1 (* Title : HSeries.ML |
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2 Author : Jacques D. Fleuriot |
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3 Copyright : 1998 University of Cambridge |
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4 Description : Finite summation and infinite series |
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5 for hyperreals |
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6 *) |
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7 |
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8 Goalw [sumhr_def] |
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9 "sumhr(M,N,f) = Abs_hypreal(UN X:Rep_hypnat(M). \ |
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10 \ UN Y: Rep_hypnat(N). \ |
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11 \ hyprel ^^{%n::nat. sumr (X n) (Y n) f})"; |
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12 by (Auto_tac); |
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13 qed "sumhr_iff"; |
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14 |
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15 Goalw [sumhr_def] |
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16 "sumhr(Abs_hypnat(hypnatrel^^{%n. M n}), \ |
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17 \ Abs_hypnat(hypnatrel^^{%n. N n}),f) = \ |
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18 \ Abs_hypreal(hyprel ^^ {%n. sumr (M n) (N n) f})"; |
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19 by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); |
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20 by (Auto_tac THEN Ultra_tac 1); |
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21 qed "sumhr"; |
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22 |
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23 (*------------------------------------------------------- |
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24 lcp's suggestion: exploit pattern matching |
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25 facilities and use as definition instead (to do) |
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26 -------------------------------------------------------*) |
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27 Goalw [sumhr_def] |
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28 "sumhr p = (%(M,N,f). Abs_hypreal(UN X:Rep_hypnat(M). \ |
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29 \ UN Y: Rep_hypnat(N). \ |
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30 \ hyprel ^^{%n::nat. sumr (X n) (Y n) f})) p"; |
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31 by (res_inst_tac [("p","p")] PairE 1); |
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32 by (res_inst_tac [("p","y")] PairE 1); |
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33 by (Auto_tac); |
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34 qed "sumhr_iff2"; |
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35 |
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36 (* Theorem corresponding to base case in def of sumr *) |
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37 Goalw [hypnat_zero_def,hypreal_zero_def] |
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38 "sumhr (m,0,f) = 0"; |
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39 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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40 by (auto_tac (claset(),simpset() addsimps [sumhr])); |
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41 qed "sumhr_zero"; |
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42 Addsimps [sumhr_zero]; |
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43 |
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44 (* Theorem corresponding to recursive case in def of sumr *) |
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45 Goalw [hypnat_one_def,hypreal_zero_def] |
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46 "sumhr(m,n+1hn,f) = (if n + 1hn <= m then 0 \ |
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47 \ else sumhr(m,n,f) + (*fNat* f) n)"; |
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48 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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49 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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50 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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51 hypnat_add,hypnat_le,starfunNat,hypreal_add])); |
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52 by (ALLGOALS(Ultra_tac)); |
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53 qed "sumhr_if"; |
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54 |
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55 Goalw [hypnat_one_def,hypreal_zero_def] |
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56 "sumhr (n + 1hn, n, f) = 0"; |
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57 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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58 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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59 hypnat_add])); |
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60 qed "sumhr_Suc_zero"; |
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61 Addsimps [sumhr_Suc_zero]; |
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62 |
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63 Goalw [hypreal_zero_def] |
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64 "sumhr (n,n,f) = 0"; |
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65 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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66 by (auto_tac (claset(), simpset() addsimps [sumhr])); |
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67 qed "sumhr_eq_bounds"; |
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68 Addsimps [sumhr_eq_bounds]; |
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69 |
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70 Goalw [hypnat_one_def] |
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71 "sumhr (m,m + 1hn,f) = (*fNat* f) m"; |
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72 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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73 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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74 hypnat_add,starfunNat])); |
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75 qed "sumhr_Suc"; |
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76 Addsimps [sumhr_Suc]; |
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77 |
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78 Goalw [hypreal_zero_def] |
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79 "sumhr(m+k,k,f) = 0"; |
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80 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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81 by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
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82 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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83 hypnat_add])); |
