abs_real (realrel ^^ {(x,y)}) = %#m"; |
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618 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
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619 simpset() addsimps preal_add_ac)); |
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620 qed "real_preal_ExI"; |
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621 |
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622 Goalw [real_preal_def] |
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623 "!!(x::preal). ? m. Abs_real (realrel ^^ {(x,y)}) = %#m ==> y < x"; |
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624 by (auto_tac (claset(), |
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625 simpset() addsimps |
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626 [preal_add_commute,preal_add_assoc])); |
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627 by (asm_full_simp_tac (simpset() addsimps |
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628 [preal_add_assoc RS sym,preal_self_less_add_left]) 1); |
|
629 qed "real_preal_ExD"; |
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630 |
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631 Goal "(? m. Abs_real (realrel ^^ {(x,y)}) = %#m) = (y < x)"; |
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632 by (blast_tac (claset() addSIs [real_preal_ExI,real_preal_ExD]) 1); |
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633 qed "real_preal_iff"; |
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634 |
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635 (*** Gleason prop 9-4.4 p 127 ***) |
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636 Goalw [real_preal_def,real_zero_def] |
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637 "? m. (x::real) = %#m | x = 0r | x = -(%#m)"; |
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638 by (res_inst_tac [("z","x")] eq_Abs_real 1); |
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639 by (auto_tac (claset(),simpset() addsimps [real_minus] @ preal_add_ac)); |
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640 by (cut_inst_tac [("r1.0","x"),("r2.0","y")] preal_linear 1); |
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641 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
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642 simpset() addsimps [preal_add_assoc RS sym])); |
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643 by (auto_tac (claset(),simpset() addsimps [preal_add_commute])); |
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644 qed "real_preal_trichotomy"; |
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645 |
|
646 Goal "!!P. [| !!m. x = %#m ==> P; \ |
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647 \ x = 0r ==> P; \ |
|
648 \ !!m. x = -(%#m) ==> P |] ==> P"; |
|
649 by (cut_inst_tac [("x","x")] real_preal_trichotomy 1); |
|
650 by Auto_tac; |
|
651 qed "real_preal_trichotomyE"; |
|
652 |
|
653 Goalw [real_preal_def] "%#m1 < %#m2 ==> m1 < m2"; |
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654 by (auto_tac (claset(),simpset() addsimps [real_less_def] @ preal_add_ac)); |
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655 by (auto_tac (claset(),simpset() addsimps [preal_add_assoc RS sym])); |
|
656 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
|
657 qed "real_preal_lessD"; |
|
658 |
|
659 Goal "m1 < m2 ==> %#m1 < %#m2"; |
|
660 by (dtac preal_less_add_left_Ex 1); |
|
661 by (auto_tac (claset(), |
|
662 simpset() addsimps [real_preal_add, |
|
663 real_preal_def,real_less_def])); |
|
664 by (REPEAT(rtac exI 1)); |
|
665 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
666 by (REPEAT(Blast_tac 2)); |
|
667 by (simp_tac (simpset() addsimps [preal_self_less_add_left] |
|
668 delsimps [preal_add_less_iff2]) 1); |
|
669 qed "real_preal_lessI"; |
|
670 |
|
671 Goal "(%#m1 < %#m2) = (m1 < m2)"; |
|
672 by (blast_tac (claset() addIs [real_preal_lessI,real_preal_lessD]) 1); |
|
673 qed "real_preal_less_iff1"; |
|
674 |
|
675 Addsimps [real_preal_less_iff1]; |
|
676 |
|
677 Goal "- %#m < %#m"; |
|
678 by (auto_tac (claset(), |
|
679 simpset() addsimps |
|
680 [real_preal_def,real_less_def,real_minus])); |
|
681 by (REPEAT(rtac exI 1)); |
|
682 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
683 by (REPEAT(Blast_tac 2)); |
|
684 by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
685 by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
686 preal_add_assoc RS sym]) 1); |
|
687 qed "real_preal_minus_less_self"; |
