1 (* Title: HOL/Datatype_Universe.ML |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1991 University of Cambridge |
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5 *) |
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6 |
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7 (** apfst -- can be used in similar type definitions **) |
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8 |
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9 Goalw [apfst_def] "apfst f (a,b) = (f(a),b)"; |
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10 by (rtac split_conv 1); |
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11 qed "apfst_conv"; |
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12 |
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13 val [major,minor] = Goal |
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14 "[| q = apfst f p; !!x y. [| p = (x,y); q = (f(x),y) |] ==> R \ |
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15 \ |] ==> R"; |
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16 by (rtac PairE 1); |
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17 by (rtac minor 1); |
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18 by (assume_tac 1); |
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19 by (rtac (major RS trans) 1); |
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20 by (etac ssubst 1); |
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21 by (rtac apfst_conv 1); |
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22 qed "apfst_convE"; |
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23 |
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24 (** Push -- an injection, analogous to Cons on lists **) |
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25 |
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26 Goalw [Push_def] "Push i f = Push j g ==> i=j"; |
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27 by (etac (fun_cong RS box_equals) 1); |
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28 by (rtac nat_case_0 1); |
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29 by (rtac nat_case_0 1); |
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30 qed "Push_inject1"; |
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31 |
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32 Goalw [Push_def] "Push i f = Push j g ==> f=g"; |
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33 by (rtac (ext RS box_equals) 1); |
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34 by (etac fun_cong 1); |
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35 by (rtac (nat_case_Suc RS ext) 1); |
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36 by (rtac (nat_case_Suc RS ext) 1); |
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37 qed "Push_inject2"; |
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38 |
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39 val [major,minor] = Goal |
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40 "[| Push i f =Push j g; [| i=j; f=g |] ==> P \ |
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41 \ |] ==> P"; |
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42 by (rtac ((major RS Push_inject2) RS ((major RS Push_inject1) RS minor)) 1); |
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43 qed "Push_inject"; |
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44 |
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45 Goalw [Push_def] "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P"; |
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46 by (rtac Suc_neq_Zero 1); |
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47 by (etac (fun_cong RS box_equals RS Inr_inject) 1); |
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48 by (rtac nat_case_0 1); |
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49 by (rtac refl 1); |
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50 qed "Push_neq_K0"; |
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51 |
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52 (*** Isomorphisms ***) |
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53 |
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54 Goal "inj(Rep_Node)"; |
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55 by (rtac inj_inverseI 1); (*cannot combine by RS: multiple unifiers*) |
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56 by (rtac Rep_Node_inverse 1); |
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57 qed "inj_Rep_Node"; |
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58 |
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59 Goal "inj_on Abs_Node Node"; |
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60 by (rtac inj_on_inverseI 1); |
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61 by (etac Abs_Node_inverse 1); |
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62 qed "inj_on_Abs_Node"; |
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63 |
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64 bind_thm ("Abs_Node_inj", inj_on_Abs_Node RS inj_onD); |
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65 |
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66 |
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67 (*** Introduction rules for Node ***) |
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68 |
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69 Goalw [Node_def] "(%k. Inr 0, a) : Node"; |
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70 by (Blast_tac 1); |
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71 qed "Node_K0_I"; |
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72 |
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73 Goalw [Node_def,Push_def] |
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74 "p: Node ==> apfst (Push i) p : Node"; |
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75 by (fast_tac (claset() addSIs [apfst_conv, nat_case_Suc RS trans]) 1); |
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76 qed "Node_Push_I"; |
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77 |
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78 |
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79 (*** Distinctness of constructors ***) |
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80 |
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81 (** Scons vs Atom **) |
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82 |
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83 Goalw [Atom_def,Scons_def,Push_Node_def,One_nat_def] |
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84 "Scons M N ~= Atom(a)"; |
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85 by (rtac notI 1); |
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86 by (etac (equalityD2 RS subsetD RS UnE) 1); |
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87 by (rtac singletonI 1); |
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88 by (REPEAT (eresolve_tac [imageE, Abs_Node_inj RS apfst_convE, |
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89 Pair_inject, sym RS Push_neq_K0] 1 |
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90 ORELSE resolve_tac [Node_K0_I, Rep_Node RS Node_Push_I] 1)); |
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91 qed "Scons_not_Atom"; |
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92 bind_thm ("Atom_not_Scons", Scons_not_Atom RS not_sym); |
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93 |
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94 |
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95 (*** Injectiveness ***) |
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96 |
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97 (** Atomic nodes **) |
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98 |
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99 Goalw [Atom_def] "inj(Atom)"; |
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100 by (blast_tac (claset() addSIs [injI, Node_K0_I] addSDs [Abs_Node_inj]) 1); |
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101 qed "inj_Atom"; |
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102 bind_thm ("Atom_inject", inj_Atom RS injD); |
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103 |
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104 Goal "(Atom(a)=Atom(b)) = (a=b)"; |
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105 by (blast_tac (claset() addSDs [Atom_inject]) 1); |
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106 qed "Atom_Atom_eq"; |
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107 AddIffs [Atom_Atom_eq]; |
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108 |