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84 qed "sumhr_add_lbound_zero"; |
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85 Addsimps [sumhr_add_lbound_zero]; |
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86 |
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87 Goal "sumhr (m,n,f) + sumhr(m,n,g) = sumhr(m,n,%i. f i + g i)"; |
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88 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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89 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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90 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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91 hypreal_add,sumr_add])); |
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92 qed "sumhr_add"; |
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93 |
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94 Goalw [hypreal_of_real_def] |
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95 "hypreal_of_real r * sumhr(m,n,f) = sumhr(m,n,%n. r * f n)"; |
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96 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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97 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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98 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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99 hypreal_mult,sumr_mult])); |
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100 qed "sumhr_mult"; |
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101 |
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102 Goalw [hypnat_zero_def] |
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103 "n < p ==> sumhr (0,n,f) + sumhr (n,p,f) = sumhr (0,p,f)"; |
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104 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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105 by (res_inst_tac [("z","p")] eq_Abs_hypnat 1); |
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106 by (auto_tac (claset() addSEs [FreeUltrafilterNat_subset], |
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107 simpset() addsimps [sumhr,hypreal_add,hypnat_less, |
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108 sumr_split_add])); |
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109 qed "sumhr_split_add"; |
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110 |
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111 Goal |
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112 "n < p ==> sumhr (0, p, f) + \ |
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113 \ - sumhr (0, n, f) = sumhr (n,p,f)"; |
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114 by (dres_inst_tac [("f1","f")] (sumhr_split_add RS sym) 1); |
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115 by (asm_simp_tac (simpset() addsimps hypreal_add_ac) 1); |
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116 qed "sumhr_split_add_minus"; |
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117 |
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118 Goal "abs(sumhr(m,n,f)) <= sumhr(m,n,%i. abs(f i))"; |
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119 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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120 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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121 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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122 hypreal_le,hypreal_hrabs,sumr_rabs])); |
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123 qed "sumhr_hrabs"; |
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124 |
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125 (* other general version also needed *) |
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126 Goalw [hypnat_of_nat_def] |
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127 "!!f g. (ALL r. m <= r & r < n --> f r = g r) \ |
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128 \ --> sumhr(hypnat_of_nat m,hypnat_of_nat n,f) = sumhr(hypnat_of_nat m,hypnat_of_nat n,g)"; |
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129 by (Step_tac 1 THEN dtac sumr_fun_eq 1); |
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130 by (auto_tac (claset(),simpset() addsimps [sumhr])); |
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131 qed "sumhr_fun_hypnat_eq"; |
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132 |
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133 Goalw [hypnat_zero_def,hypreal_of_real_def] |
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134 "sumhr(0,n,%i. r) = hypreal_of_hypnat n*hypreal_of_real r"; |
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135 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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136 by (asm_simp_tac (simpset() addsimps [sumhr, |
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137 hypreal_of_hypnat,hypreal_mult]) 1); |
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138 qed "sumhr_const"; |
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139 |
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140 Goalw [hypnat_zero_def,hypreal_of_real_def] |
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141 "sumhr(0,n,f) + -(hypreal_of_hypnat n*hypreal_of_real r) = \ |
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142 \ sumhr(0,n,%i. f i + -r)"; |
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143 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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144 by (asm_simp_tac (simpset() addsimps [sumhr, |
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145 hypreal_of_hypnat,hypreal_mult,hypreal_add, |
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146 hypreal_minus,sumr_add RS sym]) 1); |
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147 qed "sumhr_add_mult_const"; |
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148 |
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149 Goalw [hypreal_zero_def] |
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150 "n < m ==> sumhr (m,n,f) = 0"; |
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151 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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152 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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153 by (auto_tac (claset() addEs [FreeUltrafilterNat_subset], |
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154 simpset() addsimps [sumhr,hypnat_less])); |
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155 qed "sumhr_less_bounds_zero"; |
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156 Addsimps [sumhr_less_bounds_zero]; |
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157 |
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158 Goal "sumhr(m,n,%i. - f i) = - sumhr(m,n,f)"; |
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159 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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160 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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161 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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162 hypreal_minus,sumr_minus])); |
