1 (* Title : HyperNat.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 : Explicit construction of hypernaturals using |
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5 ultrafilters |
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6 *) |
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7 |
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8 (*------------------------------------------------------------------------ |
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9 Properties of hypnatrel |
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10 ------------------------------------------------------------------------*) |
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11 |
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12 (** Proving that hyprel is an equivalence relation **) |
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13 (** Natural deduction for hypnatrel - similar to hyprel! **) |
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14 |
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15 Goalw [hypnatrel_def] |
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16 "((X,Y): hypnatrel) = ({n. X n = Y n}: FreeUltrafilterNat)"; |
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17 by (Fast_tac 1); |
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18 qed "hypnatrel_iff"; |
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19 |
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20 Goalw [hypnatrel_def] |
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21 "{n. X n = Y n}: FreeUltrafilterNat ==> (X,Y): hypnatrel"; |
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22 by (Fast_tac 1); |
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23 qed "hypnatrelI"; |
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24 |
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25 Goalw [hypnatrel_def] |
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26 "p: hypnatrel --> (EX X Y. \ |
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27 \ p = (X,Y) & {n. X n = Y n} : FreeUltrafilterNat)"; |
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28 by (Fast_tac 1); |
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29 qed "hypnatrelE_lemma"; |
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30 |
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31 val [major,minor] = goal thy |
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32 "[| p: hypnatrel; \ |
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33 \ !!X Y. [| p = (X,Y); {n. X n = Y n}: FreeUltrafilterNat\ |
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34 \ |] ==> Q |] ==> Q"; |
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35 by (cut_facts_tac [major RS (hypnatrelE_lemma RS mp)] 1); |
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36 by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1)); |
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37 qed "hypnatrelE"; |
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38 |
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39 AddSIs [hypnatrelI]; |
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40 AddSEs [hypnatrelE]; |
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41 |
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42 Goalw [hypnatrel_def] "(x,x): hypnatrel"; |
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43 by (Auto_tac); |
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44 qed "hypnatrel_refl"; |
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45 |
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46 Goalw [hypnatrel_def] "(x,y): hypnatrel --> (y,x):hypnatrel"; |
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47 by (auto_tac (claset() addIs [lemma_perm RS subst],simpset())); |
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48 qed_spec_mp "hypnatrel_sym"; |
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49 |
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50 Goalw [hypnatrel_def] |
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51 "(x,y): hypnatrel --> (y,z):hypnatrel --> (x,z):hypnatrel"; |
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52 by (Auto_tac); |
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53 by (Fuf_tac 1); |
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54 qed_spec_mp "hypnatrel_trans"; |
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55 |
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56 Goalw [equiv_def, refl_def, sym_def, trans_def] |
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57 "equiv {x::nat=>nat. True} hypnatrel"; |
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58 by (auto_tac (claset() addSIs [hypnatrel_refl] addSEs |
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59 [hypnatrel_sym,hypnatrel_trans] delrules [hypnatrelI,hypnatrelE], |
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60 simpset())); |
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61 qed "equiv_hypnatrel"; |
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62 |
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63 val equiv_hypnatrel_iff = |
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64 [TrueI, TrueI] MRS |
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65 ([CollectI, CollectI] MRS |
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66 (equiv_hypnatrel RS eq_equiv_class_iff)); |
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67 |
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68 Goalw [hypnat_def,hypnatrel_def,quotient_def] "hypnatrel^^{x}:hypnat"; |
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69 by (Blast_tac 1); |
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70 qed "hypnatrel_in_hypnat"; |
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71 |
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72 Goal "inj_on Abs_hypnat hypnat"; |
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73 by (rtac inj_on_inverseI 1); |
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74 by (etac Abs_hypnat_inverse 1); |
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75 qed "inj_on_Abs_hypnat"; |
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76 |
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77 Addsimps [equiv_hypnatrel_iff,inj_on_Abs_hypnat RS inj_on_iff, |
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78 hypnatrel_iff, hypnatrel_in_hypnat, Abs_hypnat_inverse]; |
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79 |
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80 Addsimps [equiv_hypnatrel RS eq_equiv_class_iff]; |
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81 val eq_hypnatrelD = equiv_hypnatrel RSN (2,eq_equiv_class); |
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82 |
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83 Goal "inj(Rep_hypnat)"; |
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84 by (rtac inj_inverseI 1); |
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85 by (rtac Rep_hypnat_inverse 1); |
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86 qed "inj_Rep_hypnat"; |
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87 |
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88 Goalw [hypnatrel_def] "x: hypnatrel ^^ {x}"; |
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89 by (Step_tac 1); |
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90 by (Auto_tac); |
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91 qed "lemma_hypnatrel_refl"; |
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92 |
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93 Addsimps [lemma_hypnatrel_refl]; |
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94 |
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95 Goalw [hypnat_def] "{} ~: hypnat"; |
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96 by (auto_tac (claset() addSEs [quotientE],simpset())); |
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97 qed "hypnat_empty_not_mem"; |
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98 |
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99 Addsimps [hypnat_empty_not_mem]; |
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100 |
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101 Goal "Rep_hypnat x ~= {}"; |
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102 by (cut_inst_tac [("x","x")] Rep_hypnat 1); |
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103 by (Auto_tac); |
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104 qed "Rep_hypnat_nonempty"; |
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105 |
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106 Addsimps [Rep_hypnat_nonempty]; |
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107 |
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108 (*------------------------------------------------------------------------ |
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109 hypnat_of_nat: the injection from nat to hypnat |
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110 ------------------------------------------------------------------------*) |
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111 Goal "inj(hypnat_of_nat)"; |
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112 by (rtac injI 1); |
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113 by (rewtac hypnat_of_nat_def); |
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114 by (dtac (inj_on_Abs_hypnat RS inj_onD) 1); |
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115 by (REPEAT (rtac hypnatrel_in_hypnat 1)); |
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116 by (dtac eq_equiv_class 1); |
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117 by (rtac equiv_hypnatrel 1); |
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118 by (Fast_tac 1); |
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119 by (rtac ccontr 1 THEN rotate_tac 1 1); |
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120 by (Auto_tac); |
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121 qed "inj_hypnat_of_nat"; |
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122 |
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123 val [prem] = goal thy |
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124 "(!!x. z = Abs_hypnat(hypnatrel^^{x}) ==> P) ==> P"; |
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125 by (res_inst_tac [("x1","z")] |
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126 (rewrite_rule [hypnat_def] Rep_hypnat RS quotientE) 1); |
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127 by (dres_inst_tac [("f","Abs_hypnat")] arg_cong 1); |
