1 (* Title : NSCA.thy |
1 (* Title : NSCA.thy |
2 Author : Jacques D. Fleuriot |
2 Author : Jacques D. Fleuriot |
3 Copyright : 2001,2002 University of Edinburgh |
3 Copyright : 2001,2002 University of Edinburgh |
4 Description : Infinite, infinitesimal complex number etc! |
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5 *) |
4 *) |
6 |
5 |
7 NSCA = NSComplex + |
6 header{*Non-Standard Complex Analysis*} |
8 |
7 |
9 consts |
8 theory NSCA = NSComplex: |
10 |
9 |
11 (* infinitely close *) |
10 constdefs |
12 "@c=" :: [hcomplex,hcomplex] => bool (infixl 50) |
11 |
13 |
12 capprox :: "[hcomplex,hcomplex] => bool" (infixl "@c=" 50) |
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13 --{*the ``infinitely close'' relation*} |
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14 "x @c= y == (x - y) \<in> CInfinitesimal" |
14 |
15 |
15 constdefs |
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16 (* standard complex numbers reagarded as an embedded subset of NS complex *) |
16 (* standard complex numbers reagarded as an embedded subset of NS complex *) |
17 SComplex :: "hcomplex set" |
17 SComplex :: "hcomplex set" |
18 "SComplex == {x. EX r. x = hcomplex_of_complex r}" |
18 "SComplex == {x. \<exists>r. x = hcomplex_of_complex r}" |
19 |
19 |
20 CInfinitesimal :: "hcomplex set" |
20 CInfinitesimal :: "hcomplex set" |
21 "CInfinitesimal == {x. ALL r: Reals. 0 < r --> hcmod x < r}" |
21 "CInfinitesimal == {x. \<forall>r \<in> Reals. 0 < r --> hcmod x < r}" |
22 |
22 |
23 CFinite :: "hcomplex set" |
23 CFinite :: "hcomplex set" |
24 "CFinite == {x. EX r: Reals. hcmod x < r}" |
24 "CFinite == {x. \<exists>r \<in> Reals. hcmod x < r}" |
25 |
25 |
26 CInfinite :: "hcomplex set" |
26 CInfinite :: "hcomplex set" |
27 "CInfinite == {x. ALL r: Reals. r < hcmod x}" |
27 "CInfinite == {x. \<forall>r \<in> Reals. r < hcmod x}" |
28 |
28 |
29 (* standard part map *) |
29 stc :: "hcomplex => hcomplex" |
30 stc :: hcomplex => hcomplex |
30 --{* standard part map*} |
31 "stc x == (@r. x : CFinite & r:SComplex & r @c= x)" |
31 "stc x == (@r. x \<in> CFinite & r:SComplex & r @c= x)" |
32 |
32 |
33 cmonad :: hcomplex => hcomplex set |
33 cmonad :: "hcomplex => hcomplex set" |
34 "cmonad x == {y. x @c= y}" |
34 "cmonad x == {y. x @c= y}" |
35 |
35 |
36 cgalaxy :: hcomplex => hcomplex set |
36 cgalaxy :: "hcomplex => hcomplex set" |
37 "cgalaxy x == {y. (x - y) : CFinite}" |
37 "cgalaxy x == {y. (x - y) \<in> CFinite}" |
38 |
38 |
39 |
39 |
40 defs |
40 |
41 |
41 subsection{*Closure Laws for SComplex, the Standard Complex Numbers*} |
42 capprox_def "x @c= y == (x - y) : CInfinitesimal" |
42 |
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43 lemma SComplex_add: "[| x \<in> SComplex; y \<in> SComplex |] ==> x + y \<in> SComplex" |
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44 apply (simp add: SComplex_def, safe) |
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45 apply (rule_tac x = "r + ra" in exI, simp) |
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46 done |
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47 |
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48 lemma SComplex_mult: "[| x \<in> SComplex; y \<in> SComplex |] ==> x * y \<in> SComplex" |
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49 apply (simp add: SComplex_def, safe) |
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50 apply (rule_tac x = "r * ra" in exI, simp) |
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51 done |
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52 |
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53 lemma SComplex_inverse: "x \<in> SComplex ==> inverse x \<in> SComplex" |
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54 apply (simp add: SComplex_def) |
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55 apply (blast intro: hcomplex_of_complex_inverse [symmetric]) |
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56 done |
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57 |
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58 lemma SComplex_divide: "[| x \<in> SComplex; y \<in> SComplex |] ==> x/y \<in> SComplex" |
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59 by (simp add: SComplex_mult SComplex_inverse divide_inverse_zero) |
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60 |
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61 lemma SComplex_minus: "x \<in> SComplex ==> -x \<in> SComplex" |
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62 apply (simp add: SComplex_def) |
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63 apply (blast intro: hcomplex_of_complex_minus [symmetric]) |
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64 done |
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65 |
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66 lemma SComplex_minus_iff [simp]: "(-x \<in> SComplex) = (x \<in> SComplex)" |
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67 apply auto |
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68 apply (erule_tac [2] SComplex_minus) |
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69 apply (drule SComplex_minus, auto) |
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70 done |
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71 |
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72 lemma SComplex_diff: "[| x \<in> SComplex; y \<in> SComplex |] ==> x - y \<in> SComplex" |
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73 by (simp add: diff_minus SComplex_add) |
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74 |
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75 lemma SComplex_add_cancel: |
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76 "[| x + y \<in> SComplex; y \<in> SComplex |] ==> x \<in> SComplex" |
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77 by (drule SComplex_diff, assumption, simp) |
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78 |
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79 lemma SReal_hcmod_hcomplex_of_complex [simp]: |
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80 "hcmod (hcomplex_of_complex r) \<in> Reals" |
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81 by (simp add: hcomplex_of_complex_def hcmod SReal_def hypreal_of_real_def) |
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82 |
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83 lemma SReal_hcmod_number_of [simp]: "hcmod (number_of w ::hcomplex) \<in> Reals" |
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84 apply (subst hcomplex_number_of [symmetric]) |
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85 apply (rule SReal_hcmod_hcomplex_of_complex) |
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86 done |
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87 |
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88 lemma SReal_hcmod_SComplex: "x \<in> SComplex ==> hcmod x \<in> Reals" |
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89 by (auto simp add: SComplex_def) |
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90 |
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91 lemma SComplex_hcomplex_of_complex [simp]: "hcomplex_of_complex x \<in> SComplex" |
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92 by (simp add: SComplex_def) |
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93 |
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94 lemma SComplex_number_of [simp]: "(number_of w ::hcomplex) \<in> SComplex" |
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95 apply (subst hcomplex_number_of [symmetric]) |
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96 apply (rule SComplex_hcomplex_of_complex) |
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97 done |
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98 |
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99 lemma SComplex_divide_number_of: |
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100 "r \<in> SComplex ==> r/(number_of w::hcomplex) \<in> SComplex" |
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101 apply (simp only: divide_inverse_zero) |
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102 apply (blast intro!: SComplex_number_of SComplex_mult SComplex_inverse) |
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103 done |
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104 |
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105 lemma SComplex_UNIV_complex: |
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106 "{x. hcomplex_of_complex x \<in> SComplex} = (UNIV::complex set)" |
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107 by (simp add: SComplex_def) |
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108 |
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109 lemma SComplex_iff: "(x \<in> SComplex) = (\<exists>y. x = hcomplex_of_complex y)" |
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110 by (simp add: SComplex_def) |
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111 |
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112 lemma hcomplex_of_complex_image: |
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113 "hcomplex_of_complex `(UNIV::complex set) = SComplex" |
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114 by (auto simp add: SComplex_def) |
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115 |
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116 lemma inv_hcomplex_of_complex_image: "inv hcomplex_of_complex `SComplex = UNIV" |
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117 apply (auto simp add: SComplex_def) |
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118 apply (rule inj_hcomplex_of_complex [THEN inv_f_f, THEN subst], blast) |
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119 done |
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120 |
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121 lemma SComplex_hcomplex_of_complex_image: |
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122 "[| \<exists>x. x: P; P \<le> SComplex |] ==> \<exists>Q. P = hcomplex_of_complex ` Q" |
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123 apply (simp add: SComplex_def, blast) |
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124 done |
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125 |
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126 lemma SComplex_SReal_dense: |
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127 "[| x \<in> SComplex; y \<in> SComplex; hcmod x < hcmod y |
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128 |] ==> \<exists>r \<in> Reals. hcmod x< r & r < hcmod y" |
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129 apply (auto intro: SReal_dense simp add: SReal_hcmod_SComplex) |
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130 done |
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131 |
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132 lemma SComplex_hcmod_SReal: |
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133 "z \<in> SComplex ==> hcmod z \<in> Reals" |
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134 apply (simp add: SComplex_def SReal_def) |
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135 apply (rule_tac z = z in eq_Abs_hcomplex) |
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136 apply (auto simp add: hcmod hypreal_of_real_def hcomplex_of_complex_def cmod_def) |
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137 apply (rule_tac x = "cmod r" in exI) |
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138 apply (simp add: cmod_def, ultra) |
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139 done |
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140 |
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141 lemma SComplex_zero [simp]: "0 \<in> SComplex" |
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142 by (simp add: SComplex_def hcomplex_of_complex_zero_iff) |
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143 |
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144 lemma SComplex_one [simp]: "1 \<in> SComplex" |
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145 by (simp add: SComplex_def hcomplex_of_complex_def hcomplex_one_def) |
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146 |
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147 (* |
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148 Goalw [SComplex_def,SReal_def] "hcmod z \<in> Reals ==> z \<in> SComplex" |
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149 by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1); |
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150 by (auto_tac (claset(),simpset() addsimps [hcmod,hypreal_of_real_def, |
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151 hcomplex_of_complex_def,cmod_def])); |
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152 *) |
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153 |
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154 |
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155 subsection{*The Finite Elements form a Subring*} |
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156 |
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157 lemma CFinite_add: "[|x \<in> CFinite; y \<in> CFinite|] ==> (x+y) \<in> CFinite" |
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158 apply (simp add: CFinite_def) |