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163 qed "sumhr_minus"; |
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164 |
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165 Goalw [hypnat_of_nat_def] |
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166 "sumhr(m+hypnat_of_nat k,n+hypnat_of_nat k,f) = sumhr(m,n,%i. f(i + k))"; |
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167 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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168 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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169 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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170 hypnat_add,sumr_shift_bounds])); |
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171 qed "sumhr_shift_bounds"; |
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172 |
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173 (*------------------------------------------------------------------ |
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174 Theorems about NS sums - infinite sums are obtained |
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175 by summing to some infinite hypernatural (such as whn) |
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176 -----------------------------------------------------------------*) |
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177 Goalw [hypnat_omega_def,hypnat_zero_def] |
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178 "sumhr(0,whn,%i. #1) = hypreal_of_hypnat whn"; |
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179 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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180 hypreal_of_hypnat])); |
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181 qed "sumhr_hypreal_of_hypnat_omega"; |
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182 |
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183 Goalw [hypnat_omega_def,hypnat_zero_def, |
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184 hypreal_one_def,omega_def] |
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185 "sumhr(0,whn,%i. #1) = whr + -1hr"; |
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186 by (auto_tac (claset(),simpset() addsimps [sumhr, |
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187 real_of_nat_def,hypreal_minus,hypreal_add])); |
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188 qed "sumhr_hypreal_omega_minus_one"; |
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189 |
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190 Goalw [hypnat_zero_def, hypnat_omega_def, hypreal_zero_def] |
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191 "sumhr(0, whn + whn, %i. (-#1) ^ (i+1)) = 0"; |
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192 by (simp_tac (simpset() addsimps [sumhr,hypnat_add] |
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193 delsimps [realpow_Suc]) 1); |
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194 qed "sumhr_minus_one_realpow_zero"; |
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195 Addsimps [sumhr_minus_one_realpow_zero]; |
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196 |
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197 Goalw [hypnat_of_nat_def,hypreal_of_real_def] |
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198 "(ALL n. m <= Suc n --> f n = r) & m <= na \ |
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199 \ ==> sumhr(hypnat_of_nat m,hypnat_of_nat na,f) = \ |
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200 \ (hypreal_of_nat (na - m) * hypreal_of_real r)"; |
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201 by (auto_tac (claset() addSDs [sumr_interval_const], |
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202 simpset() addsimps [sumhr,hypreal_of_nat_real_of_nat, |
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203 hypreal_of_real_def,hypreal_mult])); |
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204 qed "sumhr_interval_const"; |
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205 |
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206 Goalw [hypnat_zero_def] |
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207 "(*fNat* (%n. sumr 0 n f)) N = sumhr(0,N,f)"; |
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208 by (res_inst_tac [("z","N")] eq_Abs_hypnat 1); |
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209 by (asm_full_simp_tac (simpset() addsimps [starfunNat,sumhr]) 1); |
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210 qed "starfunNat_sumr"; |
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211 |
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212 Goal "sumhr (0, M, f) @= sumhr (0, N, f) \ |
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213 \ ==> abs (sumhr (M, N, f)) @= 0"; |
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214 by (cut_inst_tac [("x","M"),("y","N")] hypnat_linear 1); |
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215 by (auto_tac (claset(),simpset() addsimps [inf_close_refl])); |
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216 by (dtac (inf_close_sym RS (inf_close_minus_iff RS iffD1)) 1); |
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217 by (auto_tac (claset() addDs [inf_close_hrabs], |
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218 simpset() addsimps [sumhr_split_add_minus])); |
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219 qed "sumhr_hrabs_inf_close"; |
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220 Addsimps [sumhr_hrabs_inf_close]; |
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221 |
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222 (*---------------------------------------------------------------- |
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223 infinite sums: Standard and NS theorems |
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224 ----------------------------------------------------------------*) |
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225 Goalw [sums_def,NSsums_def] "(f sums l) = (f NSsums l)"; |
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226 by (simp_tac (simpset() addsimps [LIMSEQ_NSLIMSEQ_iff]) 1); |
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227 qed "sums_NSsums_iff"; |
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228 |
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229 Goalw [summable_def,NSsummable_def] |
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230 "(summable f) = (NSsummable f)"; |
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231 by (simp_tac (simpset() addsimps [sums_NSsums_iff]) 1); |
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232 qed "summable_NSsummable_iff"; |
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233 |
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234 Goalw [suminf_def,NSsuminf_def] |
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235 "(suminf f) = (NSsuminf f)"; |
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236 by (simp_tac (simpset() addsimps [sums_NSsums_iff]) 1); |
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237 qed "suminf_NSsuminf_iff"; |