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128 by (res_inst_tac [("x","x")] prem 1); |
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129 by (asm_full_simp_tac (simpset() addsimps [Rep_hypnat_inverse]) 1); |
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130 qed "eq_Abs_hypnat"; |
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131 |
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132 (*----------------------------------------------------------- |
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133 Addition for hyper naturals: hypnat_add |
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134 -----------------------------------------------------------*) |
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135 Goalw [congruent2_def] |
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136 "congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n + Y n})"; |
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137 by Safe_tac; |
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138 by (ALLGOALS(Fuf_tac)); |
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139 qed "hypnat_add_congruent2"; |
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140 |
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141 Goalw [hypnat_add_def] |
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142 "Abs_hypnat(hypnatrel^^{%n. X n}) + Abs_hypnat(hypnatrel^^{%n. Y n}) = \ |
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143 \ Abs_hypnat(hypnatrel^^{%n. X n + Y n})"; |
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144 by (asm_simp_tac |
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145 (simpset() addsimps [[equiv_hypnatrel, hypnat_add_congruent2] |
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146 MRS UN_equiv_class2]) 1); |
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147 qed "hypnat_add"; |
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148 |
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149 Goal "(z::hypnat) + w = w + z"; |
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150 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
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151 by (res_inst_tac [("z","w")] eq_Abs_hypnat 1); |
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152 by (asm_simp_tac (simpset() addsimps (add_ac @ [hypnat_add])) 1); |
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153 qed "hypnat_add_commute"; |
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154 |
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155 Goal "((z1::hypnat) + z2) + z3 = z1 + (z2 + z3)"; |
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156 by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1); |
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157 by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1); |
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158 by (res_inst_tac [("z","z3")] eq_Abs_hypnat 1); |
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159 by (asm_simp_tac (simpset() addsimps [hypnat_add,add_assoc]) 1); |
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160 qed "hypnat_add_assoc"; |
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161 |
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162 (*For AC rewriting*) |
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163 Goal "(x::hypnat)+(y+z)=y+(x+z)"; |
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164 by (rtac (hypnat_add_commute RS trans) 1); |
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165 by (rtac (hypnat_add_assoc RS trans) 1); |
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166 by (rtac (hypnat_add_commute RS arg_cong) 1); |
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167 qed "hypnat_add_left_commute"; |
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168 |
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169 (* hypnat addition is an AC operator *) |
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170 val hypnat_add_ac = [hypnat_add_assoc,hypnat_add_commute, |
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171 hypnat_add_left_commute]; |
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172 |
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173 Goalw [hypnat_zero_def] "(0::hypnat) + z = z"; |
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174 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
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175 by (asm_full_simp_tac (simpset() addsimps |
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176 [hypnat_add]) 1); |
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177 qed "hypnat_add_zero_left"; |
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178 |
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179 Goal "z + (0::hypnat) = z"; |
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180 by (simp_tac (simpset() addsimps |
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181 [hypnat_add_zero_left,hypnat_add_commute]) 1); |
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182 qed "hypnat_add_zero_right"; |
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183 |
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184 Addsimps [hypnat_add_zero_left,hypnat_add_zero_right]; |
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185 |
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186 (*----------------------------------------------------------- |
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187 Subtraction for hyper naturals: hypnat_minus |
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188 -----------------------------------------------------------*) |
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189 Goalw [congruent2_def] |
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190 "congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n - Y n})"; |
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191 by Safe_tac; |
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192 by (ALLGOALS(Fuf_tac)); |
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193 qed "hypnat_minus_congruent2"; |
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194 |
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195 Goalw [hypnat_minus_def] |
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196 "Abs_hypnat(hypnatrel^^{%n. X n}) - Abs_hypnat(hypnatrel^^{%n. Y n}) = \ |
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197 \ Abs_hypnat(hypnatrel^^{%n. X n - Y n})"; |
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198 by (asm_simp_tac |
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199 (simpset() addsimps [[equiv_hypnatrel, hypnat_minus_congruent2] |
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200 MRS UN_equiv_class2]) 1); |
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201 qed "hypnat_minus"; |
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202 |
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203 Goalw [hypnat_zero_def] "z - z = (0::hypnat)"; |
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204 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
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205 by (asm_full_simp_tac (simpset() addsimps [hypnat_minus]) 1); |
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206 qed "hypnat_minus_zero"; |
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207 |
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208 Goalw [hypnat_zero_def] "(0::hypnat) - n = 0"; |
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209 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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210 by (asm_full_simp_tac (simpset() addsimps [hypnat_minus]) 1); |
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211 qed "hypnat_diff_0_eq_0"; |
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212 |
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213 Addsimps [hypnat_minus_zero,hypnat_diff_0_eq_0]; |
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214 |
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215 Goalw [hypnat_zero_def] "(m+n = (0::hypnat)) = (m=0 & n=0)"; |
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216 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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217 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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218 by (auto_tac (claset() addIs [FreeUltrafilterNat_subset] |
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219 addDs [FreeUltrafilterNat_Int], |
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220 simpset() addsimps [hypnat_add] )); |
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221 qed "hypnat_add_is_0"; |
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222 |
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223 AddIffs [hypnat_add_is_0]; |
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224 |
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225 Goal "!!i::hypnat. i-j-k = i - (j+k)"; |
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226 by (res_inst_tac [("z","i")] eq_Abs_hypnat 1); |
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227 by (res_inst_tac [("z","j")] eq_Abs_hypnat 1); |
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228 by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
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229 by (asm_full_simp_tac (simpset() addsimps |
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230 [hypnat_minus,hypnat_add,diff_diff_left]) 1); |
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231 qed "hypnat_diff_diff_left"; |
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232 |
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233 Goal "!!i::hypnat. i-j-k = i-k-j"; |
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234 by (simp_tac (simpset() addsimps |
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235 [hypnat_diff_diff_left, hypnat_add_commute]) 1); |
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236 qed "hypnat_diff_commute"; |
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237 |
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238 Goal "!!n::hypnat. (n+m) - n = m"; |
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239 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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240 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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241 by (asm_full_simp_tac (simpset() addsimps |