|
688 |
|
689 Goalw [real_zero_def] "- %#m < 0r"; |
|
690 by (auto_tac (claset(), |
|
691 simpset() addsimps [real_preal_def,real_less_def,real_minus])); |
|
692 by (REPEAT(rtac exI 1)); |
|
693 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
694 by (REPEAT(Blast_tac 2)); |
|
695 by (full_simp_tac (simpset() addsimps |
|
696 [preal_self_less_add_right] @ preal_add_ac) 1); |
|
697 qed "real_preal_minus_less_zero"; |
|
698 |
|
699 Goal "~ 0r < - %#m"; |
|
700 by (cut_facts_tac [real_preal_minus_less_zero] 1); |
|
701 by (fast_tac (claset() addDs [real_less_trans] |
|
702 addEs [real_less_irrefl]) 1); |
|
703 qed "real_preal_not_minus_gt_zero"; |
|
704 |
|
705 Goalw [real_zero_def] "0r < %#m"; |
|
706 by (auto_tac (claset(), |
|
707 simpset() addsimps [real_preal_def,real_less_def,real_minus])); |
|
708 by (REPEAT(rtac exI 1)); |
|
709 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
710 by (REPEAT(Blast_tac 2)); |
|
711 by (full_simp_tac (simpset() addsimps |
|
712 [preal_self_less_add_right] @ preal_add_ac) 1); |
|
713 qed "real_preal_zero_less"; |
|
714 |
|
715 Goal "~ %#m < 0r"; |
|
716 by (cut_facts_tac [real_preal_zero_less] 1); |
|
717 by (blast_tac (claset() addDs [real_less_trans] |
|
718 addEs [real_less_irrefl]) 1); |
|
719 qed "real_preal_not_less_zero"; |
|
720 |
|
721 Goal "0r < - - %#m"; |
|
722 by (simp_tac (simpset() addsimps |
|
723 [real_preal_zero_less]) 1); |
|
724 qed "real_minus_minus_zero_less"; |
|
725 |
|
726 (* another lemma *) |
|
727 Goalw [real_zero_def] "0r < %#m + %#m1"; |
|
728 by (auto_tac (claset(), |
|
729 simpset() addsimps [real_preal_def,real_less_def,real_add])); |
|
730 by (REPEAT(rtac exI 1)); |
|
731 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
732 by (REPEAT(Blast_tac 2)); |
|
733 by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
734 by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
735 preal_add_assoc RS sym]) 1); |
|
736 qed "real_preal_sum_zero_less"; |
|
737 |
|
738 Goal "- %#m < %#m1"; |
|
739 by (auto_tac (claset(), |
|
740 simpset() addsimps [real_preal_def,real_less_def,real_minus])); |
|
741 by (REPEAT(rtac exI 1)); |
|
742 by (EVERY[rtac conjI 1, rtac conjI 2]); |
|
743 by (REPEAT(Blast_tac 2)); |
|
744 by (full_simp_tac (simpset() addsimps preal_add_ac) 1); |
|
745 by (full_simp_tac (simpset() addsimps [preal_self_less_add_right, |
|
746 preal_add_assoc RS sym]) 1); |
|
747 qed "real_preal_minus_less_all"; |
|
748 |
|
749 Goal "~ %#m < - %#m1"; |
|
750 by (cut_facts_tac [real_preal_minus_less_all] 1); |
|
751 by (blast_tac (claset() addDs [real_less_trans] |
|
752 addEs [real_less_irrefl]) 1); |
|
753 qed "real_preal_not_minus_gt_all"; |
|
754 |
|
755 Goal "- %#m1 < - %#m2 ==> %#m2 < %#m1"; |
|
756 by (auto_tac (claset(), |
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757 simpset() addsimps [real_preal_def,real_less_def,real_minus])); |
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758 by (REPEAT(rtac exI 1)); |
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759 by (EVERY[rtac conjI 1, rtac conjI 2]); |
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760 by (REPEAT(Blast_tac 2)); |
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761 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
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762 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
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763 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
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764 qed "real_preal_minus_less_rev1"; |
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765 |
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766 Goal "%#m1 < %#m2 ==> - %#m2 < - %#m1"; |
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767 by (auto_tac (claset(), |
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768 simpset() addsimps [real_preal_def,real_less_def,real_minus])); |
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769 by (REPEAT(rtac exI 1)); |
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770 by (EVERY[rtac conjI 1, rtac conjI 2]); |
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771 by (REPEAT(Blast_tac 2)); |
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772 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
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773 by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1); |
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774 by (auto_tac (claset(),simpset() addsimps preal_add_ac)); |