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238 |
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239 Goalw [NSsums_def,NSsummable_def] |
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240 "f NSsums l ==> NSsummable f"; |
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241 by (Blast_tac 1); |
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242 qed "NSsums_NSsummable"; |
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243 |
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244 Goalw [NSsummable_def,NSsuminf_def] |
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245 "NSsummable f ==> f NSsums (NSsuminf f)"; |
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246 by (blast_tac (claset() addIs [someI2]) 1); |
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247 qed "NSsummable_NSsums"; |
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248 |
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249 Goal "f NSsums s ==> (s = NSsuminf f)"; |
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250 by (asm_full_simp_tac |
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251 (simpset() addsimps [suminf_NSsuminf_iff RS sym,sums_NSsums_iff, |
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252 sums_unique]) 1); |
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253 qed "NSsums_unique"; |
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254 |
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255 Goal "ALL m. n <= Suc m --> f(m) = #0 ==> f NSsums (sumr 0 n f)"; |
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256 by (asm_simp_tac (simpset() addsimps [sums_NSsums_iff RS sym, series_zero]) 1); |
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257 qed "NSseries_zero"; |
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258 |
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259 Goal "NSsummable f = \ |
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260 \ (ALL M: HNatInfinite. ALL N: HNatInfinite. abs (sumhr(M,N,f)) @= 0)"; |
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261 by (auto_tac (claset(), |
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262 simpset() addsimps [summable_NSsummable_iff RS sym, |
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263 summable_convergent_sumr_iff, convergent_NSconvergent_iff, |
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264 NSCauchy_NSconvergent_iff RS sym, NSCauchy_def, |
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265 starfunNat_sumr])); |
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266 by (cut_inst_tac [("x","M"),("y","N")] hypnat_linear 1); |
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267 by (auto_tac (claset(), simpset() addsimps [inf_close_refl])); |
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268 by (rtac ((inf_close_minus_iff RS iffD2) RS inf_close_sym) 1); |
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269 by (rtac (inf_close_minus_iff RS iffD2) 2); |
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270 by (auto_tac (claset() addDs [inf_close_hrabs_zero_cancel], |
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271 simpset() addsimps [sumhr_split_add_minus])); |
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272 qed "NSsummable_NSCauchy"; |
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273 |
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274 (*------------------------------------------------------------------- |
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275 Terms of a convergent series tend to zero |
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276 -------------------------------------------------------------------*) |
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277 Goalw [NSLIMSEQ_def] "NSsummable f ==> f ----NS> #0"; |
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278 by (auto_tac (claset(), simpset() addsimps [NSsummable_NSCauchy])); |
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279 by (dtac bspec 1 THEN Auto_tac); |
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280 by (dres_inst_tac [("x","N + 1hn")] bspec 1); |
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281 by (auto_tac (claset() addIs [HNatInfinite_add_one, |
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282 inf_close_hrabs_zero_cancel], |
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283 simpset() addsimps [rename_numerals hypreal_of_real_zero])); |
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284 qed "NSsummable_NSLIMSEQ_zero"; |
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285 |
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286 (* Easy to prove stsandard case now *) |
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287 Goal "summable f ==> f ----> #0"; |
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288 by (auto_tac (claset(), |
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289 simpset() addsimps [summable_NSsummable_iff, |
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290 LIMSEQ_NSLIMSEQ_iff, NSsummable_NSLIMSEQ_zero])); |
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291 qed "summable_LIMSEQ_zero"; |
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292 |
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293 (*------------------------------------------------------------------- |
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294 NS Comparison test |
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295 -------------------------------------------------------------------*) |
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296 |
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297 Goal "[| EX N. ALL n. N <= n --> abs(f n) <= g n; \ |
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298 \ NSsummable g \ |
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299 \ |] ==> NSsummable f"; |
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300 by (auto_tac (claset() addIs [summable_comparison_test], |
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301 simpset() addsimps [summable_NSsummable_iff RS sym])); |
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302 qed "NSsummable_comparison_test"; |
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303 |
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304 Goal "[| EX N. ALL n. N <= n --> abs(f n) <= g n; \ |
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305 \ NSsummable g \ |
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306 \ |] ==> NSsummable (%k. abs (f k))"; |
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307 by (rtac NSsummable_comparison_test 1); |
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308 by (auto_tac (claset(), simpset() addsimps [abs_idempotent])); |
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309 qed "NSsummable_rabs_comparison_test"; |
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310 |
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311 |
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312 |
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313 |
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314 |
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315 |
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316 |
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