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159 apply (blast intro!: SReal_add hcmod_add_less) |
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160 done |
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161 |
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162 lemma CFinite_mult: "[|x \<in> CFinite; y \<in> CFinite|] ==> x*y \<in> CFinite" |
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163 apply (simp add: CFinite_def) |
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164 apply (blast intro!: SReal_mult hcmod_mult_less) |
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165 done |
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166 |
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167 lemma CFinite_minus_iff [simp]: "(-x \<in> CFinite) = (x \<in> CFinite)" |
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168 by (simp add: CFinite_def) |
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169 |
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170 lemma SComplex_subset_CFinite [simp]: "SComplex \<le> CFinite" |
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171 apply (auto simp add: SComplex_def CFinite_def) |
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172 apply (rule_tac x = "1 + hcmod (hcomplex_of_complex r) " in bexI) |
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173 apply (auto intro: SReal_add) |
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174 done |
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175 |
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176 lemma HFinite_hcmod_hcomplex_of_complex [simp]: |
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177 "hcmod (hcomplex_of_complex r) \<in> HFinite" |
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178 by (auto intro!: SReal_subset_HFinite [THEN subsetD]) |
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179 |
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180 lemma CFinite_hcomplex_of_complex [simp]: "hcomplex_of_complex x \<in> CFinite" |
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181 by (auto intro!: SComplex_subset_CFinite [THEN subsetD]) |
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182 |
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183 lemma CFiniteD: "x \<in> CFinite ==> \<exists>t \<in> Reals. hcmod x < t" |
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184 by (simp add: CFinite_def) |
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185 |
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186 lemma CFinite_hcmod_iff: "(x \<in> CFinite) = (hcmod x \<in> HFinite)" |
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187 by (simp add: CFinite_def HFinite_def) |
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188 |
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189 lemma CFinite_number_of [simp]: "number_of w \<in> CFinite" |
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190 by (rule SComplex_number_of [THEN SComplex_subset_CFinite [THEN subsetD]]) |
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191 |
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192 lemma CFinite_bounded: "[|x \<in> CFinite; y \<le> hcmod x; 0 \<le> y |] ==> y: HFinite" |
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193 by (auto intro: HFinite_bounded simp add: CFinite_hcmod_iff) |
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194 |
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195 |
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196 subsection{*The Complex Infinitesimals form a Subring*} |
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197 |
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198 lemma CInfinitesimal_zero [iff]: "0 \<in> CInfinitesimal" |
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199 by (simp add: CInfinitesimal_def) |
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200 |
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201 lemma hcomplex_sum_of_halves: "x/(2::hcomplex) + x/(2::hcomplex) = x" |
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202 by auto |
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203 |
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204 lemma CInfinitesimal_hcmod_iff: |
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205 "(z \<in> CInfinitesimal) = (hcmod z \<in> Infinitesimal)" |
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206 by (simp add: CInfinitesimal_def Infinitesimal_def) |
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207 |
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208 lemma one_not_CInfinitesimal [simp]: "1 \<notin> CInfinitesimal" |
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209 by (simp add: CInfinitesimal_hcmod_iff) |
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210 |
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211 lemma CInfinitesimal_add: |
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212 "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> (x+y) \<in> CInfinitesimal" |
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213 apply (auto simp add: CInfinitesimal_hcmod_iff) |
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214 apply (rule hrabs_le_Infinitesimal) |
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215 apply (rule_tac y = "hcmod y" in Infinitesimal_add, auto) |
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216 done |
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217 |
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218 lemma CInfinitesimal_minus_iff [simp]: |
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219 "(-x:CInfinitesimal) = (x:CInfinitesimal)" |
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220 by (simp add: CInfinitesimal_def) |
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221 |
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222 lemma CInfinitesimal_diff: |
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223 "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> x-y \<in> CInfinitesimal" |
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224 by (simp add: diff_minus CInfinitesimal_add) |
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225 |
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226 lemma CInfinitesimal_mult: |
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227 "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> x * y \<in> CInfinitesimal" |
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228 by (auto intro: Infinitesimal_mult simp add: CInfinitesimal_hcmod_iff hcmod_mult) |
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229 |
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230 lemma CInfinitesimal_CFinite_mult: |
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231 "[| x \<in> CInfinitesimal; y \<in> CFinite |] ==> (x * y) \<in> CInfinitesimal" |
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232 by (auto intro: Infinitesimal_HFinite_mult simp add: CInfinitesimal_hcmod_iff CFinite_hcmod_iff hcmod_mult) |
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233 |
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234 lemma CInfinitesimal_CFinite_mult2: |
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235 "[| x \<in> CInfinitesimal; y \<in> CFinite |] ==> (y * x) \<in> CInfinitesimal" |
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236 by (auto dest: CInfinitesimal_CFinite_mult simp add: hcomplex_mult_commute) |
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237 |
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238 lemma CInfinite_hcmod_iff: "(z \<in> CInfinite) = (hcmod z \<in> HInfinite)" |
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239 by (simp add: CInfinite_def HInfinite_def) |
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240 |
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241 lemma CInfinite_inverse_CInfinitesimal: |
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242 "x \<in> CInfinite ==> inverse x \<in> CInfinitesimal" |
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243 by (auto intro: HInfinite_inverse_Infinitesimal simp add: CInfinitesimal_hcmod_iff CInfinite_hcmod_iff hcmod_hcomplex_inverse) |
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244 |
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245 lemma CInfinite_mult: "[|x \<in> CInfinite; y \<in> CInfinite|] ==> (x*y): CInfinite" |
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246 by (auto intro: HInfinite_mult simp add: CInfinite_hcmod_iff hcmod_mult) |
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247 |
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248 lemma CInfinite_minus_iff [simp]: "(-x \<in> CInfinite) = (x \<in> CInfinite)" |
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249 by (simp add: CInfinite_def) |
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250 |
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251 lemma CFinite_sum_squares: |
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252 "[|a \<in> CFinite; b \<in> CFinite; c \<in> CFinite|] |
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253 ==> a*a + b*b + c*c \<in> CFinite" |
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254 by (auto intro: CFinite_mult CFinite_add) |
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255 |
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256 lemma not_CInfinitesimal_not_zero: "x \<notin> CInfinitesimal ==> x \<noteq> 0" |
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257 by auto |
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258 |
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259 lemma not_CInfinitesimal_not_zero2: "x \<in> CFinite - CInfinitesimal ==> x \<noteq> 0" |
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260 by auto |
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261 |
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262 lemma CFinite_diff_CInfinitesimal_hcmod: |
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263 "x \<in> CFinite - CInfinitesimal ==> hcmod x \<in> HFinite - Infinitesimal" |
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264 by (simp add: CFinite_hcmod_iff CInfinitesimal_hcmod_iff) |
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265 |
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266 lemma hcmod_less_CInfinitesimal: |
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267 "[| e \<in> CInfinitesimal; hcmod x < hcmod e |] ==> x \<in> CInfinitesimal" |
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268 by (auto intro: hrabs_less_Infinitesimal simp add: CInfinitesimal_hcmod_iff) |
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269 |
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270 lemma hcmod_le_CInfinitesimal: |
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271 "[| e \<in> CInfinitesimal; hcmod x \<le> hcmod e |] ==> x \<in> CInfinitesimal" |
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272 by (auto intro: hrabs_le_Infinitesimal simp add: CInfinitesimal_hcmod_iff) |
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273 |
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274 lemma CInfinitesimal_interval: |
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275 "[| e \<in> CInfinitesimal; |
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276 e' \<in> CInfinitesimal; |
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277 hcmod e' < hcmod x ; hcmod x < hcmod e |
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278 |] ==> x \<in> CInfinitesimal" |
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279 by (auto intro: Infinitesimal_interval simp add: CInfinitesimal_hcmod_iff) |
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280 |
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281 lemma CInfinitesimal_interval2: |
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282 "[| e \<in> CInfinitesimal; |
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283 e' \<in> CInfinitesimal; |
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284 hcmod e' \<le> hcmod x ; hcmod x \<le> hcmod e |
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285 |] ==> x \<in> CInfinitesimal" |
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286 by (auto intro: Infinitesimal_interval2 simp add: CInfinitesimal_hcmod_iff) |
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287 |
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288 lemma not_CInfinitesimal_mult: |
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289 "[| x \<notin> CInfinitesimal; y \<notin> CInfinitesimal|] ==> (x*y) \<notin> CInfinitesimal" |
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290 apply (auto simp add: CInfinitesimal_hcmod_iff hcmod_mult) |
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291 apply (drule not_Infinitesimal_mult, auto) |
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292 done |
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293 |
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294 lemma CInfinitesimal_mult_disj: |
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295 "x*y \<in> CInfinitesimal ==> x \<in> CInfinitesimal | y \<in> CInfinitesimal" |
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296 by (auto dest: Infinitesimal_mult_disj simp add: CInfinitesimal_hcmod_iff hcmod_mult) |
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297 |