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242 [hypnat_minus,hypnat_add]) 1); |
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243 qed "hypnat_diff_add_inverse"; |
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244 Addsimps [hypnat_diff_add_inverse]; |
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245 |
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246 Goal "!!n::hypnat.(m+n) - n = m"; |
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247 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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248 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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249 by (asm_full_simp_tac (simpset() addsimps |
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250 [hypnat_minus,hypnat_add]) 1); |
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251 qed "hypnat_diff_add_inverse2"; |
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252 Addsimps [hypnat_diff_add_inverse2]; |
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253 |
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254 Goal "!!k::hypnat. (k+m) - (k+n) = m - n"; |
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255 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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256 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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257 by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
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258 by (asm_full_simp_tac (simpset() addsimps |
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259 [hypnat_minus,hypnat_add]) 1); |
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260 qed "hypnat_diff_cancel"; |
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261 Addsimps [hypnat_diff_cancel]; |
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262 |
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263 Goal "!!m::hypnat. (m+k) - (n+k) = m - n"; |
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264 val hypnat_add_commute_k = read_instantiate [("w","k")] hypnat_add_commute; |
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265 by (asm_simp_tac (simpset() addsimps ([hypnat_add_commute_k])) 1); |
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266 qed "hypnat_diff_cancel2"; |
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267 Addsimps [hypnat_diff_cancel2]; |
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268 |
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269 Goalw [hypnat_zero_def] "!!n::hypnat. n - (n+m) = (0::hypnat)"; |
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270 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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271 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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272 by (asm_full_simp_tac (simpset() addsimps |
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273 [hypnat_minus,hypnat_add]) 1); |
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274 qed "hypnat_diff_add_0"; |
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275 Addsimps [hypnat_diff_add_0]; |
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276 |
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277 (*----------------------------------------------------------- |
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278 Multiplication for hyper naturals: hypnat_mult |
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279 -----------------------------------------------------------*) |
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280 Goalw [congruent2_def] |
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281 "congruent2 hypnatrel (%X Y. hypnatrel^^{%n. X n * Y n})"; |
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282 by Safe_tac; |
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283 by (ALLGOALS(Fuf_tac)); |
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284 qed "hypnat_mult_congruent2"; |
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285 |
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286 Goalw [hypnat_mult_def] |
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287 "Abs_hypnat(hypnatrel^^{%n. X n}) * Abs_hypnat(hypnatrel^^{%n. Y n}) = \ |
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288 \ Abs_hypnat(hypnatrel^^{%n. X n * Y n})"; |
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289 by (asm_simp_tac |
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290 (simpset() addsimps [[equiv_hypnatrel,hypnat_mult_congruent2] MRS |
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291 UN_equiv_class2]) 1); |
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292 qed "hypnat_mult"; |
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293 |
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294 Goal "(z::hypnat) * w = w * z"; |
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295 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
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296 by (res_inst_tac [("z","w")] eq_Abs_hypnat 1); |
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297 by (asm_simp_tac (simpset() addsimps ([hypnat_mult] @ mult_ac)) 1); |
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298 qed "hypnat_mult_commute"; |
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299 |
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300 Goal "((z1::hypnat) * z2) * z3 = z1 * (z2 * z3)"; |
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301 by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1); |
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302 by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1); |
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303 by (res_inst_tac [("z","z3")] eq_Abs_hypnat 1); |
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304 by (asm_simp_tac (simpset() addsimps [hypnat_mult,mult_assoc]) 1); |
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305 qed "hypnat_mult_assoc"; |
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306 |
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307 qed_goal "hypnat_mult_left_commute" thy |
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308 "(z1::hypnat) * (z2 * z3) = z2 * (z1 * z3)" |
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309 (fn _ => [rtac (hypnat_mult_commute RS trans) 1, rtac (hypnat_mult_assoc RS trans) 1, |
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310 rtac (hypnat_mult_commute RS arg_cong) 1]); |
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311 |
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312 (* hypnat multiplication is an AC operator *) |
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313 val hypnat_mult_ac = [hypnat_mult_assoc, hypnat_mult_commute, |
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314 hypnat_mult_left_commute]; |
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315 |
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316 Goalw [hypnat_one_def] "1hn * z = z"; |
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317 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
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318 by (asm_full_simp_tac (simpset() addsimps [hypnat_mult]) 1); |
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319 qed "hypnat_mult_1"; |
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320 Addsimps [hypnat_mult_1]; |
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321 |
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322 Goal "z * 1hn = z"; |
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323 by (simp_tac (simpset() addsimps [hypnat_mult_commute]) 1); |
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324 qed "hypnat_mult_1_right"; |
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325 Addsimps [hypnat_mult_1_right]; |
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326 |
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327 Goalw [hypnat_zero_def] "(0::hypnat) * z = 0"; |
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328 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
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329 by (asm_full_simp_tac (simpset() addsimps [hypnat_mult]) 1); |
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330 qed "hypnat_mult_0"; |
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331 Addsimps [hypnat_mult_0]; |
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332 |
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333 Goal "z * (0::hypnat) = 0"; |
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334 by (simp_tac (simpset() addsimps [hypnat_mult_commute]) 1); |
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335 qed "hypnat_mult_0_right"; |
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336 Addsimps [hypnat_mult_0_right]; |
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337 |
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338 Goal "!!m::hypnat. (m - n) * k = (m * k) - (n * k)"; |
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339 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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340 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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341 by (res_inst_tac [("z","k")] eq_Abs_hypnat 1); |
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342 by (asm_simp_tac (simpset() addsimps [hypnat_mult, |
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343 hypnat_minus,diff_mult_distrib]) 1); |
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344 qed "hypnat_diff_mult_distrib" ; |
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345 |
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346 Goal "!!m::hypnat. k * (m - n) = (k * m) - (k * n)"; |
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347 val hypnat_mult_commute_k = read_instantiate [("z","k")] hypnat_mult_commute; |
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348 by (simp_tac (simpset() addsimps [hypnat_diff_mult_distrib, |
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349 hypnat_mult_commute_k]) 1); |
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350 qed "hypnat_diff_mult_distrib2" ; |
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351 |
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352 (*-------------------------- |
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353 A Few more theorems |
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354 -------------------------*) |