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775 qed "real_preal_minus_less_rev2"; |
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776 |
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777 Goal "(- %#m1 < - %#m2) = (%#m2 < %#m1)"; |
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778 by (blast_tac (claset() addSIs [real_preal_minus_less_rev1, |
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779 real_preal_minus_less_rev2]) 1); |
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780 qed "real_preal_minus_less_rev_iff"; |
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781 |
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782 Addsimps [real_preal_minus_less_rev_iff]; |
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783 |
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784 (*** linearity ***) |
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785 Goal "(R1::real) < R2 | R1 = R2 | R2 < R1"; |
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786 by (res_inst_tac [("x","R1")] real_preal_trichotomyE 1); |
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787 by (ALLGOALS(res_inst_tac [("x","R2")] real_preal_trichotomyE)); |
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788 by (auto_tac (claset() addSDs [preal_le_anti_sym], |
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789 simpset() addsimps [preal_less_le_iff,real_preal_minus_less_zero, |
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790 real_preal_zero_less,real_preal_minus_less_all])); |
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791 qed "real_linear"; |
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792 |
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793 Goal "!!w::real. (w ~= z) = (w<z | z<w)"; |
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794 by (cut_facts_tac [real_linear] 1); |
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795 by (Blast_tac 1); |
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796 qed "real_neq_iff"; |
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797 |
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798 Goal "!!(R1::real). [| R1 < R2 ==> P; R1 = R2 ==> P; \ |
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799 \ R2 < R1 ==> P |] ==> P"; |
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800 by (cut_inst_tac [("R1.0","R1"),("R2.0","R2")] real_linear 1); |
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801 by Auto_tac; |
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802 qed "real_linear_less2"; |
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803 |
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804 (*** Properties of <= ***) |
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805 |
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806 Goalw [real_le_def] "~(w < z) ==> z <= (w::real)"; |
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807 by (assume_tac 1); |
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808 qed "real_leI"; |
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809 |
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810 Goalw [real_le_def] "z<=w ==> ~(w<(z::real))"; |
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811 by (assume_tac 1); |
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812 qed "real_leD"; |
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813 |
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814 val real_leE = make_elim real_leD; |
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815 |
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816 Goal "(~(w < z)) = (z <= (w::real))"; |
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817 by (blast_tac (claset() addSIs [real_leI,real_leD]) 1); |
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818 qed "real_less_le_iff"; |
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819 |
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820 Goalw [real_le_def] "~ z <= w ==> w<(z::real)"; |
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821 by (Blast_tac 1); |
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822 qed "not_real_leE"; |
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823 |
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824 Goalw [real_le_def] "z < w ==> z <= (w::real)"; |
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825 by (blast_tac (claset() addEs [real_less_asym]) 1); |