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298 lemma CFinite_CInfinitesimal_diff_mult: |
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299 "[| x \<in> CFinite - CInfinitesimal; y \<in> CFinite - CInfinitesimal |] |
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300 ==> x*y \<in> CFinite - CInfinitesimal" |
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301 by (blast dest: CFinite_mult not_CInfinitesimal_mult) |
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302 |
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303 lemma CInfinitesimal_subset_CFinite: "CInfinitesimal \<le> CFinite" |
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304 by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] |
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305 simp add: CInfinitesimal_hcmod_iff CFinite_hcmod_iff) |
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306 |
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307 lemma CInfinitesimal_hcomplex_of_complex_mult: |
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308 "x \<in> CInfinitesimal ==> x * hcomplex_of_complex r \<in> CInfinitesimal" |
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309 by (auto intro!: Infinitesimal_HFinite_mult simp add: CInfinitesimal_hcmod_iff hcmod_mult) |
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310 |
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311 lemma CInfinitesimal_hcomplex_of_complex_mult2: |
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312 "x \<in> CInfinitesimal ==> hcomplex_of_complex r * x \<in> CInfinitesimal" |
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313 by (auto intro!: Infinitesimal_HFinite_mult2 simp add: CInfinitesimal_hcmod_iff hcmod_mult) |
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314 |
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315 |
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316 subsection{*The ``Infinitely Close'' Relation*} |
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317 |
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318 (* |
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319 Goalw [capprox_def,approx_def] "(z @c= w) = (hcmod z @= hcmod w)" |
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320 by (auto_tac (claset(),simpset() addsimps [CInfinitesimal_hcmod_iff])); |
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321 *) |
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322 |
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323 lemma mem_cinfmal_iff: "x:CInfinitesimal = (x @c= 0)" |
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324 by (simp add: CInfinitesimal_hcmod_iff capprox_def) |
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325 |
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326 lemma capprox_minus_iff: "(x @c= y) = (x + -y @c= 0)" |
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327 by (simp add: capprox_def diff_minus) |
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328 |
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329 lemma capprox_minus_iff2: "(x @c= y) = (-y + x @c= 0)" |
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330 by (simp add: capprox_def diff_minus add_commute) |
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331 |
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332 lemma capprox_refl [simp]: "x @c= x" |
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333 by (simp add: capprox_def) |
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334 |
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335 lemma capprox_sym: "x @c= y ==> y @c= x" |
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336 by (simp add: capprox_def CInfinitesimal_def hcmod_diff_commute) |
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337 |
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338 lemma capprox_trans: "[| x @c= y; y @c= z |] ==> x @c= z" |
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339 apply (simp add: capprox_def) |
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340 apply (drule CInfinitesimal_add, assumption) |
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341 apply (simp add: diff_minus) |
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342 done |
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343 |
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344 lemma capprox_trans2: "[| r @c= x; s @c= x |] ==> r @c= s" |
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345 by (blast intro: capprox_sym capprox_trans) |
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346 |
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347 lemma capprox_trans3: "[| x @c= r; x @c= s|] ==> r @c= s" |
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348 by (blast intro: capprox_sym capprox_trans) |
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349 |
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350 lemma number_of_capprox_reorient [simp]: |
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351 "(number_of w @c= x) = (x @c= number_of w)" |
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352 by (blast intro: capprox_sym) |
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353 |
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354 lemma CInfinitesimal_capprox_minus: "(x-y \<in> CInfinitesimal) = (x @c= y)" |
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355 by (simp add: diff_minus capprox_minus_iff [symmetric] mem_cinfmal_iff) |
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356 |
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357 lemma capprox_monad_iff: "(x @c= y) = (cmonad(x)=cmonad(y))" |
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358 by (auto simp add: cmonad_def dest: capprox_sym elim!: capprox_trans equalityCE) |
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359 |
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360 lemma Infinitesimal_capprox: |
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361 "[| x \<in> CInfinitesimal; y \<in> CInfinitesimal |] ==> x @c= y" |
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362 apply (simp add: mem_cinfmal_iff) |
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363 apply (blast intro: capprox_trans capprox_sym) |
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364 done |
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365 |
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366 lemma capprox_add: "[| a @c= b; c @c= d |] ==> a+c @c= b+d" |
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367 apply (simp add: capprox_def diff_minus) |
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368 apply (rule minus_add_distrib [THEN ssubst]) |
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369 apply (rule add_assoc [THEN ssubst]) |
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370 apply (rule_tac b1 = c in add_left_commute [THEN subst]) |
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371 apply (rule add_assoc [THEN subst]) |
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372 apply (blast intro: CInfinitesimal_add) |
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373 done |
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374 |
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375 lemma capprox_minus: "a @c= b ==> -a @c= -b" |
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376 apply (rule capprox_minus_iff [THEN iffD2, THEN capprox_sym]) |
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377 apply (drule capprox_minus_iff [THEN iffD1]) |
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378 apply (simp add: add_commute) |
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379 done |
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380 |
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381 lemma capprox_minus2: "-a @c= -b ==> a @c= b" |
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382 by (auto dest: capprox_minus) |
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383 |
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384 lemma capprox_minus_cancel [simp]: "(-a @c= -b) = (a @c= b)" |
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385 by (blast intro: capprox_minus capprox_minus2) |
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386 |
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387 lemma capprox_add_minus: "[| a @c= b; c @c= d |] ==> a + -c @c= b + -d" |
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388 by (blast intro!: capprox_add capprox_minus) |
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389 |
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390 lemma capprox_mult1: |
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391 "[| a @c= b; c \<in> CFinite|] ==> a*c @c= b*c" |
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392 apply (simp add: capprox_def diff_minus) |
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393 apply (simp only: CInfinitesimal_CFinite_mult minus_mult_left hcomplex_add_mult_distrib [symmetric]) |
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394 done |
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395 |
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396 lemma capprox_mult2: "[|a @c= b; c \<in> CFinite|] ==> c*a @c= c*b" |
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397 by (simp add: capprox_mult1 hcomplex_mult_commute) |
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398 |
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399 lemma capprox_mult_subst: |
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400 "[|u @c= v*x; x @c= y; v \<in> CFinite|] ==> u @c= v*y" |
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401 by (blast intro: capprox_mult2 capprox_trans) |
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402 |
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403 lemma capprox_mult_subst2: |
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404 "[| u @c= x*v; x @c= y; v \<in> CFinite |] ==> u @c= y*v" |
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405 by (blast intro: capprox_mult1 capprox_trans) |
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406 |
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407 lemma capprox_mult_subst_SComplex: |
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408 "[| u @c= x*hcomplex_of_complex v; x @c= y |] |
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409 ==> u @c= y*hcomplex_of_complex v" |
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410 by (auto intro: capprox_mult_subst2) |
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411 |
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412 lemma capprox_eq_imp: "a = b ==> a @c= b" |
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413 by (simp add: capprox_def) |
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414 |
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415 lemma CInfinitesimal_minus_capprox: "x \<in> CInfinitesimal ==> -x @c= x" |
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416 by (blast intro: CInfinitesimal_minus_iff [THEN iffD2] mem_cinfmal_iff [THEN iffD1] capprox_trans2) |
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417 |
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418 lemma bex_CInfinitesimal_iff: "(\<exists>y \<in> CInfinitesimal. x - z = y) = (x @c= z)" |
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419 by (unfold capprox_def, blast) |
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420 |
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421 lemma bex_CInfinitesimal_iff2: "(\<exists>y \<in> CInfinitesimal. x = z + y) = (x @c= z)" |
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422 by (simp add: bex_CInfinitesimal_iff [symmetric], force) |
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423 |
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424 lemma CInfinitesimal_add_capprox: |
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425 "[| y \<in> CInfinitesimal; x + y = z |] ==> x @c= z" |
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426 apply (rule bex_CInfinitesimal_iff [THEN iffD1]) |
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427 apply (drule CInfinitesimal_minus_iff [THEN iffD2]) |
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428 apply (simp add: eq_commute compare_rls) |
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429 done |
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430 |
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431 lemma CInfinitesimal_add_capprox_self: "y \<in> CInfinitesimal ==> x @c= x + y" |
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432 apply (rule bex_CInfinitesimal_iff [THEN iffD1]) |
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433 apply (drule CInfinitesimal_minus_iff [THEN iffD2]) |
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434 apply (simp add: eq_commute compare_rls) |
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435 done |
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436 |
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437 lemma CInfinitesimal_add_capprox_self2: "y \<in> CInfinitesimal ==> x @c= y + x" |
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438 by (auto dest: CInfinitesimal_add_capprox_self simp add: add_commute) |