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355 qed_goal "hypnat_add_assoc_cong" thy |
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356 "!!z. (z::hypnat) + v = z' + v' ==> z + (v + w) = z' + (v' + w)" |
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357 (fn _ => [(asm_simp_tac (simpset() addsimps [hypnat_add_assoc RS sym]) 1)]); |
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358 |
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359 qed_goal "hypnat_add_assoc_swap" thy |
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360 "(z::hypnat) + (v + w) = v + (z + w)" |
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361 (fn _ => [(REPEAT (ares_tac [hypnat_add_commute RS hypnat_add_assoc_cong] 1))]); |
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362 |
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363 Goal "((z1::hypnat) + z2) * w = (z1 * w) + (z2 * w)"; |
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364 by (res_inst_tac [("z","z1")] eq_Abs_hypnat 1); |
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365 by (res_inst_tac [("z","z2")] eq_Abs_hypnat 1); |
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366 by (res_inst_tac [("z","w")] eq_Abs_hypnat 1); |
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367 by (asm_simp_tac (simpset() addsimps [hypnat_mult,hypnat_add, |
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368 add_mult_distrib]) 1); |
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369 qed "hypnat_add_mult_distrib"; |
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370 Addsimps [hypnat_add_mult_distrib]; |
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371 |
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372 val hypnat_mult_commute'= read_instantiate [("z","w")] hypnat_mult_commute; |
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373 |
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374 Goal "(w::hypnat) * (z1 + z2) = (w * z1) + (w * z2)"; |
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375 by (simp_tac (simpset() addsimps [hypnat_mult_commute']) 1); |
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376 qed "hypnat_add_mult_distrib2"; |
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377 Addsimps [hypnat_add_mult_distrib2]; |
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378 |
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379 (*** (hypnat) one and zero are distinct ***) |
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380 Goalw [hypnat_zero_def,hypnat_one_def] "(0::hypnat) ~= 1hn"; |
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381 by (Auto_tac); |
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382 qed "hypnat_zero_not_eq_one"; |
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383 Addsimps [hypnat_zero_not_eq_one]; |
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384 Addsimps [hypnat_zero_not_eq_one RS not_sym]; |
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385 |
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386 Goalw [hypnat_zero_def] "(m*n = (0::hypnat)) = (m=0 | n=0)"; |
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387 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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388 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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389 by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem], |
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390 simpset() addsimps [hypnat_mult])); |
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391 by (ALLGOALS(Fuf_tac)); |
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392 qed "hypnat_mult_is_0"; |
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393 Addsimps [hypnat_mult_is_0]; |
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394 |
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395 (*------------------------------------------------------------------ |
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396 Theorems for ordering |
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397 ------------------------------------------------------------------*) |
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398 |
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399 (* prove introduction and elimination rules for hypnat_less *) |
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400 |
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401 Goalw [hypnat_less_def] |
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402 "P < (Q::hypnat) = (EX X Y. X : Rep_hypnat(P) & \ |
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403 \ Y : Rep_hypnat(Q) & \ |
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404 \ {n. X n < Y n} : FreeUltrafilterNat)"; |
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405 by (Fast_tac 1); |
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406 qed "hypnat_less_iff"; |
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407 |
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408 Goalw [hypnat_less_def] |
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409 "!!P. [| {n. X n < Y n} : FreeUltrafilterNat; \ |
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410 \ X : Rep_hypnat(P); \ |
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411 \ Y : Rep_hypnat(Q) |] ==> P < (Q::hypnat)"; |
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412 by (Fast_tac 1); |
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413 qed "hypnat_lessI"; |
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414 |
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415 Goalw [hypnat_less_def] |
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416 "!! R1. [| R1 < (R2::hypnat); \ |
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417 \ !!X Y. {n. X n < Y n} : FreeUltrafilterNat ==> P; \ |
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418 \ !!X. X : Rep_hypnat(R1) ==> P; \ |
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419 \ !!Y. Y : Rep_hypnat(R2) ==> P |] \ |
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420 \ ==> P"; |
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421 by (Auto_tac); |
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422 qed "hypnat_lessE"; |
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423 |
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424 Goalw [hypnat_less_def] |
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425 "!!R1. R1 < (R2::hypnat) ==> (EX X Y. {n. X n < Y n} : FreeUltrafilterNat & \ |
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426 \ X : Rep_hypnat(R1) & \ |
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427 \ Y : Rep_hypnat(R2))"; |
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428 by (Fast_tac 1); |
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429 qed "hypnat_lessD"; |
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430 |
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431 Goal "~ (R::hypnat) < R"; |
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432 by (res_inst_tac [("z","R")] eq_Abs_hypnat 1); |
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433 by (auto_tac (claset(),simpset() addsimps [hypnat_less_def])); |
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434 by (Fuf_empty_tac 1); |
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435 qed "hypnat_less_not_refl"; |
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436 Addsimps [hypnat_less_not_refl]; |
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437 |
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438 bind_thm("hypnat_less_irrefl",hypnat_less_not_refl RS notE); |
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439 |
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440 qed_goal "hypnat_not_refl2" thy |
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441 "!!(x::hypnat). x < y ==> x ~= y" (fn _ => [Auto_tac]); |
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442 |
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443 Goalw [hypnat_less_def,hypnat_zero_def] "~ n<(0::hypnat)"; |
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444 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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445 by (Auto_tac); |
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446 by (Fuf_empty_tac 1); |
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447 qed "hypnat_not_less0"; |
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448 AddIffs [hypnat_not_less0]; |
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449 |
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450 (* n<(0::hypnat) ==> R *) |
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451 bind_thm ("hypnat_less_zeroE", hypnat_not_less0 RS notE); |
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452 |
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453 Goalw [hypnat_less_def,hypnat_zero_def,hypnat_one_def] |
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454 "(n<1hn) = (n=0)"; |
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455 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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456 by (auto_tac (claset() addSIs [exI] addEs |
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457 [FreeUltrafilterNat_subset],simpset())); |
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458 by (Fuf_tac 1); |
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459 qed "hypnat_less_one"; |
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460 AddIffs [hypnat_less_one]; |
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461 |
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462 Goal "!!(R1::hypnat). [| R1 < R2; R2 < R3 |] ==> R1 < R3"; |
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463 by (res_inst_tac [("z","R1")] eq_Abs_hypnat 1); |
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464 by (res_inst_tac [("z","R2")] eq_Abs_hypnat 1); |
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465 by (res_inst_tac [("z","R3")] eq_Abs_hypnat 1); |
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466 by (auto_tac (claset(),simpset() addsimps [hypnat_less_def])); |
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467 by (res_inst_tac [("x","X")] exI 1); |
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468 by (Auto_tac); |
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469 by (res_inst_tac [("x","Ya")] exI 1); |
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470 by (Auto_tac); |
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471 by (Fuf_tac 1); |
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472 qed "hypnat_less_trans"; |