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826 qed "real_less_imp_le"; |
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827 |
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828 Goalw [real_le_def] "!!(x::real). x <= y ==> x < y | x = y"; |
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829 by (cut_facts_tac [real_linear] 1); |
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830 by (blast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
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831 qed "real_le_imp_less_or_eq"; |
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832 |
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833 Goalw [real_le_def] "z<w | z=w ==> z <=(w::real)"; |
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834 by (cut_facts_tac [real_linear] 1); |
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835 by (fast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1); |
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836 qed "real_less_or_eq_imp_le"; |
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837 |
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838 Goal "(x <= (y::real)) = (x < y | x=y)"; |
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839 by (REPEAT(ares_tac [iffI, real_less_or_eq_imp_le, real_le_imp_less_or_eq] 1)); |
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840 qed "real_le_less"; |
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841 |
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842 Goal "w <= (w::real)"; |
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843 by (simp_tac (simpset() addsimps [real_le_less]) 1); |
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844 qed "real_le_refl"; |
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845 |
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846 AddIffs [real_le_refl]; |
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847 |
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848 (* Axiom 'linorder_linear' of class 'linorder': *) |
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849 Goal "(z::real) <= w | w <= z"; |
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850 by (simp_tac (simpset() addsimps [real_le_less]) 1); |
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851 by (cut_facts_tac [real_linear] 1); |
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852 by (Blast_tac 1); |
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853 qed "real_le_linear"; |
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854 |
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855 Goal "[| i <= j; j < k |] ==> i < (k::real)"; |
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856 by (dtac real_le_imp_less_or_eq 1); |
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857 by (blast_tac (claset() addIs [real_less_trans]) 1); |
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858 qed "real_le_less_trans"; |
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859 |
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860 Goal "!! (i::real). [| i < j; j <= k |] ==> i < k"; |
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861 by (dtac real_le_imp_less_or_eq 1); |
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862 by (blast_tac (claset() addIs [real_less_trans]) 1); |
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863 qed "real_less_le_trans"; |
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864 |
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865 Goal "[| i <= j; j <= k |] ==> i <= (k::real)"; |
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866 by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
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867 rtac real_less_or_eq_imp_le, blast_tac (claset() addIs [real_less_trans])]); |
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868 qed "real_le_trans"; |
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869 |
|
870 Goal "[| z <= w; w <= z |] ==> z = (w::real)"; |
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871 by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq, |
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872 fast_tac (claset() addEs [real_less_irrefl,real_less_asym])]); |
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873 qed "real_le_anti_sym"; |
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874 |
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875 Goal "[| ~ y < x; y ~= x |] ==> x < (y::real)"; |
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876 by (rtac not_real_leE 1); |