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439 |
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440 lemma CInfinitesimal_add_minus_capprox_self: |
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441 "y \<in> CInfinitesimal ==> x @c= x + -y" |
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442 by (blast intro!: CInfinitesimal_add_capprox_self CInfinitesimal_minus_iff [THEN iffD2]) |
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443 |
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444 lemma CInfinitesimal_add_cancel: |
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445 "[| y \<in> CInfinitesimal; x+y @c= z|] ==> x @c= z" |
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446 apply (drule_tac x = x in CInfinitesimal_add_capprox_self [THEN capprox_sym]) |
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447 apply (erule capprox_trans3 [THEN capprox_sym], assumption) |
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448 done |
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449 |
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450 lemma CInfinitesimal_add_right_cancel: |
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451 "[| y \<in> CInfinitesimal; x @c= z + y|] ==> x @c= z" |
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452 apply (drule_tac x = z in CInfinitesimal_add_capprox_self2 [THEN capprox_sym]) |
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453 apply (erule capprox_trans3 [THEN capprox_sym]) |
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454 apply (simp add: add_commute) |
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455 apply (erule capprox_sym) |
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456 done |
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457 |
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458 lemma capprox_add_left_cancel: "d + b @c= d + c ==> b @c= c" |
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459 apply (drule capprox_minus_iff [THEN iffD1]) |
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460 apply (simp add: minus_add_distrib capprox_minus_iff [symmetric] add_ac) |
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461 done |
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462 |
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463 lemma capprox_add_right_cancel: "b + d @c= c + d ==> b @c= c" |
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464 apply (rule capprox_add_left_cancel) |
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465 apply (simp add: add_commute) |
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466 done |
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467 |
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468 lemma capprox_add_mono1: "b @c= c ==> d + b @c= d + c" |
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469 apply (rule capprox_minus_iff [THEN iffD2]) |
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470 apply (simp add: capprox_minus_iff [symmetric] add_ac) |
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471 done |
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472 |
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473 lemma capprox_add_mono2: "b @c= c ==> b + a @c= c + a" |
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474 apply (simp (no_asm_simp) add: add_commute capprox_add_mono1) |
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475 done |
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476 |
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477 lemma capprox_add_left_iff [iff]: "(a + b @c= a + c) = (b @c= c)" |
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478 by (fast elim: capprox_add_left_cancel capprox_add_mono1) |
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479 |
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480 lemma capprox_add_right_iff [iff]: "(b + a @c= c + a) = (b @c= c)" |
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481 by (simp add: add_commute) |
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482 |
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483 lemma capprox_CFinite: "[| x \<in> CFinite; x @c= y |] ==> y \<in> CFinite" |
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484 apply (drule bex_CInfinitesimal_iff2 [THEN iffD2], safe) |
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485 apply (drule CInfinitesimal_subset_CFinite [THEN subsetD, THEN CFinite_minus_iff [THEN iffD2]]) |
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486 apply (drule CFinite_add) |
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487 apply (assumption, auto) |
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488 done |
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489 |
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490 lemma capprox_hcomplex_of_complex_CFinite: |
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491 "x @c= hcomplex_of_complex D ==> x \<in> CFinite" |
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492 by (rule capprox_sym [THEN [2] capprox_CFinite], auto) |
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493 |
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494 lemma capprox_mult_CFinite: |
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495 "[|a @c= b; c @c= d; b \<in> CFinite; d \<in> CFinite|] ==> a*c @c= b*d" |
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496 apply (rule capprox_trans) |
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497 apply (rule_tac [2] capprox_mult2) |
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498 apply (rule capprox_mult1) |
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499 prefer 2 apply (blast intro: capprox_CFinite capprox_sym, auto) |
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500 done |
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501 |
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502 lemma capprox_mult_hcomplex_of_complex: |
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503 "[|a @c= hcomplex_of_complex b; c @c= hcomplex_of_complex d |] |
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504 ==> a*c @c= hcomplex_of_complex b * hcomplex_of_complex d" |
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505 apply (blast intro!: capprox_mult_CFinite capprox_hcomplex_of_complex_CFinite CFinite_hcomplex_of_complex) |
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506 done |
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507 |
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508 lemma capprox_SComplex_mult_cancel_zero: |
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509 "[| a \<in> SComplex; a \<noteq> 0; a*x @c= 0 |] ==> x @c= 0" |
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510 apply (drule SComplex_inverse [THEN SComplex_subset_CFinite [THEN subsetD]]) |
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511 apply (auto dest: capprox_mult2 simp add: hcomplex_mult_assoc [symmetric]) |
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512 done |
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513 |
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514 lemma capprox_mult_SComplex1: "[| a \<in> SComplex; x @c= 0 |] ==> x*a @c= 0" |
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515 by (auto dest: SComplex_subset_CFinite [THEN subsetD] capprox_mult1) |
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516 |
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517 lemma capprox_mult_SComplex2: "[| a \<in> SComplex; x @c= 0 |] ==> a*x @c= 0" |
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518 by (auto dest: SComplex_subset_CFinite [THEN subsetD] capprox_mult2) |
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519 |
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520 lemma capprox_mult_SComplex_zero_cancel_iff [simp]: |
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521 "[|a \<in> SComplex; a \<noteq> 0 |] ==> (a*x @c= 0) = (x @c= 0)" |
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522 by (blast intro: capprox_SComplex_mult_cancel_zero capprox_mult_SComplex2) |
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523 |
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524 lemma capprox_SComplex_mult_cancel: |
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525 "[| a \<in> SComplex; a \<noteq> 0; a* w @c= a*z |] ==> w @c= z" |
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526 apply (drule SComplex_inverse [THEN SComplex_subset_CFinite [THEN subsetD]]) |
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527 apply (auto dest: capprox_mult2 simp add: hcomplex_mult_assoc [symmetric]) |
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528 done |
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529 |
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530 lemma capprox_SComplex_mult_cancel_iff1 [simp]: |
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531 "[| a \<in> SComplex; a \<noteq> 0|] ==> (a* w @c= a*z) = (w @c= z)" |
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532 by (auto intro!: capprox_mult2 SComplex_subset_CFinite [THEN subsetD] |
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533 intro: capprox_SComplex_mult_cancel) |
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534 |
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535 lemma capprox_hcmod_approx_zero: "(x @c= y) = (hcmod (y - x) @= 0)" |
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536 apply (rule capprox_minus_iff [THEN ssubst]) |
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537 apply (simp add: capprox_def CInfinitesimal_hcmod_iff mem_infmal_iff diff_minus [symmetric] hcmod_diff_commute) |
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538 done |
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539 |
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540 lemma capprox_approx_zero_iff: "(x @c= 0) = (hcmod x @= 0)" |
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541 by (simp add: capprox_hcmod_approx_zero) |
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542 |
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543 lemma capprox_minus_zero_cancel_iff [simp]: "(-x @c= 0) = (x @c= 0)" |
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544 by (simp add: capprox_hcmod_approx_zero) |
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545 |
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546 lemma Infinitesimal_hcmod_add_diff: |
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547 "u @c= 0 ==> hcmod(x + u) - hcmod x \<in> Infinitesimal" |
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548 apply (rule_tac e = "hcmod u" and e' = "- hcmod u" in Infinitesimal_interval2) |
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549 apply (auto dest: capprox_approx_zero_iff [THEN iffD1] |
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550 simp add: mem_infmal_iff [symmetric] hypreal_diff_def) |
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551 apply (rule_tac c1 = "hcmod x" in add_le_cancel_left [THEN iffD1]) |
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552 apply (auto simp add: diff_minus [symmetric]) |
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553 done |
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554 |
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555 lemma approx_hcmod_add_hcmod: "u @c= 0 ==> hcmod(x + u) @= hcmod x" |
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556 apply (rule approx_minus_iff [THEN iffD2]) |
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557 apply (auto intro: Infinitesimal_hcmod_add_diff simp add: mem_infmal_iff [symmetric] diff_minus [symmetric]) |
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558 done |
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559 |
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560 lemma capprox_hcmod_approx: "x @c= y ==> hcmod x @= hcmod y" |
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561 by (auto intro: approx_hcmod_add_hcmod |
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562 dest!: bex_CInfinitesimal_iff2 [THEN iffD2] |
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563 simp add: mem_cinfmal_iff) |
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564 |
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565 |
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566 subsection{*Zero is the Only Infinitesimal Complex Number*} |
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567 |
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568 lemma CInfinitesimal_less_SComplex: |
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569 "[| x \<in> SComplex; y \<in> CInfinitesimal; 0 < hcmod x |] ==> hcmod y < hcmod x" |
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570 by (auto intro!: Infinitesimal_less_SReal SComplex_hcmod_SReal simp add: CInfinitesimal_hcmod_iff) |
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571 |
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572 lemma SComplex_Int_CInfinitesimal_zero: "SComplex Int CInfinitesimal = {0}" |
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573 apply (auto simp add: SComplex_def CInfinitesimal_hcmod_iff) |
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574 apply (cut_tac r = r in SReal_hcmod_hcomplex_of_complex) |
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575 apply (drule_tac A = Reals in IntI, assumption) |