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473 |
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474 Goal "!! (R1::hypnat). [| R1 < R2; R2 < R1 |] ==> P"; |
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475 by (dtac hypnat_less_trans 1 THEN assume_tac 1); |
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476 by (Asm_full_simp_tac 1); |
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477 qed "hypnat_less_asym"; |
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478 |
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479 (*----- hypnat less iff less a.e -----*) |
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480 (* See comments in HYPER for corresponding thm *) |
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481 |
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482 Goalw [hypnat_less_def] |
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483 "(Abs_hypnat(hypnatrel^^{%n. X n}) < \ |
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484 \ Abs_hypnat(hypnatrel^^{%n. Y n})) = \ |
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485 \ ({n. X n < Y n} : FreeUltrafilterNat)"; |
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486 by (auto_tac (claset() addSIs [lemma_hypnatrel_refl],simpset())); |
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487 by (Fuf_tac 1); |
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488 qed "hypnat_less"; |
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489 |
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490 Goal "~ m<n --> n+(m-n) = (m::hypnat)"; |
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491 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
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492 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
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493 by (auto_tac (claset(),simpset() addsimps |
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494 [hypnat_minus,hypnat_add,hypnat_less])); |
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495 by (dtac FreeUltrafilterNat_Compl_mem 1); |
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496 by (Fuf_tac 1); |
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497 qed_spec_mp "hypnat_add_diff_inverse"; |
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498 |
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499 Goal "n<=m ==> n+(m-n) = (m::hypnat)"; |
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500 by (asm_full_simp_tac (simpset() addsimps |
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501 [hypnat_add_diff_inverse, hypnat_le_def]) 1); |
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502 qed "hypnat_le_add_diff_inverse"; |
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503 |
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504 Goal "n<=m ==> (m-n)+n = (m::hypnat)"; |
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505 by (asm_simp_tac (simpset() addsimps [hypnat_le_add_diff_inverse, |
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506 hypnat_add_commute]) 1); |
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507 qed "hypnat_le_add_diff_inverse2"; |
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508 |
|
509 (*--------------------------------------------------------------------------------- |
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510 Hyper naturals as a linearly ordered field |
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511 ---------------------------------------------------------------------------------*) |
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512 Goalw [hypnat_zero_def] |
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513 "[| (0::hypnat) < x; 0 < y |] ==> 0 < x + y"; |
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514 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
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515 by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
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516 by (auto_tac (claset(),simpset() addsimps |
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517 [hypnat_less_def,hypnat_add])); |
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518 by (REPEAT(Step_tac 1)); |
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519 by (Fuf_tac 1); |
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520 qed "hypnat_add_order"; |
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521 |
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522 Goalw [hypnat_zero_def] |
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523 "!!(x::hypnat). [| (0::hypnat) < x; 0 < y |] ==> 0 < x * y"; |
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524 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
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525 by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
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526 by (auto_tac (claset(),simpset() addsimps |
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527 [hypnat_less_def,hypnat_mult])); |
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528 by (REPEAT(Step_tac 1)); |
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529 by (Fuf_tac 1); |
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530 qed "hypnat_mult_order"; |
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531 |
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532 (*--------------------------------------------------------------------------------- |
|
533 Trichotomy of the hyper naturals |
|
534 --------------------------------------------------------------------------------*) |
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535 Goalw [hypnatrel_def] "EX x. x: hypnatrel ^^ {%n. 0}"; |
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536 by (res_inst_tac [("x","%n. 0")] exI 1); |
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537 by (Step_tac 1); |
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538 by (Auto_tac); |
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539 qed "lemma_hypnatrel_0_mem"; |
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540 |
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541 (* linearity is actually proved further down! *) |
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542 Goalw [hypnat_zero_def, |
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543 hypnat_less_def]"(0::hypnat) < x | x = 0 | x < 0"; |
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544 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
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545 by (Auto_tac ); |
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546 by (REPEAT(Step_tac 1)); |
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547 by (REPEAT(dtac FreeUltrafilterNat_Compl_mem 1)); |
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548 by (Fuf_tac 1); |
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549 qed "hypnat_trichotomy"; |
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550 |
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551 Goal "!!x. [| (0::hypnat) < x ==> P; \ |
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552 \ x = 0 ==> P; \ |
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553 \ x < 0 ==> P |] ==> P"; |
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554 by (cut_inst_tac [("x","x")] hypnat_trichotomy 1); |
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555 by (Auto_tac); |
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556 qed "hypnat_trichotomyE"; |
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557 |
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558 (*------------------------------------------------------------------------------ |
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559 More properties of < |
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560 ------------------------------------------------------------------------------*) |
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561 Goal "!!(A::hypnat). A < B ==> A + C < B + C"; |
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562 by (res_inst_tac [("z","A")] eq_Abs_hypnat 1); |
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563 by (res_inst_tac [("z","B")] eq_Abs_hypnat 1); |
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564 by (res_inst_tac [("z","C")] eq_Abs_hypnat 1); |
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565 by (auto_tac (claset(),simpset() addsimps |
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566 [hypnat_less_def,hypnat_add])); |
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567 by (REPEAT(Step_tac 1)); |
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568 by (Fuf_tac 1); |
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569 qed "hypnat_add_less_mono1"; |
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570 |
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571 Goal "!!(A::hypnat). A < B ==> C + A < C + B"; |
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572 by (auto_tac (claset() addIs [hypnat_add_less_mono1], |
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573 simpset() addsimps [hypnat_add_commute])); |
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574 qed "hypnat_add_less_mono2"; |
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575 |
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576 Goal "!!k l::hypnat. [|i<j; k<l |] ==> i + k < j + l"; |
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577 by (etac (hypnat_add_less_mono1 RS hypnat_less_trans) 1); |
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578 by (simp_tac (simpset() addsimps [hypnat_add_commute]) 1); |
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579 (*j moves to the end because it is free while k, l are bound*) |
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580 by (etac hypnat_add_less_mono1 1); |
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581 qed "hypnat_add_less_mono"; |
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582 |
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583 (*--------------------------------------- |
|
584 hypnat_minus_less |
|
585 ---------------------------------------*) |
|
586 Goalw [hypnat_less_def,hypnat_zero_def] |
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587 "((x::hypnat) < y) = ((0::hypnat) < y - x)"; |
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588 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
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589 by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