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877 by (blast_tac (claset() addDs [real_le_imp_less_or_eq]) 1); |
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878 qed "not_less_not_eq_real_less"; |
|
879 |
|
880 (* Axiom 'order_less_le' of class 'order': *) |
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881 Goal "(w::real) < z = (w <= z & w ~= z)"; |
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882 by (simp_tac (simpset() addsimps [real_le_def, real_neq_iff]) 1); |
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883 by (blast_tac (claset() addSEs [real_less_asym]) 1); |
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884 qed "real_less_le"; |
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885 |
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886 |
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887 Goal "(0r < -R) = (R < 0r)"; |
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888 by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
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889 by (auto_tac (claset(), |
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890 simpset() addsimps [real_preal_not_minus_gt_zero, |
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891 real_preal_not_less_zero,real_preal_zero_less, |
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892 real_preal_minus_less_zero])); |
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893 qed "real_minus_zero_less_iff"; |
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894 |
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895 Addsimps [real_minus_zero_less_iff]; |
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896 |
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897 Goal "(-R < 0r) = (0r < R)"; |
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898 by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
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899 by (auto_tac (claset(), |
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900 simpset() addsimps [real_preal_not_minus_gt_zero, |
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901 real_preal_not_less_zero,real_preal_zero_less, |
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902 real_preal_minus_less_zero])); |
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903 qed "real_minus_zero_less_iff2"; |
|
904 |
|
905 |
|
906 (*Alternative definition for real_less*) |
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907 Goal "!!(R::real). R < S ==> ? T. 0r < T & R + T = S"; |
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908 by (res_inst_tac [("x","R")] real_preal_trichotomyE 1); |
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909 by (ALLGOALS(res_inst_tac [("x","S")] real_preal_trichotomyE)); |
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910 by (auto_tac (claset() addSDs [preal_less_add_left_Ex], |
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911 simpset() addsimps [real_preal_not_minus_gt_all, |
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912 real_preal_add, real_preal_not_less_zero, |
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913 real_less_not_refl, |
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914 real_preal_not_minus_gt_zero])); |
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915 by (res_inst_tac [("x","%#D")] exI 1); |
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916 by (res_inst_tac [("x","%#m+%#ma")] exI 2); |
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917 by (res_inst_tac [("x","%#m")] exI 3); |
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918 by (res_inst_tac [("x","%#D")] exI 4); |
|
919 by (auto_tac (claset(), |
|
920 simpset() addsimps [real_preal_zero_less, |
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921 real_preal_sum_zero_less,real_add_assoc])); |
|
922 qed "real_less_add_positive_left_Ex"; |
|
923 |
|
924 |
|
925 |
|
926 (** change naff name(s)! **) |
|
927 Goal "(W < S) ==> (0r < S + -W)"; |
|
928 by (dtac real_less_add_positive_left_Ex 1); |
|
929 by (auto_tac (claset(), |
|
930 simpset() addsimps [real_add_minus, |
|
931 real_add_zero_right] @ real_add_ac)); |
|
932 qed "real_less_sum_gt_zero"; |
|
933 |
|
934 Goal "!!S::real. T = S + W ==> S = T + -W"; |
|
935 by (asm_simp_tac (simpset() addsimps real_add_ac) 1); |
|
936 qed "real_lemma_change_eq_subj"; |
|
937 |
|
938 (* FIXME: long! *) |
|
939 Goal "(0r < S + -W) ==> (W < S)"; |
|
940 by (rtac ccontr 1); |