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576 apply (subgoal_tac "hcmod (hcomplex_of_complex r) = 0") |
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577 apply simp |
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578 apply (simp add: SReal_Int_Infinitesimal_zero) |
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579 done |
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580 |
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581 lemma SComplex_CInfinitesimal_zero: |
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582 "[| x \<in> SComplex; x \<in> CInfinitesimal|] ==> x = 0" |
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583 by (cut_tac SComplex_Int_CInfinitesimal_zero, blast) |
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584 |
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585 lemma SComplex_CFinite_diff_CInfinitesimal: |
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586 "[| x \<in> SComplex; x \<noteq> 0 |] ==> x \<in> CFinite - CInfinitesimal" |
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587 by (auto dest: SComplex_CInfinitesimal_zero SComplex_subset_CFinite [THEN subsetD]) |
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588 |
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589 lemma hcomplex_of_complex_CFinite_diff_CInfinitesimal: |
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590 "hcomplex_of_complex x \<noteq> 0 |
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591 ==> hcomplex_of_complex x \<in> CFinite - CInfinitesimal" |
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592 by (rule SComplex_CFinite_diff_CInfinitesimal, auto) |
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593 |
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594 lemma hcomplex_of_complex_CInfinitesimal_iff_0 [iff]: |
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595 "(hcomplex_of_complex x \<in> CInfinitesimal) = (x=0)" |
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596 apply (auto simp add: hcomplex_of_complex_zero) |
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597 apply (rule ccontr) |
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598 apply (rule hcomplex_of_complex_CFinite_diff_CInfinitesimal [THEN DiffD2], auto) |
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599 done |
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600 |
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601 lemma number_of_not_CInfinitesimal [simp]: |
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602 "number_of w \<noteq> (0::hcomplex) ==> number_of w \<notin> CInfinitesimal" |
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603 by (fast dest: SComplex_number_of [THEN SComplex_CInfinitesimal_zero]) |
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604 |
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605 lemma capprox_SComplex_not_zero: |
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606 "[| y \<in> SComplex; x @c= y; y\<noteq> 0 |] ==> x \<noteq> 0" |
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607 by (auto dest: SComplex_CInfinitesimal_zero capprox_sym [THEN mem_cinfmal_iff [THEN iffD2]]) |
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608 |
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609 lemma CFinite_diff_CInfinitesimal_capprox: |
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610 "[| x @c= y; y \<in> CFinite - CInfinitesimal |] |
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611 ==> x \<in> CFinite - CInfinitesimal" |
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612 apply (auto intro: capprox_sym [THEN [2] capprox_CFinite] |
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613 simp add: mem_cinfmal_iff) |
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614 apply (drule capprox_trans3, assumption) |
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615 apply (blast dest: capprox_sym) |
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616 done |
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617 |
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618 lemma CInfinitesimal_ratio: |
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619 "[| y \<noteq> 0; y \<in> CInfinitesimal; x/y \<in> CFinite |] ==> x \<in> CInfinitesimal" |
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620 apply (drule CInfinitesimal_CFinite_mult2, assumption) |
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621 apply (simp add: divide_inverse_zero hcomplex_mult_assoc) |
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622 done |
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623 |
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624 lemma SComplex_capprox_iff: |
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625 "[|x \<in> SComplex; y \<in> SComplex|] ==> (x @c= y) = (x = y)" |
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626 apply auto |
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627 apply (simp add: capprox_def) |
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628 apply (subgoal_tac "x-y = 0", simp) |
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629 apply (rule SComplex_CInfinitesimal_zero) |
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630 apply (simp add: SComplex_diff, assumption) |
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631 done |
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632 |
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633 lemma number_of_capprox_iff [simp]: |
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634 "(number_of v @c= number_of w) = (number_of v = (number_of w :: hcomplex))" |
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635 by (rule SComplex_capprox_iff, auto) |
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636 |
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637 lemma number_of_CInfinitesimal_iff [simp]: |
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638 "(number_of w \<in> CInfinitesimal) = (number_of w = (0::hcomplex))" |
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639 apply (rule iffI) |
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640 apply (fast dest: SComplex_number_of [THEN SComplex_CInfinitesimal_zero]) |
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641 apply (simp (no_asm_simp)) |
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642 done |
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643 |
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644 lemma hcomplex_of_complex_approx_iff [simp]: |
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645 "(hcomplex_of_complex k @c= hcomplex_of_complex m) = (k = m)" |
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646 apply auto |
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647 apply (rule inj_hcomplex_of_complex [THEN injD]) |
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648 apply (rule SComplex_capprox_iff [THEN iffD1], auto) |
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649 done |
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650 |
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651 lemma hcomplex_of_complex_capprox_number_of_iff [simp]: |
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652 "(hcomplex_of_complex k @c= number_of w) = (k = number_of w)" |
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653 by (subst hcomplex_of_complex_approx_iff [symmetric], auto) |
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654 |
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655 lemma capprox_unique_complex: |
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656 "[| r \<in> SComplex; s \<in> SComplex; r @c= x; s @c= x|] ==> r = s" |
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657 by (blast intro: SComplex_capprox_iff [THEN iffD1] capprox_trans2) |
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658 |
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659 lemma hcomplex_capproxD1: |
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660 "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n}) |
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661 ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) @= |
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662 Abs_hypreal(hyprel `` {%n. Re(Y n)})" |
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663 apply (auto simp add: approx_FreeUltrafilterNat_iff) |
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664 apply (drule capprox_minus_iff [THEN iffD1]) |
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665 apply (auto simp add: hcomplex_minus hcomplex_add mem_cinfmal_iff [symmetric] CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff2) |
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666 apply (drule_tac x = m in spec, ultra) |
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667 apply (rename_tac Z x) |
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668 apply (case_tac "X x") |
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669 apply (case_tac "Y x") |
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670 apply (auto simp add: complex_minus complex_add complex_mod |
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671 simp del: realpow_Suc) |
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672 apply (rule_tac y="abs(Z x)" in order_le_less_trans) |
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673 apply (drule_tac t = "Z x" in sym) |
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674 apply (auto simp add: abs_eqI1 simp del: realpow_Suc) |
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675 done |
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676 |
|
677 (* same proof *) |
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678 lemma hcomplex_capproxD2: |
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679 "Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n}) |
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680 ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) @= |
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681 Abs_hypreal(hyprel `` {%n. Im(Y n)})" |
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682 apply (auto simp add: approx_FreeUltrafilterNat_iff) |
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683 apply (drule capprox_minus_iff [THEN iffD1]) |
|
684 apply (auto simp add: hcomplex_minus hcomplex_add mem_cinfmal_iff [symmetric] CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff2) |
|
685 apply (drule_tac x = m in spec, ultra) |
|
686 apply (rename_tac Z x) |
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687 apply (case_tac "X x") |
|
688 apply (case_tac "Y x") |
|
689 apply (auto simp add: complex_minus complex_add complex_mod simp del: realpow_Suc) |
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690 apply (rule_tac y="abs(Z x)" in order_le_less_trans) |
|
691 apply (drule_tac t = "Z x" in sym) |
|
692 apply (auto simp add: abs_eqI1 simp del: realpow_Suc) |
|
693 done |
|
694 |
|
695 lemma hcomplex_capproxI: |
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696 "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) @= |
|
697 Abs_hypreal(hyprel `` {%n. Re(Y n)}); |
|
698 Abs_hypreal(hyprel `` {%n. Im(X n)}) @= |
|
699 Abs_hypreal(hyprel `` {%n. Im(Y n)}) |
|
700 |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})" |
|
701 apply (drule approx_minus_iff [THEN iffD1]) |
|
702 apply (drule approx_minus_iff [THEN iffD1]) |
|
703 apply (rule capprox_minus_iff [THEN iffD2]) |
|
704 apply (auto simp add: mem_cinfmal_iff [symmetric] mem_infmal_iff [symmetric] hypreal_minus hypreal_add hcomplex_minus hcomplex_add CInfinitesimal_hcmod_iff hcmod Infinitesimal_FreeUltrafilterNat_iff) |
|
705 apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto) |
|
706 apply (drule_tac x = "u/2" in spec) |
|
707 apply (drule_tac x = "u/2" in spec, auto, ultra) |
|
708 apply (drule sym, drule sym) |
|
709 apply (case_tac "X x") |
|
710 apply (case_tac "Y x") |
|
711 apply (auto simp add: complex_minus complex_add complex_mod snd_conv fst_conv numeral_2_eq_2) |
|
712 apply (rename_tac a b c d) |
|
713 apply (subgoal_tac "sqrt (abs (a + - c) ^ 2 + abs (b + - d) ^ 2) < u") |
|
714 apply (rule_tac [2] lemma_sqrt_hcomplex_capprox, auto) |
|
715 apply (simp add: power2_eq_square) |
|
716 done |
|
717 |
|
718 lemma capprox_approx_iff: |
|
719 "(Abs_hcomplex(hcomplexrel ``{%n. X n}) @c= Abs_hcomplex(hcomplexrel``{%n. Y n})) = |
|
720 (Abs_hypreal(hyprel `` {%n. Re(X n)}) @= Abs_hypreal(hyprel `` {%n. Re(Y n)}) & |
|
721 Abs_hypreal(hyprel `` {%n. Im(X n)}) @= Abs_hypreal(hyprel `` {%n. Im(Y n)}))" |
|
722 apply (blast intro: hcomplex_capproxI hcomplex_capproxD1 hcomplex_capproxD2) |
|
723 done |
|
724 |
|
725 lemma hcomplex_of_hypreal_capprox_iff [simp]: |
|
726 "(hcomplex_of_hypreal x @c= hcomplex_of_hypreal z) = (x @= z)" |
|
727 apply (rule eq_Abs_hypreal [of x]) |
|
728 apply (rule eq_Abs_hypreal [of z]) |
|
729 apply (simp add: hcomplex_of_hypreal capprox_approx_iff) |
|
730 done |
|
731 |
|
732 lemma CFinite_HFinite_Re: |
|
733 "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite |
|
734 ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HFinite" |
|
735 apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff) |
|
736 apply (rule bexI, rule_tac [2] lemma_hyprel_refl) |
|
737 apply (rule_tac x = u in exI, ultra) |