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590 by (auto_tac (claset(),simpset() addsimps |
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591 [hypnat_minus])); |
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592 by (REPEAT(Step_tac 1)); |
|
593 by (REPEAT(Step_tac 2)); |
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594 by (ALLGOALS(fuf_tac (claset() addDs [sym],simpset()))); |
|
595 |
|
596 (*** linearity ***) |
|
597 Goalw [hypnat_less_def] "(x::hypnat) < y | x = y | y < x"; |
|
598 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
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599 by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
|
600 by (Auto_tac ); |
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601 by (REPEAT(Step_tac 1)); |
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602 by (REPEAT(dtac FreeUltrafilterNat_Compl_mem 1)); |
|
603 by (Fuf_tac 1); |
|
604 qed "hypnat_linear"; |
|
605 |
|
606 Goal |
|
607 "!!(x::hypnat). [| x < y ==> P; x = y ==> P; \ |
|
608 \ y < x ==> P |] ==> P"; |
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609 by (cut_inst_tac [("x","x"),("y","y")] hypnat_linear 1); |
|
610 by (Auto_tac); |
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611 qed "hypnat_linear_less2"; |
|
612 |
|
613 (*------------------------------------------------------------------------------ |
|
614 Properties of <= |
|
615 ------------------------------------------------------------------------------*) |
|
616 (*------ hypnat le iff nat le a.e ------*) |
|
617 Goalw [hypnat_le_def,le_def] |
|
618 "(Abs_hypnat(hypnatrel^^{%n. X n}) <= \ |
|
619 \ Abs_hypnat(hypnatrel^^{%n. Y n})) = \ |
|
620 \ ({n. X n <= Y n} : FreeUltrafilterNat)"; |
|
621 by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem], |
|
622 simpset() addsimps [hypnat_less])); |
|
623 by (Fuf_tac 1 THEN Fuf_empty_tac 1); |
|
624 qed "hypnat_le"; |
|
625 |
|
626 (*---------------------------------------------------------*) |
|
627 (*---------------------------------------------------------*) |
|
628 Goalw [hypnat_le_def] "!!w. ~(w < z) ==> z <= (w::hypnat)"; |
|
629 by (assume_tac 1); |
|
630 qed "hypnat_leI"; |
|
631 |
|
632 Goalw [hypnat_le_def] "!!w. z<=w ==> ~(w<(z::hypnat))"; |
|
633 by (assume_tac 1); |
|
634 qed "hypnat_leD"; |
|
635 |
|
636 val hypnat_leE = make_elim hypnat_leD; |
|
637 |
|
638 Goal "!!w. (~(w < z)) = (z <= (w::hypnat))"; |
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639 by (fast_tac (claset() addSIs [hypnat_leI,hypnat_leD]) 1); |
|
640 qed "hypnat_less_le_iff"; |
|
641 |
|
642 Goalw [hypnat_le_def] "!!z. ~ z <= w ==> w<(z::hypnat)"; |
|
643 by (Fast_tac 1); |
|
644 qed "not_hypnat_leE"; |
|
645 |
|
646 Goalw [hypnat_le_def] "!!z. z < w ==> z <= (w::hypnat)"; |
|
647 by (fast_tac (claset() addEs [hypnat_less_asym]) 1); |
|
648 qed "hypnat_less_imp_le"; |
|
649 |
|
650 Goalw [hypnat_le_def] "!!(x::hypnat). x <= y ==> x < y | x = y"; |
|
651 by (cut_facts_tac [hypnat_linear] 1); |
|
652 by (fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym]) 1); |
|
653 qed "hypnat_le_imp_less_or_eq"; |
|
654 |
|
655 Goalw [hypnat_le_def] "!!z. z<w | z=w ==> z <=(w::hypnat)"; |
|
656 by (cut_facts_tac [hypnat_linear] 1); |
|
657 by (fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym]) 1); |
|
658 qed "hypnat_less_or_eq_imp_le"; |
|
659 |
|
660 Goal "(x <= (y::hypnat)) = (x < y | x=y)"; |
|
661 by (REPEAT(ares_tac [iffI, hypnat_less_or_eq_imp_le, hypnat_le_imp_less_or_eq] 1)); |
|
662 qed "hypnat_le_eq_less_or_eq"; |
|
663 |
|
664 Goal "w <= (w::hypnat)"; |
|
665 by (simp_tac (simpset() addsimps [hypnat_le_eq_less_or_eq]) 1); |
|
666 qed "hypnat_le_refl"; |
|
667 Addsimps [hypnat_le_refl]; |
|
668 |
|
669 val prems = goal thy "!!i. [| i <= j; j < k |] ==> i < (k::hypnat)"; |
|
670 by (dtac hypnat_le_imp_less_or_eq 1); |
|
671 by (fast_tac (claset() addIs [hypnat_less_trans]) 1); |
|
672 qed "hypnat_le_less_trans"; |
|
673 |
|
674 Goal "!! (i::hypnat). [| i < j; j <= k |] ==> i < k"; |
|
675 by (dtac hypnat_le_imp_less_or_eq 1); |
|
676 by (fast_tac (claset() addIs [hypnat_less_trans]) 1); |
|
677 qed "hypnat_less_le_trans"; |
|
678 |
|
679 Goal "!!i. [| i <= j; j <= k |] ==> i <= (k::hypnat)"; |
|
680 by (EVERY1 [dtac hypnat_le_imp_less_or_eq, dtac hypnat_le_imp_less_or_eq, |
|
681 rtac hypnat_less_or_eq_imp_le, fast_tac (claset() addIs [hypnat_less_trans])]); |
|
682 qed "hypnat_le_trans"; |
|
683 |
|
684 Goal "!!z. [| z <= w; w <= z |] ==> z = (w::hypnat)"; |
|
685 by (EVERY1 [dtac hypnat_le_imp_less_or_eq, dtac hypnat_le_imp_less_or_eq, |
|
686 fast_tac (claset() addEs [hypnat_less_irrefl,hypnat_less_asym])]); |
|
687 qed "hypnat_le_anti_sym"; |
|
688 |
|
689 Goal "!!x. [| ~ y < x; y ~= x |] ==> x < (y::hypnat)"; |
|
690 by (rtac not_hypnat_leE 1); |
|
691 by (fast_tac (claset() addDs [hypnat_le_imp_less_or_eq]) 1); |
|
692 qed "not_less_not_eq_hypnat_less"; |
|
693 |
|
694 Goal "!!x. [| (0::hypnat) <= x; 0 <= y |] ==> 0 <= x * y"; |
|
695 by (REPEAT(dtac hypnat_le_imp_less_or_eq 1)); |
|
696 by (auto_tac (claset() addIs [hypnat_mult_order, |
|
697 hypnat_less_imp_le],simpset() addsimps [hypnat_le_refl])); |
|
698 qed "hypnat_le_mult_order"; |
|
699 |
|
700 Goalw [hypnat_one_def,hypnat_zero_def,hypnat_less_def] |
|
701 "(0::hypnat) < 1hn"; |
|
702 by (res_inst_tac [("x","%n. 0")] exI 1); |
|
703 by (res_inst_tac [("x","%n. 1")] exI 1); |
|
704 by (Auto_tac); |
|
705 qed "hypnat_zero_less_one"; |
|
706 |
|
707 Goal "!!x. [| (0::hypnat) <= x; 0 <= y |] ==> 0 <= x + y"; |
|
708 by (REPEAT(dtac hypnat_le_imp_less_or_eq 1)); |
|
709 by (auto_tac (claset() addIs [hypnat_add_order, |
|
710 hypnat_less_imp_le],simpset() addsimps [hypnat_le_refl])); |
|
711 qed "hypnat_le_add_order"; |
|
712 |
|
713 Goal "!!(q1::hypnat). q1 <= q2 ==> x + q1 <= x + q2"; |
|
714 by (dtac hypnat_le_imp_less_or_eq 1); |
|
715 by (Step_tac 1); |
|
716 by (auto_tac (claset() addSIs [hypnat_le_refl, |
|
717 hypnat_less_imp_le,hypnat_add_less_mono1], |
|
718 simpset() addsimps [hypnat_add_commute])); |
|
719 qed "hypnat_add_le_mono2"; |
|
720 |
|
721 Goal "!!(q1::hypnat). q1 <= q2 ==> q1 + x <= q2 + x"; |
|
722 by (auto_tac (claset() addDs [hypnat_add_le_mono2], |
|
723 simpset() addsimps [hypnat_add_commute])); |
|
724 qed "hypnat_add_le_mono1"; |
|
725 |
|
726 Goal "!!k l::hypnat. [|i<=j; k<=l |] ==> i + k <= j + l"; |
|
727 by (etac (hypnat_add_le_mono1 RS hypnat_le_trans) 1); |
|
728 by (simp_tac (simpset() addsimps [hypnat_add_commute]) 1); |
|
729 (*j moves to the end because it is free while k, l are bound*) |
|
730 by (etac hypnat_add_le_mono1 1); |
|
731 qed "hypnat_add_le_mono"; |
|
732 |
|
733 Goalw [hypnat_zero_def] |
|
734 "!!x::hypnat. [| (0::hypnat) < z; x < y |] ==> x * z < y * z"; |
|
735 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
|
736 by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
|
737 by (res_inst_tac [("z","z")] eq_Abs_hypnat 1); |
|
738 by (auto_tac (claset(),simpset() addsimps |
|
739 [hypnat_less,hypnat_mult])); |
|
740 by (Fuf_tac 1); |
|
741 qed "hypnat_mult_less_mono1"; |
|
742 |
|
743 Goal "!!x::hypnat. [| 0 < z; x < y |] ==> z * x < z * y"; |
|
744 by (auto_tac (claset() addIs [hypnat_mult_less_mono1], |
|
745 simpset() addsimps [hypnat_mult_commute])); |
|
746 qed "hypnat_mult_less_mono2"; |
|
747 |
|
748 Addsimps [hypnat_mult_less_mono2,hypnat_mult_less_mono1]; |
|
749 |
|
750 Goal "(x::hypnat) <= n + x"; |
|
751 by (res_inst_tac [("x","n")] hypnat_trichotomyE 1); |
|
752 by (auto_tac (claset() addDs [(hypnat_less_imp_le RS |
|
753 hypnat_add_le_mono1)],simpset() addsimps [hypnat_le_refl])); |
|
754 qed "hypnat_add_self_le"; |
|
755 Addsimps [hypnat_add_self_le]; |
|
756 |
|
757 Goal "(x::hypnat) < x + 1hn"; |
|
758 by (cut_facts_tac [hypnat_zero_less_one |
|
759 RS hypnat_add_less_mono2] 1); |
|
760 by (Auto_tac); |
|
761 qed "hypnat_add_one_self_less"; |
|
762 Addsimps [hypnat_add_one_self_less]; |
|
763 |
|
764 Goalw [hypnat_le_def] "~ x + 1hn <= x"; |
|
765 by (Simp_tac 1); |
|
766 qed "not_hypnat_add_one_le_self"; |
|
767 Addsimps [not_hypnat_add_one_le_self]; |
|
768 |
|
769 Goal "((0::hypnat) < n) = (1hn <= n)"; |
|
770 by (res_inst_tac [("x","n")] hypnat_trichotomyE 1); |
|
771 by (auto_tac (claset(),simpset() addsimps [hypnat_le_def])); |
|
772 qed "hypnat_gt_zero_iff"; |
|
773 |
|
774 Addsimps [hypnat_le_add_diff_inverse, hypnat_le_add_diff_inverse2, |
|
775 hypnat_less_imp_le RS hypnat_le_add_diff_inverse2]; |
|
776 |
|
777 Goal "(0 < n) = (EX m. n = m + 1hn)"; |
|
778 by (Step_tac 1); |
|
779 by (res_inst_tac [("x","m")] hypnat_trichotomyE 2); |
|
780 by (rtac hypnat_less_trans 2); |
|
781 by (res_inst_tac [("x","n - 1hn")] exI 1); |
|
782 by (auto_tac (claset(),simpset() addsimps |
|
783 [hypnat_gt_zero_iff,hypnat_add_commute])); |
|
784 qed "hypnat_gt_zero_iff2"; |
|
785 |
|
786 Goalw [hypnat_zero_def] "(0::hypnat) <= n"; |
|
787 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
|
788 by (asm_simp_tac (simpset() addsimps [hypnat_le]) 1); |
|
789 qed "hypnat_le_zero"; |
|
790 Addsimps [hypnat_le_zero]; |
|
791 |
|
792 (*------------------------------------------------------------------ |
|
793 hypnat_of_nat: properties embedding of naturals in hypernaturals |
|
794 -----------------------------------------------------------------*) |
|
795 (** hypnat_of_nat preserves field and order properties **) |
|
796 |
|
797 Goalw [hypnat_of_nat_def] |
|
798 "hypnat_of_nat ((z1::nat) + z2) = \ |
|
799 \ hypnat_of_nat z1 + hypnat_of_nat z2"; |
|
800 by (asm_simp_tac (simpset() addsimps [hypnat_add]) 1); |
|
801 qed "hypnat_of_nat_add"; |
|
802 |
|
803 Goalw [hypnat_of_nat_def] |
|
804 "hypnat_of_nat ((z1::nat) - z2) = \ |
|
805 \ hypnat_of_nat z1 - hypnat_of_nat z2"; |
|
806 by (asm_simp_tac (simpset() addsimps [hypnat_minus]) 1); |
|
807 qed "hypnat_of_nat_minus"; |
|
808 |
|
809 Goalw [hypnat_of_nat_def] |
|
810 "hypnat_of_nat (z1 * z2) = hypnat_of_nat z1 * hypnat_of_nat z2"; |
|
811 by (full_simp_tac (simpset() addsimps [hypnat_mult]) 1); |
|
812 qed "hypnat_of_nat_mult"; |
|
813 |
|
814 Goalw [hypnat_less_def,hypnat_of_nat_def] |
|
815 "(z1 < z2) = (hypnat_of_nat z1 < hypnat_of_nat z2)"; |
|
816 by (auto_tac (claset() addSIs [exI] addIs |
|
817 [FreeUltrafilterNat_all],simpset())); |
|
818 by (rtac FreeUltrafilterNat_P 1 THEN Fuf_tac 1); |
|
819 qed "hypnat_of_nat_less_iff"; |
|
820 Addsimps [hypnat_of_nat_less_iff RS sym]; |
|
821 |
|
822 Goalw [hypnat_le_def,le_def] |
|
823 "(z1 <= z2) = (hypnat_of_nat z1 <= hypnat_of_nat z2)"; |
|
824 by (Auto_tac); |
|
825 qed "hypnat_of_nat_le_iff"; |
|
826 |
|
827 Goalw [hypnat_of_nat_def,hypnat_one_def] "hypnat_of_nat 1 = 1hn"; |
|
828 by (Simp_tac 1); |
|
829 qed "hypnat_of_nat_one"; |