|
941 by (dtac (real_leI RS real_le_imp_less_or_eq) 1); |
|
942 by (auto_tac (claset(), |
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943 simpset() addsimps [real_less_not_refl])); |
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944 by (EVERY1[dtac real_less_add_positive_left_Ex, etac exE, etac conjE]); |
|
945 by (Asm_full_simp_tac 1); |
|
946 by (dtac real_lemma_change_eq_subj 1); |
|
947 by Auto_tac; |
|
948 by (dtac real_less_sum_gt_zero 1); |
|
949 by (asm_full_simp_tac (simpset() addsimps real_add_ac) 1); |
|
950 by (EVERY1[rotate_tac 1, dtac (real_add_left_commute RS ssubst)]); |
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951 by (auto_tac (claset() addEs [real_less_asym], simpset())); |
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952 qed "real_sum_gt_zero_less"; |
|
953 |
|
954 Goal "(0r < S + -W) = (W < S)"; |
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955 by (blast_tac (claset() addIs [real_less_sum_gt_zero, |
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956 real_sum_gt_zero_less]) 1); |
|
957 qed "real_less_sum_gt_0_iff"; |
|
958 |
|
959 |
|
960 Goalw [real_diff_def] "(x<y) = (x-y < 0r)"; |
|
961 by (stac (real_minus_zero_less_iff RS sym) 1); |
|
962 by (simp_tac (simpset() addsimps [real_add_commute, |
|
963 real_less_sum_gt_0_iff]) 1); |
|
964 qed "real_less_eq_diff"; |
|
965 |
|
966 |
|
967 (*** Subtraction laws ***) |
|
968 |
|
969 Goal "x + (y - z) = (x + y) - (z::real)"; |
|
970 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
971 qed "real_add_diff_eq"; |
|
972 |
|
973 Goal "(x - y) + z = (x + z) - (y::real)"; |
|
974 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
975 qed "real_diff_add_eq"; |
|
976 |
|
977 Goal "(x - y) - z = x - (y + (z::real))"; |
|
978 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
979 qed "real_diff_diff_eq"; |
|
980 |
|
981 Goal "x - (y - z) = (x + z) - (y::real)"; |
|
982 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
983 qed "real_diff_diff_eq2"; |
|
984 |
|
985 Goal "(x-y < z) = (x < z + (y::real))"; |
|
986 by (stac real_less_eq_diff 1); |
|
987 by (res_inst_tac [("y1", "z")] (real_less_eq_diff RS ssubst) 1); |
|
988 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
989 qed "real_diff_less_eq"; |
|
990 |
|
991 Goal "(x < z-y) = (x + (y::real) < z)"; |
|
992 by (stac real_less_eq_diff 1); |
|
993 by (res_inst_tac [("y1", "z-y")] (real_less_eq_diff RS ssubst) 1); |
|
994 by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1); |
|
995 qed "real_less_diff_eq"; |
|
996 |
|
997 Goalw [real_le_def] "(x-y <= z) = (x <= z + (y::real))"; |
|
998 by (simp_tac (simpset() addsimps [real_less_diff_eq]) 1); |
|
999 qed "real_diff_le_eq"; |
|
1000 |
|
1001 Goalw [real_le_def] "(x <= z-y) = (x + (y::real) <= z)"; |
|
1002 by (simp_tac (simpset() addsimps [real_diff_less_eq]) 1); |
|
1003 qed "real_le_diff_eq"; |
|
1004 |
|
1005 Goalw [real_diff_def] "(x-y = z) = (x = z + (y::real))"; |
|
1006 by (auto_tac (claset(), simpset() addsimps [real_add_assoc])); |
|
1007 qed "real_diff_eq_eq"; |
|
1008 |
|
1009 Goalw [real_diff_def] "(x = z-y) = (x + (y::real) = z)"; |
|
1010 by (auto_tac (claset(), simpset() addsimps [real_add_assoc])); |
|
1011 qed "real_eq_diff_eq"; |
|
1012 |
|
1013 (*This list of rewrites simplifies (in)equalities by bringing subtractions |
|
1014 to the top and then moving negative terms to the other side. |
|
1015 Use with real_add_ac*) |
|
1016 val real_compare_rls = |
|
1017 [symmetric real_diff_def, |
|
1018 real_add_diff_eq, real_diff_add_eq, real_diff_diff_eq, real_diff_diff_eq2, |
|
1019 real_diff_less_eq, real_less_diff_eq, real_diff_le_eq, real_le_diff_eq, |
|
1020 real_diff_eq_eq, real_eq_diff_eq]; |
|
1021 |
|
1022 |
|
1023 (** For the cancellation simproc. |
|
1024 The idea is to cancel like terms on opposite sides by subtraction **) |
|
1025 |
|
1026 Goal "(x::real) - y = x' - y' ==> (x<y) = (x'<y')"; |
|
1027 by (stac real_less_eq_diff 1); |
|
1028 by (res_inst_tac [("y1", "y")] (real_less_eq_diff RS ssubst) 1); |
|
1029 by (Asm_simp_tac 1); |
|
1030 qed "real_less_eqI"; |
|
1031 |
|
1032 Goal "(x::real) - y = x' - y' ==> (y<=x) = (y'<=x')"; |
|
1033 by (dtac real_less_eqI 1); |
|
1034 by (asm_simp_tac (simpset() addsimps [real_le_def]) 1); |
|
1035 qed "real_le_eqI"; |
|
1036 |
|
1037 Goal "(x::real) - y = x' - y' ==> (x=y) = (x'=y')"; |
|
1038 by Safe_tac; |
|
1039 by (ALLGOALS |
|
1040 (asm_full_simp_tac |
|
1041 (simpset() addsimps [real_eq_diff_eq, real_diff_eq_eq]))); |
|
1042 qed "real_eq_eqI"; |