|
738 apply (drule sym, case_tac "X x") |
|
739 apply (auto simp add: complex_mod numeral_2_eq_2 simp del: realpow_Suc) |
|
740 apply (rule ccontr, drule linorder_not_less [THEN iffD1]) |
|
741 apply (drule order_less_le_trans, assumption) |
|
742 apply (drule real_sqrt_ge_abs1 [THEN [2] order_less_le_trans]) |
|
743 apply (auto simp add: numeral_2_eq_2 [symmetric]) |
|
744 done |
|
745 |
|
746 lemma CFinite_HFinite_Im: |
|
747 "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite |
|
748 ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HFinite" |
|
749 apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff) |
|
750 apply (rule bexI, rule_tac [2] lemma_hyprel_refl) |
|
751 apply (rule_tac x = u in exI, ultra) |
|
752 apply (drule sym, case_tac "X x") |
|
753 apply (auto simp add: complex_mod simp del: realpow_Suc) |
|
754 apply (rule ccontr, drule linorder_not_less [THEN iffD1]) |
|
755 apply (drule order_less_le_trans, assumption) |
|
756 apply (drule real_sqrt_ge_abs2 [THEN [2] order_less_le_trans], auto) |
|
757 done |
|
758 |
|
759 lemma HFinite_Re_Im_CFinite: |
|
760 "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HFinite; |
|
761 Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HFinite |
|
762 |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite" |
|
763 apply (auto simp add: CFinite_hcmod_iff hcmod HFinite_FreeUltrafilterNat_iff) |
|
764 apply (rename_tac Y Z u v) |
|
765 apply (rule bexI, rule_tac [2] lemma_hyprel_refl) |
|
766 apply (rule_tac x = "2* (u + v) " in exI) |
|
767 apply ultra |
|
768 apply (drule sym, case_tac "X x") |
|
769 apply (auto simp add: complex_mod numeral_2_eq_2 simp del: realpow_Suc) |
|
770 apply (subgoal_tac "0 < u") |
|
771 prefer 2 apply arith |
|
772 apply (subgoal_tac "0 < v") |
|
773 prefer 2 apply arith |
|
774 apply (subgoal_tac "sqrt (abs (Y x) ^ 2 + abs (Z x) ^ 2) < 2*u + 2*v") |
|
775 apply (rule_tac [2] lemma_sqrt_hcomplex_capprox, auto) |
|
776 apply (simp add: power2_eq_square) |
|
777 done |
|
778 |
|
779 lemma CFinite_HFinite_iff: |
|
780 "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CFinite) = |
|
781 (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HFinite & |
|
782 Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HFinite)" |
|
783 by (blast intro: HFinite_Re_Im_CFinite CFinite_HFinite_Im CFinite_HFinite_Re) |
|
784 |
|
785 lemma SComplex_Re_SReal: |
|
786 "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex |
|
787 ==> Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Reals" |
|
788 apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def) |
|
789 apply (rule_tac x = "Re r" in exI, ultra) |
|
790 done |
|
791 |
|
792 lemma SComplex_Im_SReal: |
|
793 "Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex |
|
794 ==> Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Reals" |
|
795 apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def) |
|
796 apply (rule_tac x = "Im r" in exI, ultra) |
|
797 done |
|
798 |
|
799 lemma Reals_Re_Im_SComplex: |
|
800 "[| Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Reals; |
|
801 Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Reals |
|
802 |] ==> Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex" |
|
803 apply (auto simp add: SComplex_def hcomplex_of_complex_def SReal_def hypreal_of_real_def) |
|
804 apply (rule_tac x = "Complex r ra" in exI, ultra) |
|
805 done |
|
806 |
|
807 lemma SComplex_SReal_iff: |
|
808 "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> SComplex) = |
|
809 (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Reals & |
|
810 Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Reals)" |
|
811 by (blast intro: SComplex_Re_SReal SComplex_Im_SReal Reals_Re_Im_SComplex) |
|
812 |
|
813 lemma CInfinitesimal_Infinitesimal_iff: |
|
814 "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CInfinitesimal) = |
|
815 (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> Infinitesimal & |
|
816 Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> Infinitesimal)" |
|
817 by (simp add: mem_cinfmal_iff mem_infmal_iff hcomplex_zero_num hypreal_zero_num capprox_approx_iff) |
|
818 |
|
819 lemma eq_Abs_hcomplex_EX: |
|
820 "(\<exists>t. P t) = (\<exists>X. P (Abs_hcomplex(hcomplexrel `` {X})))" |
|
821 apply auto |
|
822 apply (rule_tac z = t in eq_Abs_hcomplex, auto) |
|
823 done |
|
824 |
|
825 lemma eq_Abs_hcomplex_Bex: |
|
826 "(\<exists>t \<in> A. P t) = (\<exists>X. (Abs_hcomplex(hcomplexrel `` {X})) \<in> A & |
|
827 P (Abs_hcomplex(hcomplexrel `` {X})))" |
|
828 apply auto |
|
829 apply (rule_tac z = t in eq_Abs_hcomplex, auto) |
|
830 done |
|
831 |
|
832 (* Here we go - easy proof now!! *) |
|
833 lemma stc_part_Ex: "x:CFinite ==> \<exists>t \<in> SComplex. x @c= t" |
|
834 apply (rule_tac z = x in eq_Abs_hcomplex) |
|
835 apply (auto simp add: CFinite_HFinite_iff eq_Abs_hcomplex_Bex SComplex_SReal_iff capprox_approx_iff) |
|
836 apply (drule st_part_Ex, safe)+ |
|
837 apply (rule_tac z = t in eq_Abs_hypreal) |
|
838 apply (rule_tac z = ta in eq_Abs_hypreal, auto) |
|
839 apply (rule_tac x = "%n. Complex (xa n) (xb n) " in exI) |
|
840 apply auto |
|
841 done |
|
842 |
|
843 lemma stc_part_Ex1: "x:CFinite ==> EX! t. t \<in> SComplex & x @c= t" |
|
844 apply (drule stc_part_Ex, safe) |
|
845 apply (drule_tac [2] capprox_sym, drule_tac [2] capprox_sym, drule_tac [2] capprox_sym) |
|
846 apply (auto intro!: capprox_unique_complex) |
|
847 done |
|
848 |
|
849 lemma CFinite_Int_CInfinite_empty: "CFinite Int CInfinite = {}" |
|
850 by (simp add: CFinite_def CInfinite_def, auto) |
|
851 |
|
852 lemma CFinite_not_CInfinite: "x \<in> CFinite ==> x \<notin> CInfinite" |
|
853 by (insert CFinite_Int_CInfinite_empty, blast) |
|
854 |
|
855 text{*Not sure this is a good idea!*} |
|
856 declare CFinite_Int_CInfinite_empty [simp] |
|
857 |
|
858 lemma not_CFinite_CInfinite: "x\<notin> CFinite ==> x \<in> CInfinite" |
|
859 by (auto intro: not_HFinite_HInfinite simp add: CFinite_hcmod_iff CInfinite_hcmod_iff) |
|
860 |
|
861 lemma CInfinite_CFinite_disj: "x \<in> CInfinite | x \<in> CFinite" |
|
862 by (blast intro: not_CFinite_CInfinite) |
|
863 |
|
864 lemma CInfinite_CFinite_iff: "(x \<in> CInfinite) = (x \<notin> CFinite)" |
|
865 by (blast dest: CFinite_not_CInfinite not_CFinite_CInfinite) |
|
866 |
|
867 lemma CFinite_CInfinite_iff: "(x \<in> CFinite) = (x \<notin> CInfinite)" |
|
868 by (simp add: CInfinite_CFinite_iff) |
|
869 |
|
870 lemma CInfinite_diff_CFinite_CInfinitesimal_disj: |
|
871 "x \<notin> CInfinitesimal ==> x \<in> CInfinite | x \<in> CFinite - CInfinitesimal" |
|
872 by (fast intro: not_CFinite_CInfinite) |
|
873 |
|
874 lemma CFinite_inverse: |
|
875 "[| x \<in> CFinite; x \<notin> CInfinitesimal |] ==> inverse x \<in> CFinite" |
|
876 apply (cut_tac x = "inverse x" in CInfinite_CFinite_disj) |
|
877 apply (auto dest!: CInfinite_inverse_CInfinitesimal) |
|
878 done |
|
879 |
|
880 lemma CFinite_inverse2: "x \<in> CFinite - CInfinitesimal ==> inverse x \<in> CFinite" |
|
881 by (blast intro: CFinite_inverse) |
|
882 |
|
883 lemma CInfinitesimal_inverse_CFinite: |
|
884 "x \<notin> CInfinitesimal ==> inverse(x) \<in> CFinite" |
|
885 apply (drule CInfinite_diff_CFinite_CInfinitesimal_disj) |
|
886 apply (blast intro: CFinite_inverse CInfinite_inverse_CInfinitesimal CInfinitesimal_subset_CFinite [THEN subsetD]) |
|
887 done |
|
888 |
|
889 |
|
890 lemma CFinite_not_CInfinitesimal_inverse: |
|
891 "x \<in> CFinite - CInfinitesimal ==> inverse x \<in> CFinite - CInfinitesimal" |
|
892 apply (auto intro: CInfinitesimal_inverse_CFinite) |
|
893 apply (drule CInfinitesimal_CFinite_mult2, assumption) |
|
894 apply (simp add: not_CInfinitesimal_not_zero) |
|
895 done |
|
896 |
|
897 lemma capprox_inverse: |
|
898 "[| x @c= y; y \<in> CFinite - CInfinitesimal |] ==> inverse x @c= inverse y" |
|
899 apply (frule CFinite_diff_CInfinitesimal_capprox, assumption) |
|
900 apply (frule not_CInfinitesimal_not_zero2) |
|
901 apply (frule_tac x = x in not_CInfinitesimal_not_zero2) |
|
902 apply (drule CFinite_inverse2)+ |
|
903 apply (drule capprox_mult2, assumption, auto) |
|
904 apply (drule_tac c = "inverse x" in capprox_mult1, assumption) |
|
905 apply (auto intro: capprox_sym simp add: hcomplex_mult_assoc) |
|
906 done |
|
907 |
|
908 lemmas hcomplex_of_complex_capprox_inverse = hcomplex_of_complex_CFinite_diff_CInfinitesimal [THEN [2] capprox_inverse] |
|
909 |
|
910 lemma inverse_add_CInfinitesimal_capprox: |
|
911 "[| x \<in> CFinite - CInfinitesimal; |
|
912 h \<in> CInfinitesimal |] ==> inverse(x + h) @c= inverse x" |
|
913 by (auto intro: capprox_inverse capprox_sym CInfinitesimal_add_capprox_self) |
|
914 |
|
915 lemma inverse_add_CInfinitesimal_capprox2: |
|
916 "[| x \<in> CFinite - CInfinitesimal; |
|
917 h \<in> CInfinitesimal |] ==> inverse(h + x) @c= inverse x" |
|
918 apply (rule add_commute [THEN subst]) |
|
919 apply (blast intro: inverse_add_CInfinitesimal_capprox) |
|
920 done |
|
921 |
|
922 lemma inverse_add_CInfinitesimal_approx_CInfinitesimal: |
|
923 "[| x \<in> CFinite - CInfinitesimal; |
|
924 h \<in> CInfinitesimal |] ==> inverse(x + h) - inverse x @c= h" |
|
925 apply (rule capprox_trans2) |
|
926 apply (auto intro: inverse_add_CInfinitesimal_capprox |
|
927 simp add: mem_cinfmal_iff diff_minus capprox_minus_iff [symmetric]) |
|
928 done |
|
929 |
|
930 lemma CInfinitesimal_square_iff [iff]: |
|
931 "(x*x \<in> CInfinitesimal) = (x \<in> CInfinitesimal)" |
|
932 by (simp add: CInfinitesimal_hcmod_iff hcmod_mult) |
|
933 |
|
934 lemma capprox_CFinite_mult_cancel: |
|
935 "[| a \<in> CFinite-CInfinitesimal; a*w @c= a*z |] ==> w @c= z" |
|
936 apply safe |
|
937 apply (frule CFinite_inverse, assumption) |
|
938 apply (drule not_CInfinitesimal_not_zero) |
|
939 apply (auto dest: capprox_mult2 simp add: hcomplex_mult_assoc [symmetric]) |
|
940 done |
|
941 |
|
942 lemma capprox_CFinite_mult_cancel_iff1: |
|
943 "a \<in> CFinite-CInfinitesimal ==> (a * w @c= a * z) = (w @c= z)" |
|
944 by (auto intro: capprox_mult2 capprox_CFinite_mult_cancel) |
|
945 |
|
946 |
|
947 subsection{*Theorems About Monads*} |
|
948 |
|
949 lemma capprox_cmonad_iff: "(x @c= y) = (cmonad(x)=cmonad(y))" |
|
950 apply (simp add: cmonad_def) |
|
951 apply (auto dest: capprox_sym elim!: capprox_trans equalityCE) |
|
952 done |
|
953 |
|
954 lemma CInfinitesimal_cmonad_eq: |
|
955 "e \<in> CInfinitesimal ==> cmonad (x+e) = cmonad x" |
|
956 by (fast intro!: CInfinitesimal_add_capprox_self [THEN capprox_sym] capprox_cmonad_iff [THEN iffD1]) |
|
957 |
|
958 lemma mem_cmonad_iff: "(u \<in> cmonad x) = (-u \<in> cmonad (-x))" |
|
959 by (simp add: cmonad_def) |
|
960 |
|
961 lemma CInfinitesimal_cmonad_zero_iff: "(x:CInfinitesimal) = (x \<in> cmonad 0)" |
|
962 by (auto intro: capprox_sym simp add: mem_cinfmal_iff cmonad_def) |
|
963 |
|
964 lemma cmonad_zero_minus_iff: "(x \<in> cmonad 0) = (-x \<in> cmonad 0)" |
|
965 by (simp add: CInfinitesimal_cmonad_zero_iff [symmetric]) |
|
966 |
|
967 lemma cmonad_zero_hcmod_iff: "(x \<in> cmonad 0) = (hcmod x:monad 0)" |
|
968 by (simp add: CInfinitesimal_cmonad_zero_iff [symmetric] CInfinitesimal_hcmod_iff Infinitesimal_monad_zero_iff [symmetric]) |
|
969 |
|
970 lemma mem_cmonad_self [simp]: "x \<in> cmonad x" |
|
971 by (simp add: cmonad_def) |
|
972 |
|
973 |
|
974 subsection{*Theorems About Standard Part*} |
|
975 |
|
976 lemma stc_capprox_self: "x \<in> CFinite ==> stc x @c= x" |
|
977 apply (simp add: stc_def) |
|
978 apply (frule stc_part_Ex, safe) |
|
979 apply (rule someI2) |
|
980 apply (auto intro: capprox_sym) |
|
981 done |
|
982 |
|
983 lemma stc_SComplex: "x \<in> CFinite ==> stc x \<in> SComplex" |
|
984 apply (simp add: stc_def) |
|
985 apply (frule stc_part_Ex, safe) |
|
986 apply (rule someI2) |
|
987 apply (auto intro: capprox_sym) |
|
988 done |
|
989 |
|
990 lemma stc_CFinite: "x \<in> CFinite ==> stc x \<in> CFinite" |
|
991 by (erule stc_SComplex [THEN SComplex_subset_CFinite [THEN subsetD]]) |
|
992 |
|
993 lemma stc_SComplex_eq [simp]: "x \<in> SComplex ==> stc x = x" |
|
994 apply (simp add: stc_def) |
|
995 apply (rule some_equality) |
|
996 apply (auto intro: SComplex_subset_CFinite [THEN subsetD]) |
|
997 apply (blast dest: SComplex_capprox_iff [THEN iffD1]) |
|
998 done |
|
999 |
|
1000 lemma stc_hcomplex_of_complex: |
|
1001 "stc (hcomplex_of_complex x) = hcomplex_of_complex x" |
|
1002 by auto |
|
1003 |
|
1004 lemma stc_eq_capprox: |
|
1005 "[| x \<in> CFinite; y \<in> CFinite; stc x = stc y |] ==> x @c= y" |
|
1006 by (auto dest!: stc_capprox_self elim!: capprox_trans3) |
|
1007 |
|
1008 lemma capprox_stc_eq: |
|
1009 "[| x \<in> CFinite; y \<in> CFinite; x @c= y |] ==> stc x = stc y" |
|
1010 by (blast intro: capprox_trans capprox_trans2 SComplex_capprox_iff [THEN iffD1] |
|
1011 dest: stc_capprox_self stc_SComplex) |
|
1012 |
|
1013 lemma stc_eq_capprox_iff: |
|
1014 "[| x \<in> CFinite; y \<in> CFinite|] ==> (x @c= y) = (stc x = stc y)" |
|
1015 by (blast intro: capprox_stc_eq stc_eq_capprox) |
|
1016 |
|
1017 lemma stc_CInfinitesimal_add_SComplex: |
|
1018 "[| x \<in> SComplex; e \<in> CInfinitesimal |] ==> stc(x + e) = x" |
|
1019 apply (frule stc_SComplex_eq [THEN subst]) |
|
1020 prefer 2 apply assumption |
|
1021 apply (frule SComplex_subset_CFinite [THEN subsetD]) |
|
1022 apply (frule CInfinitesimal_subset_CFinite [THEN subsetD]) |
|
1023 apply (drule stc_SComplex_eq) |
|
1024 apply (rule capprox_stc_eq) |
|
1025 apply (auto intro: CFinite_add simp add: CInfinitesimal_add_capprox_self [THEN capprox_sym]) |