|
830 |
|
831 Goalw [hypnat_of_nat_def,hypnat_zero_def] "hypnat_of_nat 0 = 0"; |
|
832 by (Simp_tac 1); |
|
833 qed "hypnat_of_nat_zero"; |
|
834 |
|
835 Goal "(hypnat_of_nat n = 0) = (n = 0)"; |
|
836 by (auto_tac (claset() addIs [FreeUltrafilterNat_P], |
|
837 simpset() addsimps [hypnat_of_nat_def, |
|
838 hypnat_zero_def])); |
|
839 qed "hypnat_of_nat_zero_iff"; |
|
840 |
|
841 Goal "(hypnat_of_nat n ~= 0) = (n ~= 0)"; |
|
842 by (full_simp_tac (simpset() addsimps [hypnat_of_nat_zero_iff]) 1); |
|
843 qed "hypnat_of_nat_not_zero_iff"; |
|
844 |
|
845 goalw HyperNat.thy [hypnat_of_nat_def,hypnat_one_def] |
|
846 "hypnat_of_nat (Suc n) = hypnat_of_nat n + 1hn"; |
|
847 by (auto_tac (claset(),simpset() addsimps [hypnat_add])); |
|
848 qed "hypnat_of_nat_Suc"; |
|
849 |
|
850 (*--------------------------------------------------------------------------------- |
|
851 Existence of infinite hypernatural number |
|
852 ---------------------------------------------------------------------------------*) |
|
853 |
|
854 Goal "hypnatrel^^{%n::nat. n} : hypnat"; |
|
855 by (Auto_tac); |
|
856 qed "hypnat_omega"; |
|
857 |
|
858 Goalw [hypnat_omega_def] "Rep_hypnat(whn) : hypnat"; |
|
859 by (rtac Rep_hypnat 1); |
|
860 qed "Rep_hypnat_omega"; |
|
861 |
|
862 (* See Hyper.thy for similar argument*) |
|
863 (* existence of infinite number not corresponding to any natural number *) |
|
864 (* use assumption that member FreeUltrafilterNat is not finite *) |
|
865 (* a few lemmas first *) |
|
866 |
|
867 Goalw [hypnat_omega_def,hypnat_of_nat_def] |
|
868 "~ (EX x. hypnat_of_nat x = whn)"; |
|
869 by (auto_tac (claset() addDs [FreeUltrafilterNat_not_finite], |
|
870 simpset())); |
|
871 qed "not_ex_hypnat_of_nat_eq_omega"; |
|
872 |
|
873 Goal "hypnat_of_nat x ~= whn"; |
|
874 by (cut_facts_tac [not_ex_hypnat_of_nat_eq_omega] 1); |
|
875 by (Auto_tac); |
|
876 qed "hypnat_of_nat_not_eq_omega"; |
|
877 Addsimps [hypnat_of_nat_not_eq_omega RS not_sym]; |
|
878 |
|
879 (*----------------------------------------------------------- |
|
880 Properties of the set SHNat of embedded natural numbers |
|
881 (cf. set SReal in NSA.thy/NSA.ML) |
|
882 ----------------------------------------------------------*) |
|
883 |
|
884 (* Infinite hypernatural not in embedded SHNat *) |
|
885 Goalw [SHNat_def] "whn ~: SHNat"; |
|
886 by (Auto_tac); |
|
887 qed "SHNAT_omega_not_mem"; |
|
888 Addsimps [SHNAT_omega_not_mem]; |
|
889 |
|
890 (*----------------------------------------------------------------------- |
|
891 Closure laws for members of (embedded) set standard naturals SHNat |
|
892 -----------------------------------------------------------------------*) |
|
893 Goalw [SHNat_def] |
|
894 "!!x. [| x: SHNat; y: SHNat |] ==> x + y: SHNat"; |
|
895 by (Step_tac 1); |
|
896 by (res_inst_tac [("x","N + Na")] exI 1); |
|
897 by (simp_tac (simpset() addsimps [hypnat_of_nat_add]) 1); |
|
898 qed "SHNat_add"; |
|
899 |
|
900 Goalw [SHNat_def] |
|
901 "!!x. [| x: SHNat; y: SHNat |] ==> x - y: SHNat"; |
|
902 by (Step_tac 1); |
|
903 by (res_inst_tac [("x","N - Na")] exI 1); |
|
904 by (simp_tac (simpset() addsimps [hypnat_of_nat_minus]) 1); |
|
905 qed "SHNat_minus"; |
|
906 |
|
907 Goalw [SHNat_def] |
|
908 "!!x. [| x: SHNat; y: SHNat |] ==> x * y: SHNat"; |
|
909 by (Step_tac 1); |
|
910 by (res_inst_tac [("x","N * Na")] exI 1); |
|
911 by (simp_tac (simpset() addsimps [hypnat_of_nat_mult]) 1); |
|
912 qed "SHNat_mult"; |
|
913 |
|
914 Goal "!!x. [| x + y : SHNat; y: SHNat |] ==> x: SHNat"; |
|
915 by (dres_inst_tac [("x","x+y")] SHNat_minus 1); |
|
916 by (Auto_tac); |
|
917 qed "SHNat_add_cancel"; |
|
918 |
|
919 Goalw [SHNat_def] "hypnat_of_nat x : SHNat"; |
|
920 by (Blast_tac 1); |
|
921 qed "SHNat_hypnat_of_nat"; |
|
922 Addsimps [SHNat_hypnat_of_nat]; |
|
923 |
|
924 Goal "hypnat_of_nat 1 : SHNat"; |
|
925 by (Simp_tac 1); |
|
926 qed "SHNat_hypnat_of_nat_one"; |
|
927 |
|
928 Goal "hypnat_of_nat 0 : SHNat"; |
|
929 by (Simp_tac 1); |
|
930 qed "SHNat_hypnat_of_nat_zero"; |
|
931 |
|
932 Goal "1hn : SHNat"; |
|
933 by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_one, |
|
934 hypnat_of_nat_one RS sym]) 1); |
|
935 qed "SHNat_one"; |
|
936 |
|
937 Goal "0 : SHNat"; |
|
938 by (simp_tac (simpset() addsimps [SHNat_hypnat_of_nat_zero, |
|
939 hypnat_of_nat_zero RS sym]) 1); |
|
940 qed "SHNat_zero"; |
|
941 |
|
942 Addsimps [SHNat_hypnat_of_nat_one,SHNat_hypnat_of_nat_zero, |
|
943 SHNat_one,SHNat_zero]; |
|
944 |
|
945 Goal "1hn + 1hn : SHNat"; |
|
946 by (rtac ([SHNat_one,SHNat_one] MRS SHNat_add) 1); |
|
947 qed "SHNat_two"; |
|
948 |
|
949 Goalw [SHNat_def] "{x. hypnat_of_nat x : SHNat} = (UNIV::nat set)"; |
|
950 by (Auto_tac); |
|
951 qed "SHNat_UNIV_nat"; |
|
952 |
|
953 Goalw [SHNat_def] "(x: SHNat) = (EX y. x = hypnat_of_nat y)"; |
|
954 by (Auto_tac); |
|
955 qed "SHNat_iff"; |
|
956 |
|
957 Goalw [SHNat_def] "hypnat_of_nat ``(UNIV::nat set) = SHNat"; |
|
958 by (Auto_tac); |
|
959 qed "hypnat_of_nat_image"; |
|
960 |
|
961 Goalw [SHNat_def] "inv hypnat_of_nat ``SHNat = (UNIV::nat set)"; |
|
962 by (Auto_tac); |
|
963 by (rtac (inj_hypnat_of_nat RS inv_f_f RS subst) 1); |
|
964 by (Blast_tac 1); |
|
965 qed "inv_hypnat_of_nat_image"; |
|
966 |
|
967 Goalw [SHNat_def] |
|
968 "!!P. [| EX x. x: P; P <= SHNat |] ==> \ |
|
969 \ EX Q. P = hypnat_of_nat `` Q"; |
|
970 by (Best_tac 1); |
|
971 qed "SHNat_hypnat_of_nat_image"; |
|
972 |
|
973 Goalw [SHNat_def] |
|
974 "SHNat = hypnat_of_nat `` (UNIV::nat set)"; |
|
975 by (Auto_tac); |
|
976 qed "SHNat_hypnat_of_nat_iff"; |
|
977 |
|
978 Goalw [SHNat_def] "SHNat <= (UNIV::hypnat set)"; |
|
979 by (Auto_tac); |
|
980 qed "SHNat_subset_UNIV"; |
|
981 |
|
982 Goal "{n. n <= Suc m} = {n. n <= m} Un {n. n = Suc m}"; |
|
983 by (auto_tac (claset(),simpset() addsimps [le_Suc_eq])); |
|
984 qed "leSuc_Un_eq"; |
|
985 |
|
986 Goal "finite {n::nat. n <= m}"; |
|
987 by (nat_ind_tac "m" 1); |
|
988 by (auto_tac (claset(),simpset() addsimps [leSuc_Un_eq])); |
|
989 qed "finite_nat_le_segment"; |
|
990 |
|
991 Goal "{n::nat. m < n} : FreeUltrafilterNat"; |
|
992 by (cut_inst_tac [("m2","m")] (finite_nat_le_segment RS |
|
993 FreeUltrafilterNat_finite RS FreeUltrafilterNat_Compl_mem) 1); |
|
994 by (Fuf_tac 1); |
|
995 qed "lemma_unbounded_set"; |
|
996 Addsimps [lemma_unbounded_set]; |
|
997 |
|
998 Goalw [SHNat_def,hypnat_of_nat_def, |
|
999 hypnat_less_def,hypnat_omega_def] |
|
1000 "ALL n: SHNat. n < whn"; |
|
1001 by (Clarify_tac 1); |
|
1002 by (auto_tac (claset() addSIs [exI],simpset())); |
|
1003 qed "hypnat_omega_gt_SHNat"; |
|
1004 |
|
1005 Goal "hypnat_of_nat n < whn"; |
|
1006 by (cut_facts_tac [hypnat_omega_gt_SHNat] 1); |
|
1007 by (dres_inst_tac [("x","hypnat_of_nat n")] bspec 1); |
|
1008 by (Auto_tac); |
|
1009 qed "hypnat_of_nat_less_whn"; |
|
1010 Addsimps [hypnat_of_nat_less_whn]; |
|
1011 |
|
1012 Goal "hypnat_of_nat n <= whn"; |
|
1013 by (rtac (hypnat_of_nat_less_whn RS hypnat_less_imp_le) 1); |
|
1014 qed "hypnat_of_nat_le_whn"; |
|
1015 Addsimps [hypnat_of_nat_le_whn]; |
|
1016 |
|
1017 Goal "0 < whn"; |
|
1018 by (rtac (hypnat_omega_gt_SHNat RS ballE) 1); |
|
1019 by (Auto_tac); |
|
1020 qed "hypnat_zero_less_hypnat_omega"; |
|
1021 Addsimps [hypnat_zero_less_hypnat_omega]; |
|
1022 |
|
1023 Goal "1hn < whn"; |
|
1024 by (rtac (hypnat_omega_gt_SHNat RS ballE) 1); |
|
1025 by (Auto_tac); |
|
1026 qed "hypnat_one_less_hypnat_omega"; |
|
1027 Addsimps [hypnat_one_less_hypnat_omega]; |
|
1028 |
|
1029 (*-------------------------------------------------------------------------- |
|
1030 Theorems about infinite hypernatural numbers -- HNatInfinite |
|
1031 -------------------------------------------------------------------------*) |
|
1032 Goalw [HNatInfinite_def,SHNat_def] "whn : HNatInfinite"; |
|
1033 by (Auto_tac); |
|
1034 qed "HNatInfinite_whn"; |
|
1035 Addsimps [HNatInfinite_whn]; |
|
1036 |
|
1037 Goalw [HNatInfinite_def] "!!x. x: SHNat ==> x ~: HNatInfinite"; |
|
1038 by (Simp_tac 1); |
|
1039 qed "SHNat_not_HNatInfinite"; |
|
1040 |
|
1041 Goalw [HNatInfinite_def] "!!x. x ~: HNatInfinite ==> x: SHNat"; |
|
1042 by (Asm_full_simp_tac 1); |
|
1043 qed "not_HNatInfinite_SHNat"; |
|
1044 |
|
1045 Goalw [HNatInfinite_def] "!!x. x ~: SHNat ==> x: HNatInfinite"; |
|
1046 by (Simp_tac 1); |
|
1047 qed "not_SHNat_HNatInfinite"; |
|
1048 |
|
1049 Goalw [HNatInfinite_def] "!!x. x: HNatInfinite ==> x ~: SHNat"; |
|
1050 by (Asm_full_simp_tac 1); |
|
1051 qed "HNatInfinite_not_SHNat"; |
|
1052 |
|
1053 Goal "(x: SHNat) = (x ~: HNatInfinite)"; |
|
1054 by (blast_tac (claset() addSIs [SHNat_not_HNatInfinite, |
|
1055 not_HNatInfinite_SHNat]) 1); |
|
1056 qed "SHNat_not_HNatInfinite_iff"; |
|
1057 |
|
1058 Goal "(x ~: SHNat) = (x: HNatInfinite)"; |
|
1059 by (blast_tac (claset() addSIs [not_SHNat_HNatInfinite, |
|
1060 HNatInfinite_not_SHNat]) 1); |
|
1061 qed "not_SHNat_HNatInfinite_iff"; |
|
1062 |
|
1063 Goal "x : SHNat | x : HNatInfinite"; |
|
1064 by (simp_tac (simpset() addsimps [SHNat_not_HNatInfinite_iff]) 1); |
|
1065 qed "SHNat_HNatInfinite_disj"; |
|
1066 |
|
1067 (*------------------------------------------------------------------- |
|
1068 Proof of alternative definition for set of Infinite hypernatural |
|
1069 numbers --- HNatInfinite = {N. ALL n: SHNat. n < N} |
|
1070 -------------------------------------------------------------------*) |
|
1071 Goal "!!N (xa::nat=>nat). \ |
|
1072 \ (ALL N. {n. xa n ~= N} : FreeUltrafilterNat) ==> \ |
|
1073 \ ({n. N < xa n} : FreeUltrafilterNat)"; |
|
1074 by (nat_ind_tac "N" 1); |
|
1075 by (dres_inst_tac [("x","0")] spec 1); |
|
1076 by (rtac ccontr 1 THEN dtac FreeUltrafilterNat_Compl_mem 1 |
|
1077 THEN dtac FreeUltrafilterNat_Int 1 THEN assume_tac 1); |
|
1078 by (Asm_full_simp_tac 1); |
|
1079 by (dres_inst_tac [("x","Suc N")] spec 1); |
|
1080 by (fuf_tac (claset() addSDs [Suc_leI],simpset() addsimps |
|
1081 [le_eq_less_or_eq]) 1); |
|
1082 qed "HNatInfinite_FreeUltrafilterNat_lemma"; |
|
1083 |
|
1084 (*** alternative definition ***) |
|
1085 Goalw [HNatInfinite_def,SHNat_def,hypnat_of_nat_def] |
|
1086 "HNatInfinite = {N. ALL n:SHNat. n < N}"; |
|
1087 by (Step_tac 1); |
|
1088 by (dres_inst_tac [("x","Abs_hypnat \ |
|
1089 \ (hypnatrel ^^ {%n. N})")] bspec 2); |
|
1090 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
|
1091 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
|
1092 by (auto_tac (claset(),simpset() addsimps [hypnat_less_iff])); |
|
1093 by (auto_tac (claset() addSIs [exI] addEs |
|
1094 [HNatInfinite_FreeUltrafilterNat_lemma], |
|
1095 simpset() addsimps [FreeUltrafilterNat_Compl_iff1, |
|
1096 CLAIM "- {n. xa n = N} = {n. xa n ~= N}"])); |
|
1097 qed "HNatInfinite_iff"; |
|
1098 |
|