|
1026 done |
|
1027 |
|
1028 lemma stc_CInfinitesimal_add_SComplex2: |
|
1029 "[| x \<in> SComplex; e \<in> CInfinitesimal |] ==> stc(e + x) = x" |
|
1030 apply (rule add_commute [THEN subst]) |
|
1031 apply (blast intro!: stc_CInfinitesimal_add_SComplex) |
|
1032 done |
|
1033 |
|
1034 lemma CFinite_stc_CInfinitesimal_add: |
|
1035 "x \<in> CFinite ==> \<exists>e \<in> CInfinitesimal. x = stc(x) + e" |
|
1036 by (blast dest!: stc_capprox_self [THEN capprox_sym] bex_CInfinitesimal_iff2 [THEN iffD2]) |
|
1037 |
|
1038 lemma stc_add: |
|
1039 "[| x \<in> CFinite; y \<in> CFinite |] ==> stc (x + y) = stc(x) + stc(y)" |
|
1040 apply (frule CFinite_stc_CInfinitesimal_add) |
|
1041 apply (frule_tac x = y in CFinite_stc_CInfinitesimal_add, safe) |
|
1042 apply (subgoal_tac "stc (x + y) = stc ((stc x + e) + (stc y + ea))") |
|
1043 apply (drule_tac [2] sym, drule_tac [2] sym) |
|
1044 prefer 2 apply simp |
|
1045 apply (simp (no_asm_simp) add: add_ac) |
|
1046 apply (drule stc_SComplex)+ |
|
1047 apply (drule SComplex_add, assumption) |
|
1048 apply (drule CInfinitesimal_add, assumption) |
|
1049 apply (rule add_assoc [THEN subst]) |
|
1050 apply (blast intro!: stc_CInfinitesimal_add_SComplex2) |
|
1051 done |
|
1052 |
|
1053 lemma stc_number_of [simp]: "stc (number_of w) = number_of w" |
|
1054 by (rule SComplex_number_of [THEN stc_SComplex_eq]) |
|
1055 |
|
1056 lemma stc_zero [simp]: "stc 0 = 0" |
|
1057 by simp |
|
1058 |
|
1059 lemma stc_one [simp]: "stc 1 = 1" |
|
1060 by simp |
|
1061 |
|
1062 lemma stc_minus: "y \<in> CFinite ==> stc(-y) = -stc(y)" |
|
1063 apply (frule CFinite_minus_iff [THEN iffD2]) |
|
1064 apply (rule hcomplex_add_minus_eq_minus) |
|
1065 apply (drule stc_add [symmetric], assumption) |
|
1066 apply (simp add: add_commute) |
|
1067 done |
|
1068 |
|
1069 lemma stc_diff: |
|
1070 "[| x \<in> CFinite; y \<in> CFinite |] ==> stc (x-y) = stc(x) - stc(y)" |
|
1071 apply (simp add: diff_minus) |
|
1072 apply (frule_tac y1 = y in stc_minus [symmetric]) |
|
1073 apply (drule_tac x1 = y in CFinite_minus_iff [THEN iffD2]) |
|
1074 apply (auto intro: stc_add) |
|
1075 done |
|
1076 |
|
1077 lemma lemma_stc_mult: |
|
1078 "[| x \<in> CFinite; y \<in> CFinite; |
|
1079 e \<in> CInfinitesimal; |
|
1080 ea: CInfinitesimal |] |
|
1081 ==> e*y + x*ea + e*ea: CInfinitesimal" |
|
1082 apply (frule_tac x = e and y = y in CInfinitesimal_CFinite_mult) |
|
1083 apply (frule_tac [2] x = ea and y = x in CInfinitesimal_CFinite_mult) |
|
1084 apply (drule_tac [3] CInfinitesimal_mult) |
|
1085 apply (auto intro: CInfinitesimal_add simp add: add_ac mult_ac) |
|
1086 done |
|
1087 |
|
1088 lemma stc_mult: |
|
1089 "[| x \<in> CFinite; y \<in> CFinite |] |
|
1090 ==> stc (x * y) = stc(x) * stc(y)" |
|
1091 apply (frule CFinite_stc_CInfinitesimal_add) |
|
1092 apply (frule_tac x = y in CFinite_stc_CInfinitesimal_add, safe) |
|
1093 apply (subgoal_tac "stc (x * y) = stc ((stc x + e) * (stc y + ea))") |
|
1094 apply (drule_tac [2] sym, drule_tac [2] sym) |
|
1095 prefer 2 apply simp |
|
1096 apply (erule_tac V = "x = stc x + e" in thin_rl) |
|
1097 apply (erule_tac V = "y = stc y + ea" in thin_rl) |
|
1098 apply (simp add: hcomplex_add_mult_distrib right_distrib) |
|
1099 apply (drule stc_SComplex)+ |
|
1100 apply (simp (no_asm_use) add: add_assoc) |
|
1101 apply (rule stc_CInfinitesimal_add_SComplex) |
|
1102 apply (blast intro!: SComplex_mult) |
|
1103 apply (drule SComplex_subset_CFinite [THEN subsetD])+ |
|
1104 apply (rule add_assoc [THEN subst]) |
|
1105 apply (blast intro!: lemma_stc_mult) |
|
1106 done |
|
1107 |
|
1108 lemma stc_CInfinitesimal: "x \<in> CInfinitesimal ==> stc x = 0" |
|
1109 apply (rule stc_zero [THEN subst]) |
|
1110 apply (rule capprox_stc_eq) |
|
1111 apply (auto intro: CInfinitesimal_subset_CFinite [THEN subsetD] |
|
1112 simp add: mem_cinfmal_iff [symmetric]) |
|
1113 done |
|
1114 |
|
1115 lemma stc_not_CInfinitesimal: "stc(x) \<noteq> 0 ==> x \<notin> CInfinitesimal" |
|
1116 by (fast intro: stc_CInfinitesimal) |
|
1117 |
|
1118 lemma stc_inverse: |
|
1119 "[| x \<in> CFinite; stc x \<noteq> 0 |] |
|
1120 ==> stc(inverse x) = inverse (stc x)" |
|
1121 apply (rule_tac c1 = "stc x" in hcomplex_mult_left_cancel [THEN iffD1]) |
|
1122 apply (auto simp add: stc_mult [symmetric] stc_not_CInfinitesimal CFinite_inverse) |
|
1123 apply (subst right_inverse, auto) |
|
1124 done |
|
1125 |
|
1126 lemma stc_divide [simp]: |
|
1127 "[| x \<in> CFinite; y \<in> CFinite; stc y \<noteq> 0 |] |
|
1128 ==> stc(x/y) = (stc x) / (stc y)" |
|
1129 by (simp add: divide_inverse_zero stc_mult stc_not_CInfinitesimal CFinite_inverse stc_inverse) |
|
1130 |
|
1131 lemma stc_idempotent [simp]: "x \<in> CFinite ==> stc(stc(x)) = stc(x)" |
|
1132 by (blast intro: stc_CFinite stc_capprox_self capprox_stc_eq) |
|
1133 |
|
1134 lemma CFinite_HFinite_hcomplex_of_hypreal: |
|
1135 "z \<in> HFinite ==> hcomplex_of_hypreal z \<in> CFinite" |
|
1136 apply (rule eq_Abs_hypreal [of z]) |
|
1137 apply (simp add: hcomplex_of_hypreal CFinite_HFinite_iff hypreal_zero_def [symmetric]) |
|
1138 done |
|
1139 |
|
1140 lemma SComplex_SReal_hcomplex_of_hypreal: |
|
1141 "x \<in> Reals ==> hcomplex_of_hypreal x \<in> SComplex" |
|
1142 apply (rule eq_Abs_hypreal [of x]) |
|
1143 apply (simp add: hcomplex_of_hypreal SComplex_SReal_iff hypreal_zero_def [symmetric]) |
|
1144 done |
|
1145 |
|
1146 lemma stc_hcomplex_of_hypreal: |
|
1147 "z \<in> HFinite ==> stc(hcomplex_of_hypreal z) = hcomplex_of_hypreal (st z)" |
|
1148 apply (simp add: st_def stc_def) |
|
1149 apply (frule st_part_Ex, safe) |
|
1150 apply (rule someI2) |
|
1151 apply (auto intro: approx_sym) |
|
1152 apply (drule CFinite_HFinite_hcomplex_of_hypreal) |
|
1153 apply (frule stc_part_Ex, safe) |
|
1154 apply (rule someI2) |
|
1155 apply (auto intro: capprox_sym intro!: capprox_unique_complex dest: SComplex_SReal_hcomplex_of_hypreal) |
|
1156 done |
|
1157 |
|
1158 (* |
|
1159 Goal "x \<in> CFinite ==> hcmod(stc x) = st(hcmod x)" |
|
1160 by (dtac stc_capprox_self 1) |
|
1161 by (auto_tac (claset(),simpset() addsimps [bex_CInfinitesimal_iff2 RS sym])); |
|
1162 |
|
1163 |
|
1164 approx_hcmod_add_hcmod |
|
1165 *) |
|
1166 |
|
1167 lemma CInfinitesimal_hcnj_iff [simp]: |
|
1168 "(hcnj z \<in> CInfinitesimal) = (z \<in> CInfinitesimal)" |
|
1169 by (simp add: CInfinitesimal_hcmod_iff) |
|
1170 |
|
1171 lemma CInfinite_HInfinite_iff: |
|
1172 "(Abs_hcomplex(hcomplexrel ``{%n. X n}) \<in> CInfinite) = |
|
1173 (Abs_hypreal(hyprel `` {%n. Re(X n)}) \<in> HInfinite | |
|
1174 Abs_hypreal(hyprel `` {%n. Im(X n)}) \<in> HInfinite)" |
|
1175 by (simp add: CInfinite_CFinite_iff HInfinite_HFinite_iff CFinite_HFinite_iff) |
|
1176 |
|
1177 text{*These theorems should probably be deleted*} |
|
1178 lemma hcomplex_split_CInfinitesimal_iff: |
|
1179 "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> CInfinitesimal) = |
|
1180 (x \<in> Infinitesimal & y \<in> Infinitesimal)" |
|
1181 apply (rule eq_Abs_hypreal [of x]) |
|
1182 apply (rule eq_Abs_hypreal [of y]) |
|
1183 apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal CInfinitesimal_Infinitesimal_iff) |
|
1184 done |
|
1185 |
|
1186 lemma hcomplex_split_CFinite_iff: |
|
1187 "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> CFinite) = |
|
1188 (x \<in> HFinite & y \<in> HFinite)" |
|
1189 apply (rule eq_Abs_hypreal [of x]) |
|
1190 apply (rule eq_Abs_hypreal [of y]) |
|
1191 apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal CFinite_HFinite_iff) |
|
1192 done |
|
1193 |
|
1194 lemma hcomplex_split_SComplex_iff: |
|
1195 "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> SComplex) = |
|
1196 (x \<in> Reals & y \<in> Reals)" |
|
1197 apply (rule eq_Abs_hypreal [of x]) |
|
1198 apply (rule eq_Abs_hypreal [of y]) |
|
1199 apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal SComplex_SReal_iff) |
|
1200 done |
|
1201 |
|
1202 lemma hcomplex_split_CInfinite_iff: |
|
1203 "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y \<in> CInfinite) = |
|
1204 (x \<in> HInfinite | y \<in> HInfinite)" |
|
1205 apply (rule eq_Abs_hypreal [of x]) |
|
1206 apply (rule eq_Abs_hypreal [of y]) |
|
1207 apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal CInfinite_HInfinite_iff) |
|
1208 done |
|
1209 |
|
1210 lemma hcomplex_split_capprox_iff: |
|
1211 "(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y @c= |
|
1212 hcomplex_of_hypreal x' + iii * hcomplex_of_hypreal y') = |
|
1213 (x @= x' & y @= y')" |
|
1214 apply (rule eq_Abs_hypreal [of x]) |
|
1215 apply (rule eq_Abs_hypreal [of y]) |
|
1216 apply (rule eq_Abs_hypreal [of x']) |
|
1217 apply (rule eq_Abs_hypreal [of y']) |
|
1218 apply (simp add: iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal capprox_approx_iff) |
|
1219 done |
|
1220 |
|
1221 lemma complex_seq_to_hcomplex_CInfinitesimal: |
|
1222 "\<forall>n. cmod (X n - x) < inverse (real (Suc n)) ==> |
|
1223 Abs_hcomplex(hcomplexrel``{X}) - hcomplex_of_complex x \<in> CInfinitesimal" |
|
1224 apply (simp add: hcomplex_diff CInfinitesimal_hcmod_iff hcomplex_of_complex_def Infinitesimal_FreeUltrafilterNat_iff hcmod) |
|
1225 apply (rule bexI, auto) |
|
1226 apply (auto dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset) |
|
1227 done |
|
1228 |
|
1229 lemma CInfinitesimal_hcomplex_of_hypreal_epsilon [simp]: |
|
1230 "hcomplex_of_hypreal epsilon \<in> CInfinitesimal" |
|
1231 by (simp add: CInfinitesimal_hcmod_iff) |
|
1232 |
|
1233 lemma hcomplex_of_complex_approx_zero_iff [simp]: |
|
1234 "(hcomplex_of_complex z @c= 0) = (z = 0)" |
|
1235 by (simp add: hcomplex_of_complex_zero [symmetric] |
|
1236 del: hcomplex_of_complex_zero) |
|
1237 |
|
1238 lemma hcomplex_of_complex_approx_zero_iff2 [simp]: |
|
1239 "(0 @c= hcomplex_of_complex z) = (z = 0)" |
|
1240 by (simp add: hcomplex_of_complex_zero [symmetric] |
|
1241 del: hcomplex_of_complex_zero) |
|
1242 |
|
1243 |
|
1244 ML |
|
1245 {* |
|
1246 val SComplex_add = thm "SComplex_add"; |
|
1247 val SComplex_mult = thm "SComplex_mult"; |
|
1248 val SComplex_inverse = thm "SComplex_inverse"; |
|
1249 val SComplex_divide = thm "SComplex_divide"; |
|
1250 val SComplex_minus = thm "SComplex_minus"; |
|
1251 val SComplex_minus_iff = thm "SComplex_minus_iff"; |
|
1252 val SComplex_diff = thm "SComplex_diff"; |
|
1253 val SComplex_add_cancel = thm "SComplex_add_cancel"; |
|
1254 val SReal_hcmod_hcomplex_of_complex = thm "SReal_hcmod_hcomplex_of_complex"; |
|
1255 val SReal_hcmod_number_of = thm "SReal_hcmod_number_of"; |
|
1256 val SReal_hcmod_SComplex = thm "SReal_hcmod_SComplex"; |
|
1257 val SComplex_hcomplex_of_complex = thm "SComplex_hcomplex_of_complex"; |
|
1258 val SComplex_number_of = thm "SComplex_number_of"; |
|
1259 val SComplex_divide_number_of = thm "SComplex_divide_number_of"; |
|
1260 val SComplex_UNIV_complex = thm "SComplex_UNIV_complex"; |
|
1261 val SComplex_iff = thm "SComplex_iff"; |
|
1262 val hcomplex_of_complex_image = thm "hcomplex_of_complex_image"; |
|
1263 val inv_hcomplex_of_complex_image = thm "inv_hcomplex_of_complex_image"; |
|
1264 val SComplex_hcomplex_of_complex_image = thm "SComplex_hcomplex_of_complex_image"; |
|
1265 val SComplex_SReal_dense = thm "SComplex_SReal_dense"; |
|
1266 val SComplex_hcmod_SReal = thm "SComplex_hcmod_SReal"; |
|
1267 val SComplex_zero = thm "SComplex_zero"; |
|
1268 val SComplex_one = thm "SComplex_one"; |
|
1269 val CFinite_add = thm "CFinite_add"; |
|
1270 val CFinite_mult = thm "CFinite_mult"; |
|
1271 val CFinite_minus_iff = thm "CFinite_minus_iff"; |
|
1272 val SComplex_subset_CFinite = thm "SComplex_subset_CFinite"; |
|
1273 val HFinite_hcmod_hcomplex_of_complex = thm "HFinite_hcmod_hcomplex_of_complex"; |
|
1274 val CFinite_hcomplex_of_complex = thm "CFinite_hcomplex_of_complex"; |
|
1275 val CFiniteD = thm "CFiniteD"; |
|
1276 val CFinite_hcmod_iff = thm "CFinite_hcmod_iff"; |
|
1277 val CFinite_number_of = thm "CFinite_number_of"; |
|
1278 val CFinite_bounded = thm "CFinite_bounded"; |
|
1279 val CInfinitesimal_zero = thm "CInfinitesimal_zero"; |
|
1280 val hcomplex_sum_of_halves = thm "hcomplex_sum_of_halves"; |
|
1281 val CInfinitesimal_hcmod_iff = thm "CInfinitesimal_hcmod_iff"; |
|
1282 val one_not_CInfinitesimal = thm "one_not_CInfinitesimal"; |
|
1283 val CInfinitesimal_add = thm "CInfinitesimal_add"; |
|
1284 val CInfinitesimal_minus_iff = thm "CInfinitesimal_minus_iff"; |
|
1285 val CInfinitesimal_diff = thm "CInfinitesimal_diff"; |
|
1286 val CInfinitesimal_mult = thm "CInfinitesimal_mult"; |
|
1287 val CInfinitesimal_CFinite_mult = thm "CInfinitesimal_CFinite_mult"; |
|
1288 val CInfinitesimal_CFinite_mult2 = thm "CInfinitesimal_CFinite_mult2"; |
|
1289 val CInfinite_hcmod_iff = thm "CInfinite_hcmod_iff"; |
|
1290 val CInfinite_inverse_CInfinitesimal = thm "CInfinite_inverse_CInfinitesimal"; |
|
1291 val CInfinite_mult = thm "CInfinite_mult"; |
|
1292 val CInfinite_minus_iff = thm "CInfinite_minus_iff"; |
|
1293 val CFinite_sum_squares = thm "CFinite_sum_squares"; |
|
1294 val not_CInfinitesimal_not_zero = thm "not_CInfinitesimal_not_zero"; |
|
1295 val not_CInfinitesimal_not_zero2 = thm "not_CInfinitesimal_not_zero2"; |
|
1296 val CFinite_diff_CInfinitesimal_hcmod = thm "CFinite_diff_CInfinitesimal_hcmod"; |
|
1297 val hcmod_less_CInfinitesimal = thm "hcmod_less_CInfinitesimal"; |
|
1298 val hcmod_le_CInfinitesimal = thm "hcmod_le_CInfinitesimal"; |
|
1299 val CInfinitesimal_interval = thm "CInfinitesimal_interval"; |
|
1300 val CInfinitesimal_interval2 = thm "CInfinitesimal_interval2"; |
|
1301 val not_CInfinitesimal_mult = thm "not_CInfinitesimal_mult"; |
|
1302 val CInfinitesimal_mult_disj = thm "CInfinitesimal_mult_disj"; |
|
1303 val CFinite_CInfinitesimal_diff_mult = thm "CFinite_CInfinitesimal_diff_mult"; |
|
1304 val CInfinitesimal_subset_CFinite = thm "CInfinitesimal_subset_CFinite"; |
|
1305 val CInfinitesimal_hcomplex_of_complex_mult = thm "CInfinitesimal_hcomplex_of_complex_mult"; |
|
1306 val CInfinitesimal_hcomplex_of_complex_mult2 = thm "CInfinitesimal_hcomplex_of_complex_mult2"; |
|
1307 val mem_cinfmal_iff = thm "mem_cinfmal_iff"; |
|
1308 val capprox_minus_iff = thm "capprox_minus_iff"; |
|
1309 val capprox_minus_iff2 = thm "capprox_minus_iff2"; |
|
1310 val capprox_refl = thm "capprox_refl"; |
|
1311 val capprox_sym = thm "capprox_sym"; |
|
1312 val capprox_trans = thm "capprox_trans"; |
|
1313 val capprox_trans2 = thm "capprox_trans2"; |
|
1314 val capprox_trans3 = thm "capprox_trans3"; |
|
1315 val number_of_capprox_reorient = thm "number_of_capprox_reorient"; |
|
1316 val CInfinitesimal_capprox_minus = thm "CInfinitesimal_capprox_minus"; |
|
1317 val capprox_monad_iff = thm "capprox_monad_iff"; |
|
1318 val Infinitesimal_capprox = thm "Infinitesimal_capprox"; |
|
1319 val capprox_add = thm "capprox_add"; |
|
1320 val capprox_minus = thm "capprox_minus"; |
|
1321 val capprox_minus2 = thm "capprox_minus2"; |
|
1322 val capprox_minus_cancel = thm "capprox_minus_cancel"; |
|
1323 val capprox_add_minus = thm "capprox_add_minus"; |
|
1324 val capprox_mult1 = thm "capprox_mult1"; |
|
1325 val capprox_mult2 = thm "capprox_mult2"; |
|
1326 val capprox_mult_subst = thm "capprox_mult_subst"; |
|
1327 val capprox_mult_subst2 = thm "capprox_mult_subst2"; |
|
1328 val capprox_mult_subst_SComplex = thm "capprox_mult_subst_SComplex"; |
|
1329 val capprox_eq_imp = thm "capprox_eq_imp"; |
|
1330 val CInfinitesimal_minus_capprox = thm "CInfinitesimal_minus_capprox"; |
|
1331 val bex_CInfinitesimal_iff = thm "bex_CInfinitesimal_iff"; |
|
1332 val bex_CInfinitesimal_iff2 = thm "bex_CInfinitesimal_iff2"; |
|
1333 val CInfinitesimal_add_capprox = thm "CInfinitesimal_add_capprox"; |
|
1334 val CInfinitesimal_add_capprox_self = thm "CInfinitesimal_add_capprox_self"; |
|
1335 val CInfinitesimal_add_capprox_self2 = thm "CInfinitesimal_add_capprox_self2"; |
|
1336 val CInfinitesimal_add_minus_capprox_self = thm "CInfinitesimal_add_minus_capprox_self"; |
|
1337 val CInfinitesimal_add_cancel = thm "CInfinitesimal_add_cancel"; |
|
1338 val CInfinitesimal_add_right_cancel = thm "CInfinitesimal_add_right_cancel"; |
|
1339 val capprox_add_left_cancel = thm "capprox_add_left_cancel"; |
|
1340 val capprox_add_right_cancel = thm "capprox_add_right_cancel"; |
|
1341 val capprox_add_mono1 = thm "capprox_add_mono1"; |
|
1342 val capprox_add_mono2 = thm "capprox_add_mono2"; |
|
1343 val capprox_add_left_iff = thm "capprox_add_left_iff"; |
|
1344 val capprox_add_right_iff = thm "capprox_add_right_iff"; |
|
1345 val capprox_CFinite = thm "capprox_CFinite"; |
|
1346 val capprox_hcomplex_of_complex_CFinite = thm "capprox_hcomplex_of_complex_CFinite"; |
|
1347 val capprox_mult_CFinite = thm "capprox_mult_CFinite"; |
|
1348 val capprox_mult_hcomplex_of_complex = thm "capprox_mult_hcomplex_of_complex"; |
|
1349 val capprox_SComplex_mult_cancel_zero = thm "capprox_SComplex_mult_cancel_zero"; |
|
1350 val capprox_mult_SComplex1 = thm "capprox_mult_SComplex1"; |
|
1351 val capprox_mult_SComplex2 = thm "capprox_mult_SComplex2"; |
|
1352 val capprox_mult_SComplex_zero_cancel_iff = thm "capprox_mult_SComplex_zero_cancel_iff"; |
|
1353 val capprox_SComplex_mult_cancel = thm "capprox_SComplex_mult_cancel"; |
|
1354 val capprox_SComplex_mult_cancel_iff1 = thm "capprox_SComplex_mult_cancel_iff1"; |
|
1355 val capprox_hcmod_approx_zero = thm "capprox_hcmod_approx_zero"; |
|
1356 val capprox_approx_zero_iff = thm "capprox_approx_zero_iff"; |
|
1357 val capprox_minus_zero_cancel_iff = thm "capprox_minus_zero_cancel_iff"; |
|
1358 val Infinitesimal_hcmod_add_diff = thm "Infinitesimal_hcmod_add_diff"; |
|
1359 val approx_hcmod_add_hcmod = thm "approx_hcmod_add_hcmod"; |
|
1360 val capprox_hcmod_approx = thm "capprox_hcmod_approx"; |
|
1361 val CInfinitesimal_less_SComplex = thm "CInfinitesimal_less_SComplex"; |
|
1362 val SComplex_Int_CInfinitesimal_zero = thm "SComplex_Int_CInfinitesimal_zero"; |
|
1363 val SComplex_CInfinitesimal_zero = thm "SComplex_CInfinitesimal_zero"; |
|
1364 val SComplex_CFinite_diff_CInfinitesimal = thm "SComplex_CFinite_diff_CInfinitesimal"; |
|
1365 val hcomplex_of_complex_CFinite_diff_CInfinitesimal = thm "hcomplex_of_complex_CFinite_diff_CInfinitesimal"; |
|
1366 val hcomplex_of_complex_CInfinitesimal_iff_0 = thm "hcomplex_of_complex_CInfinitesimal_iff_0"; |
|
1367 val number_of_not_CInfinitesimal = thm "number_of_not_CInfinitesimal"; |
|
1368 val capprox_SComplex_not_zero = thm "capprox_SComplex_not_zero"; |
|
1369 val CFinite_diff_CInfinitesimal_capprox = thm "CFinite_diff_CInfinitesimal_capprox"; |
|
1370 val CInfinitesimal_ratio = thm "CInfinitesimal_ratio"; |
|
1371 val SComplex_capprox_iff = thm "SComplex_capprox_iff"; |
|
1372 val number_of_capprox_iff = thm "number_of_capprox_iff"; |
|
1373 val number_of_CInfinitesimal_iff = thm "number_of_CInfinitesimal_iff"; |
|
1374 val hcomplex_of_complex_approx_iff = thm "hcomplex_of_complex_approx_iff"; |
|
1375 val hcomplex_of_complex_capprox_number_of_iff = thm "hcomplex_of_complex_capprox_number_of_iff"; |
|
1376 val capprox_unique_complex = thm "capprox_unique_complex"; |
|
1377 val hcomplex_capproxD1 = thm "hcomplex_capproxD1"; |
|
1378 val hcomplex_capproxD2 = thm "hcomplex_capproxD2"; |
|
1379 val hcomplex_capproxI = thm "hcomplex_capproxI"; |
|
1380 val capprox_approx_iff = thm "capprox_approx_iff"; |
|
1381 val hcomplex_of_hypreal_capprox_iff = thm "hcomplex_of_hypreal_capprox_iff"; |
|
1382 val CFinite_HFinite_Re = thm "CFinite_HFinite_Re"; |
|
1383 val CFinite_HFinite_Im = thm "CFinite_HFinite_Im"; |
|
1384 val HFinite_Re_Im_CFinite = thm "HFinite_Re_Im_CFinite"; |
|
1385 val CFinite_HFinite_iff = thm "CFinite_HFinite_iff"; |
|
1386 val SComplex_Re_SReal = thm "SComplex_Re_SReal"; |
|
1387 val SComplex_Im_SReal = thm "SComplex_Im_SReal"; |
|
1388 val Reals_Re_Im_SComplex = thm "Reals_Re_Im_SComplex"; |
|
1389 val SComplex_SReal_iff = thm "SComplex_SReal_iff"; |
|
1390 val CInfinitesimal_Infinitesimal_iff = thm "CInfinitesimal_Infinitesimal_iff"; |
|
1391 val eq_Abs_hcomplex_Bex = thm "eq_Abs_hcomplex_Bex"; |
|
1392 val stc_part_Ex = thm "stc_part_Ex"; |
|
1393 val stc_part_Ex1 = thm "stc_part_Ex1"; |
|
1394 val CFinite_Int_CInfinite_empty = thm "CFinite_Int_CInfinite_empty"; |
|
1395 val CFinite_not_CInfinite = thm "CFinite_not_CInfinite"; |
|
1396 val not_CFinite_CInfinite = thm "not_CFinite_CInfinite"; |
|
1397 val CInfinite_CFinite_disj = thm "CInfinite_CFinite_disj"; |
|
1398 val CInfinite_CFinite_iff = thm "CInfinite_CFinite_iff"; |
|
1399 val CFinite_CInfinite_iff = thm "CFinite_CInfinite_iff"; |
|
1400 val CInfinite_diff_CFinite_CInfinitesimal_disj = thm "CInfinite_diff_CFinite_CInfinitesimal_disj"; |
|
1401 val CFinite_inverse = thm "CFinite_inverse"; |
|
1402 val CFinite_inverse2 = thm "CFinite_inverse2"; |
|
1403 val CInfinitesimal_inverse_CFinite = thm "CInfinitesimal_inverse_CFinite"; |
|
1404 val CFinite_not_CInfinitesimal_inverse = thm "CFinite_not_CInfinitesimal_inverse"; |
|
1405 val capprox_inverse = thm "capprox_inverse"; |
|
1406 val hcomplex_of_complex_capprox_inverse = thms "hcomplex_of_complex_capprox_inverse"; |
|
1407 val inverse_add_CInfinitesimal_capprox = thm "inverse_add_CInfinitesimal_capprox"; |
|
1408 val inverse_add_CInfinitesimal_capprox2 = thm "inverse_add_CInfinitesimal_capprox2"; |
|
1409 val inverse_add_CInfinitesimal_approx_CInfinitesimal = thm "inverse_add_CInfinitesimal_approx_CInfinitesimal"; |
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1410 val CInfinitesimal_square_iff = thm "CInfinitesimal_square_iff"; |
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1411 val capprox_CFinite_mult_cancel = thm "capprox_CFinite_mult_cancel"; |
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1412 val capprox_CFinite_mult_cancel_iff1 = thm "capprox_CFinite_mult_cancel_iff1"; |
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1413 val capprox_cmonad_iff = thm "capprox_cmonad_iff"; |
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1414 val CInfinitesimal_cmonad_eq = thm "CInfinitesimal_cmonad_eq"; |
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1415 val mem_cmonad_iff = thm "mem_cmonad_iff"; |
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1416 val CInfinitesimal_cmonad_zero_iff = thm "CInfinitesimal_cmonad_zero_iff"; |
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1417 val cmonad_zero_minus_iff = thm "cmonad_zero_minus_iff"; |
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1418 val cmonad_zero_hcmod_iff = thm "cmonad_zero_hcmod_iff"; |
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1419 val mem_cmonad_self = thm "mem_cmonad_self"; |
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1420 val stc_capprox_self = thm "stc_capprox_self"; |
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1421 val stc_SComplex = thm "stc_SComplex"; |
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1422 val stc_CFinite = thm "stc_CFinite"; |
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1423 val stc_SComplex_eq = thm "stc_SComplex_eq"; |
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1424 val stc_hcomplex_of_complex = thm "stc_hcomplex_of_complex"; |
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1425 val stc_eq_capprox = thm "stc_eq_capprox"; |
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1426 val capprox_stc_eq = thm "capprox_stc_eq"; |
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1427 val stc_eq_capprox_iff = thm "stc_eq_capprox_iff"; |
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1428 val stc_CInfinitesimal_add_SComplex = thm "stc_CInfinitesimal_add_SComplex"; |
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1429 val stc_CInfinitesimal_add_SComplex2 = thm "stc_CInfinitesimal_add_SComplex2"; |
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1430 val CFinite_stc_CInfinitesimal_add = thm "CFinite_stc_CInfinitesimal_add"; |
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1431 val stc_add = thm "stc_add"; |
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1432 val stc_number_of = thm "stc_number_of"; |
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1433 val stc_zero = thm "stc_zero"; |
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1434 val stc_one = thm "stc_one"; |
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1435 val stc_minus = thm "stc_minus"; |
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1436 val stc_diff = thm "stc_diff"; |
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1437 val lemma_stc_mult = thm "lemma_stc_mult"; |
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1438 val stc_mult = thm "stc_mult"; |
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1439 val stc_CInfinitesimal = thm "stc_CInfinitesimal"; |
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1440 val stc_not_CInfinitesimal = thm "stc_not_CInfinitesimal"; |
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1441 val stc_inverse = thm "stc_inverse"; |
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1442 val stc_divide = thm "stc_divide"; |
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1443 val stc_idempotent = thm "stc_idempotent"; |
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1444 val CFinite_HFinite_hcomplex_of_hypreal = thm "CFinite_HFinite_hcomplex_of_hypreal"; |
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1445 val SComplex_SReal_hcomplex_of_hypreal = thm "SComplex_SReal_hcomplex_of_hypreal"; |
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1446 val stc_hcomplex_of_hypreal = thm "stc_hcomplex_of_hypreal"; |
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1447 val CInfinitesimal_hcnj_iff = thm "CInfinitesimal_hcnj_iff"; |
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1448 val CInfinite_HInfinite_iff = thm "CInfinite_HInfinite_iff"; |
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1449 val hcomplex_split_CInfinitesimal_iff = thm "hcomplex_split_CInfinitesimal_iff"; |
|
1450 val hcomplex_split_CFinite_iff = thm "hcomplex_split_CFinite_iff"; |
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1451 val hcomplex_split_SComplex_iff = thm "hcomplex_split_SComplex_iff"; |
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1452 val hcomplex_split_CInfinite_iff = thm "hcomplex_split_CInfinite_iff"; |
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1453 val hcomplex_split_capprox_iff = thm "hcomplex_split_capprox_iff"; |
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1454 val complex_seq_to_hcomplex_CInfinitesimal = thm "complex_seq_to_hcomplex_CInfinitesimal"; |
|
1455 val CInfinitesimal_hcomplex_of_hypreal_epsilon = thm "CInfinitesimal_hcomplex_of_hypreal_epsilon"; |
|
1456 val hcomplex_of_complex_approx_zero_iff = thm "hcomplex_of_complex_approx_zero_iff"; |
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1457 val hcomplex_of_complex_approx_zero_iff2 = thm "hcomplex_of_complex_approx_zero_iff2"; |
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1458 *} |
|
1459 |
43 |
1460 |
44 end |
1461 end |