1099 (*-------------------------------------------------------------------- |
|
1100 Alternative definition for HNatInfinite using Free ultrafilter |
|
1101 --------------------------------------------------------------------*) |
|
1102 Goal "!!x. x : HNatInfinite ==> EX X: Rep_hypnat x. \ |
|
1103 \ ALL u. {n. u < X n}: FreeUltrafilterNat"; |
|
1104 by (auto_tac (claset(),simpset() addsimps [hypnat_less_def, |
|
1105 HNatInfinite_iff,SHNat_iff,hypnat_of_nat_def])); |
|
1106 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
|
1107 by (EVERY[Auto_tac, rtac bexI 1, |
|
1108 rtac lemma_hypnatrel_refl 2, Step_tac 1]); |
|
1109 by (dres_inst_tac [("x","hypnat_of_nat u")] bspec 1); |
|
1110 by (Simp_tac 1); |
|
1111 by (auto_tac (claset(), |
|
1112 simpset() addsimps [hypnat_of_nat_def])); |
|
1113 by (Fuf_tac 1); |
|
1114 qed "HNatInfinite_FreeUltrafilterNat"; |
|
1115 |
|
1116 Goal "!!x. EX X: Rep_hypnat x. \ |
|
1117 \ ALL u. {n. u < X n}: FreeUltrafilterNat \ |
|
1118 \ ==> x: HNatInfinite"; |
|
1119 by (auto_tac (claset(),simpset() addsimps [hypnat_less_def, |
|
1120 HNatInfinite_iff,SHNat_iff,hypnat_of_nat_def])); |
|
1121 by (rtac exI 1 THEN Auto_tac); |
|
1122 qed "FreeUltrafilterNat_HNatInfinite"; |
|
1123 |
|
1124 Goal "!!x. (x : HNatInfinite) = (EX X: Rep_hypnat x. \ |
|
1125 \ ALL u. {n. u < X n}: FreeUltrafilterNat)"; |
|
1126 by (blast_tac (claset() addIs [HNatInfinite_FreeUltrafilterNat, |
|
1127 FreeUltrafilterNat_HNatInfinite]) 1); |
|
1128 qed "HNatInfinite_FreeUltrafilterNat_iff"; |
|
1129 |
|
1130 Goal "!!x. x : HNatInfinite ==> 1hn < x"; |
|
1131 by (auto_tac (claset(),simpset() addsimps [HNatInfinite_iff])); |
|
1132 qed "HNatInfinite_gt_one"; |
|
1133 Addsimps [HNatInfinite_gt_one]; |
|
1134 |
|
1135 Goal "!!x. 0 ~: HNatInfinite"; |
|
1136 by (auto_tac (claset(),simpset() |
|
1137 addsimps [HNatInfinite_iff])); |
|
1138 by (dres_inst_tac [("a","1hn")] equals0D 1); |
|
1139 by (Asm_full_simp_tac 1); |
|
1140 qed "zero_not_mem_HNatInfinite"; |
|
1141 Addsimps [zero_not_mem_HNatInfinite]; |
|
1142 |
|
1143 Goal "!!x. x : HNatInfinite ==> x ~= 0"; |
|
1144 by (Auto_tac); |
|
1145 qed "HNatInfinite_not_eq_zero"; |
|
1146 |
|
1147 Goal "!!x. x : HNatInfinite ==> 1hn <= x"; |
|
1148 by (blast_tac (claset() addIs [hypnat_less_imp_le, |
|
1149 HNatInfinite_gt_one]) 1); |
|
1150 qed "HNatInfinite_ge_one"; |
|
1151 Addsimps [HNatInfinite_ge_one]; |
|
1152 |
|
1153 (*-------------------------------------------------- |
|
1154 Closure Rules |
|
1155 --------------------------------------------------*) |
|
1156 Goal "!!x. [| x: HNatInfinite; y: HNatInfinite |] \ |
|
1157 \ ==> x + y: HNatInfinite"; |
|
1158 by (auto_tac (claset(),simpset() addsimps [HNatInfinite_iff])); |
|
1159 by (dtac bspec 1 THEN assume_tac 1); |
|
1160 by (dtac (SHNat_zero RSN (2,bspec)) 1); |
|
1161 by (dtac hypnat_add_less_mono 1 THEN assume_tac 1); |
|
1162 by (Asm_full_simp_tac 1); |
|
1163 qed "HNatInfinite_add"; |
|
1164 |
|
1165 Goal "!!x. [| x: HNatInfinite; y: SHNat |] \ |
|
1166 \ ==> x + y: HNatInfinite"; |
|
1167 by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1); |
|
1168 by (dres_inst_tac [("x","x + y")] SHNat_minus 1); |
|
1169 by (auto_tac (claset(),simpset() addsimps |
|
1170 [SHNat_not_HNatInfinite_iff])); |
|
1171 qed "HNatInfinite_SHNat_add"; |
|
1172 |
|
1173 goal HyperNat.thy "!!x. [| x: HNatInfinite; y: SHNat |] \ |
|
1174 \ ==> x - y: HNatInfinite"; |
|
1175 by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1); |
|
1176 by (dres_inst_tac [("x","x - y")] SHNat_add 1); |
|
1177 by (subgoal_tac "y <= x" 2); |
|
1178 by (auto_tac (claset() addSDs [hypnat_le_add_diff_inverse2], |
|
1179 simpset() addsimps [not_SHNat_HNatInfinite_iff RS sym])); |
|
1180 by (auto_tac (claset() addSIs [hypnat_less_imp_le], |
|
1181 simpset() addsimps [not_SHNat_HNatInfinite_iff,HNatInfinite_iff])); |
|
1182 qed "HNatInfinite_SHNat_diff"; |
|
1183 |
|
1184 Goal |
|
1185 "!!x. x: HNatInfinite ==> x + 1hn: HNatInfinite"; |
|
1186 by (auto_tac (claset() addIs [HNatInfinite_SHNat_add], |
|
1187 simpset())); |
|
1188 qed "HNatInfinite_add_one"; |
|
1189 |
|
1190 Goal |
|
1191 "!!x. x: HNatInfinite ==> x - 1hn: HNatInfinite"; |
|
1192 by (rtac ccontr 1 THEN dtac not_HNatInfinite_SHNat 1); |
|
1193 by (dres_inst_tac [("x","x - 1hn"),("y","1hn")] SHNat_add 1); |
|
1194 by (auto_tac (claset(),simpset() addsimps |
|
1195 [not_SHNat_HNatInfinite_iff RS sym])); |
|
1196 qed "HNatInfinite_minus_one"; |
|
1197 |
|
1198 Goal "!!x. x : HNatInfinite ==> EX y. x = y + 1hn"; |
|
1199 by (res_inst_tac [("x","x - 1hn")] exI 1); |
|
1200 by (Auto_tac); |
|
1201 qed "HNatInfinite_is_Suc"; |
|
1202 |
|
1203 (*--------------------------------------------------------------- |
|
1204 HNat : the hypernaturals embedded in the hyperreals |
|
1205 Obtained using the NS extension of the naturals |
|
1206 --------------------------------------------------------------*) |
|
1207 |
|
1208 Goalw [HNat_def,starset_def, |
|
1209 hypreal_of_posnat_def,hypreal_of_real_def] |
|
1210 "hypreal_of_posnat N : HNat"; |
|
1211 by (Auto_tac); |
|
1212 by (Ultra_tac 1); |
|
1213 by (res_inst_tac [("x","Suc N")] exI 1); |
|
1214 by (dtac sym 1 THEN auto_tac (claset(),simpset() |
|
1215 addsimps [real_of_preal_real_of_nat_Suc])); |
|
1216 qed "HNat_hypreal_of_posnat"; |
|
1217 Addsimps [HNat_hypreal_of_posnat]; |
|
1218 |
|
1219 Goalw [HNat_def,starset_def] |
|
1220 "[| x: HNat; y: HNat |] ==> x + y: HNat"; |
|
1221 by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
|
1222 by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
|
1223 by (auto_tac (claset() addSDs [bspec] addIs [lemma_hyprel_refl], |
|
1224 simpset() addsimps [hypreal_add])); |
|
1225 by (Ultra_tac 1); |
|
1226 by (dres_inst_tac [("t","Y xb")] sym 1); |
|
1227 by (auto_tac (claset(),simpset() addsimps [real_of_nat_add RS sym])); |
|
1228 qed "HNat_add"; |
|
1229 |
|
1230 Goalw [HNat_def,starset_def] |
|
1231 "[| x: HNat; y: HNat |] ==> x * y: HNat"; |
|
1232 by (res_inst_tac [("z","x")] eq_Abs_hypreal 1); |
|
1233 by (res_inst_tac [("z","y")] eq_Abs_hypreal 1); |
|
1234 by (auto_tac (claset() addSDs [bspec] addIs [lemma_hyprel_refl], |
|
1235 simpset() addsimps [hypreal_mult])); |
|
1236 by (Ultra_tac 1); |
|
1237 by (dres_inst_tac [("t","Y xb")] sym 1); |
|
1238 by (auto_tac (claset(),simpset() addsimps [real_of_nat_mult RS sym])); |
|
1239 qed "HNat_mult"; |
|
1240 |
|
1241 (*--------------------------------------------------------------- |
|
1242 Embedding of the hypernaturals into the hyperreal |
|
1243 --------------------------------------------------------------*) |
|
1244 (*** A lemma that should have been derived a long time ago! ***) |
|
1245 Goal "(Ya : hyprel ^^{%n. f(n)}) = \ |
|
1246 \ ({n. f n = Ya n} : FreeUltrafilterNat)"; |
|
1247 by (Auto_tac); |
|
1248 qed "lemma_hyprel_FUFN"; |
|
1249 |
|
1250 Goalw [hypreal_of_hypnat_def] |
|
1251 "hypreal_of_hypnat (Abs_hypnat(hypnatrel^^{%n. X n})) = \ |
|
1252 \ Abs_hypreal(hyprel ^^ {%n. real_of_nat (X n)})"; |
|
1253 by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1); |
|
1254 by (auto_tac (claset() addEs [FreeUltrafilterNat_Int RS |
|
1255 FreeUltrafilterNat_subset],simpset() addsimps |
|
1256 [lemma_hyprel_FUFN])); |
|
1257 qed "hypreal_of_hypnat"; |
|
1258 |
|
1259 Goal "inj(hypreal_of_hypnat)"; |
|
1260 by (rtac injI 1); |
|
1261 by (res_inst_tac [("z","x")] eq_Abs_hypnat 1); |
|
1262 by (res_inst_tac [("z","y")] eq_Abs_hypnat 1); |
|
1263 by (auto_tac (claset(),simpset() addsimps |
|
1264 [hypreal_of_hypnat,real_of_nat_eq_cancel])); |
|
1265 qed "inj_hypreal_of_hypnat"; |
|
1266 |
|
1267 Goal "(hypreal_of_hypnat n = hypreal_of_hypnat m) = (n = m)"; |
|
1268 by (auto_tac (claset(),simpset() addsimps [inj_hypreal_of_hypnat RS injD])); |
|
1269 qed "hypreal_of_hypnat_eq_cancel"; |
|
1270 Addsimps [hypreal_of_hypnat_eq_cancel]; |
|
1271 |
|
1272 Goal "(hypnat_of_nat n = hypnat_of_nat m) = (n = m)"; |
|
1273 by (auto_tac (claset() addDs [inj_hypnat_of_nat RS injD], |
|
1274 simpset())); |
|
1275 qed "hypnat_of_nat_eq_cancel"; |
|
1276 Addsimps [hypnat_of_nat_eq_cancel]; |
|
1277 |
|
1278 Goalw [hypreal_zero_def,hypnat_zero_def] |
|
1279 "hypreal_of_hypnat 0 = 0"; |
|
1280 by (simp_tac (simpset() addsimps [hypreal_of_hypnat, |
|
1281 real_of_nat_zero]) 1); |
|
1282 qed "hypreal_of_hypnat_zero"; |
|
1283 |
|
1284 Goalw [hypreal_one_def,hypnat_one_def] |
|
1285 "hypreal_of_hypnat 1hn = 1hr"; |
|
1286 by (simp_tac (simpset() addsimps [hypreal_of_hypnat, |
|
1287 real_of_nat_one]) 1); |
|
1288 qed "hypreal_of_hypnat_one"; |
|
1289 |
|
1290 Goal "hypreal_of_hypnat n1 + hypreal_of_hypnat n2 = hypreal_of_hypnat (n1 + n2)"; |
|
1291 by (res_inst_tac [("z","n1")] eq_Abs_hypnat 1); |
|
1292 by (res_inst_tac [("z","n2")] eq_Abs_hypnat 1); |
|
1293 by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat, |
|
1294 hypreal_add,hypnat_add,real_of_nat_add]) 1); |
|
1295 qed "hypreal_of_hypnat_add"; |
|
1296 |
|
1297 Goal "hypreal_of_hypnat n1 * hypreal_of_hypnat n2 = hypreal_of_hypnat (n1 * n2)"; |
|
1298 by (res_inst_tac [("z","n1")] eq_Abs_hypnat 1); |
|
1299 by (res_inst_tac [("z","n2")] eq_Abs_hypnat 1); |
|
1300 by (asm_simp_tac (simpset() addsimps [hypreal_of_hypnat, |
|
1301 hypreal_mult,hypnat_mult,real_of_nat_mult]) 1); |
|
1302 qed "hypreal_of_hypnat_mult"; |
|
1303 |
|
1304 Goal "(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"; |
|
1305 by (res_inst_tac [("z","n")] eq_Abs_hypnat 1); |
|
1306 by (res_inst_tac [("z","m")] eq_Abs_hypnat 1); |
|
1307 by (asm_simp_tac (simpset() addsimps |
|
1308 [hypreal_of_hypnat,hypreal_less,hypnat_less]) 1); |
|
1309 qed "hypreal_of_hypnat_less_iff"; |
|
1310 Addsimps [hypreal_of_hypnat_less_iff]; |
|
1311 |
|
1312 Goal "(hypreal_of_hypnat N = 0) = (N = 0)"; |
|
1313 by (simp_tac (simpset() addsimps [hypreal_of_hypnat_zero RS sym]) 1); |
|
1314 qed "hypreal_of_hypnat_eq_zero_iff"; |
|
1315 Addsimps [hypreal_of_hypnat_eq_zero_iff]; |
|
1316 |
|
1317 Goal "ALL n. N <= n ==> N = (0::hypnat)"; |
|
1318 by (dres_inst_tac [("x","0")] spec 1); |
|
1319 by (res_inst_tac [("z","N")] eq_Abs_hypnat 1); |
|
1320 by (auto_tac (claset(),simpset() addsimps [hypnat_le,hypnat_zero_def])); |
|
1321 qed "hypnat_eq_zero"; |
|
1322 Addsimps [hypnat_eq_zero]; |
|
1323 |
|
1324 Goal "~ (ALL n. n = (0::hypnat))"; |
|
1325 by Auto_tac; |
|
1326 by (res_inst_tac [("x","1hn")] exI 1); |
|
1327 by (Simp_tac 1); |
|
1328 qed "hypnat_not_all_eq_zero"; |
|
1329 Addsimps [hypnat_not_all_eq_zero]; |
|
1330 |
|
1331 Goal "n ~= 0 ==> (n <= 1hn) = (n = 1hn)"; |
|
1332 by (auto_tac (claset(),simpset() addsimps [hypnat_le_eq_less_or_eq])); |
|
1333 qed "hypnat_le_one_eq_one"; |
|
1334 Addsimps [hypnat_le_one_eq_one]; |
|
1335 |
|
1336 |
|
1337 |
|
1338 |
|