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109 Goalw [Leaf_def,o_def] "inj(Leaf)"; |
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110 by (rtac injI 1); |
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111 by (etac (Atom_inject RS Inl_inject) 1); |
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112 qed "inj_Leaf"; |
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113 |
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114 bind_thm ("Leaf_inject", inj_Leaf RS injD); |
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115 AddSDs [Leaf_inject]; |
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116 |
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117 Goalw [Numb_def,o_def] "inj(Numb)"; |
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118 by (rtac injI 1); |
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119 by (etac (Atom_inject RS Inr_inject) 1); |
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120 qed "inj_Numb"; |
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121 |
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122 bind_thm ("Numb_inject", inj_Numb RS injD); |
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123 AddSDs [Numb_inject]; |
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124 |
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125 (** Injectiveness of Push_Node **) |
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126 |
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127 val [major,minor] = Goalw [Push_Node_def] |
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128 "[| Push_Node i m =Push_Node j n; [| i=j; m=n |] ==> P \ |
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129 \ |] ==> P"; |
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130 by (rtac (major RS Abs_Node_inj RS apfst_convE) 1); |
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131 by (REPEAT (resolve_tac [Rep_Node RS Node_Push_I] 1)); |
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132 by (etac (sym RS apfst_convE) 1); |
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133 by (rtac minor 1); |
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134 by (etac Pair_inject 1); |
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135 by (etac (Push_inject1 RS sym) 1); |
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136 by (rtac (inj_Rep_Node RS injD) 1); |
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137 by (etac trans 1); |
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138 by (safe_tac (claset() addSEs [Push_inject,sym])); |
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139 qed "Push_Node_inject"; |
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140 |
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141 |
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142 (** Injectiveness of Scons **) |
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143 |
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144 Goalw [Scons_def,One_nat_def] "Scons M N <= Scons M' N' ==> M<=M'"; |
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145 by (blast_tac (claset() addSDs [Push_Node_inject]) 1); |
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146 qed "Scons_inject_lemma1"; |
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147 |
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148 Goalw [Scons_def,One_nat_def] "Scons M N <= Scons M' N' ==> N<=N'"; |
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149 by (blast_tac (claset() addSDs [Push_Node_inject]) 1); |
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150 qed "Scons_inject_lemma2"; |
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151 |
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152 Goal "Scons M N = Scons M' N' ==> M=M'"; |
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153 by (etac equalityE 1); |
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154 by (REPEAT (ares_tac [equalityI, Scons_inject_lemma1] 1)); |
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155 qed "Scons_inject1"; |
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156 |
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157 Goal "Scons M N = Scons M' N' ==> N=N'"; |
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158 by (etac equalityE 1); |
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159 by (REPEAT (ares_tac [equalityI, Scons_inject_lemma2] 1)); |
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160 qed "Scons_inject2"; |
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161 |
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162 val [major,minor] = Goal |
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163 "[| Scons M N = Scons M' N'; [| M=M'; N=N' |] ==> P \ |
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164 \ |] ==> P"; |
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165 by (rtac ((major RS Scons_inject2) RS ((major RS Scons_inject1) RS minor)) 1); |
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166 qed "Scons_inject"; |
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167 |
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168 Goal "(Scons M N = Scons M' N') = (M=M' & N=N')"; |
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169 by (blast_tac (claset() addSEs [Scons_inject]) 1); |
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170 qed "Scons_Scons_eq"; |
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171 |
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172 (*** Distinctness involving Leaf and Numb ***) |
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173 |
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174 (** Scons vs Leaf **) |
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175 |
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176 Goalw [Leaf_def,o_def] "Scons M N ~= Leaf(a)"; |
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177 by (rtac Scons_not_Atom 1); |
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178 qed "Scons_not_Leaf"; |
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179 bind_thm ("Leaf_not_Scons", Scons_not_Leaf RS not_sym); |
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180 |
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181 AddIffs [Scons_not_Leaf, Leaf_not_Scons]; |
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182 |
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183 |
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184 (** Scons vs Numb **) |
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185 |
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186 Goalw [Numb_def,o_def] "Scons M N ~= Numb(k)"; |
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187 by (rtac Scons_not_Atom 1); |
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188 qed "Scons_not_Numb"; |
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189 bind_thm ("Numb_not_Scons", Scons_not_Numb RS not_sym); |
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190 |
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191 AddIffs [Scons_not_Numb, Numb_not_Scons]; |
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192 |
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193 |
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194 (** Leaf vs Numb **) |
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195 |
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196 Goalw [Leaf_def,Numb_def] "Leaf(a) ~= Numb(k)"; |
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197 by (simp_tac (simpset() addsimps [Inl_not_Inr]) 1); |
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198 qed "Leaf_not_Numb"; |
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199 bind_thm ("Numb_not_Leaf", Leaf_not_Numb RS not_sym); |
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200 |
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201 AddIffs [Leaf_not_Numb, Numb_not_Leaf]; |
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202 |
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203 |
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204 (*** ndepth -- the depth of a node ***) |
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205 |
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206 Addsimps [apfst_conv]; |
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207 AddIffs [Scons_not_Atom, Atom_not_Scons, Scons_Scons_eq]; |
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208 |
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209 |
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210 Goalw [ndepth_def] "ndepth (Abs_Node(%k. Inr 0, x)) = 0"; |
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211 by (EVERY1[stac (Node_K0_I RS Abs_Node_inverse), stac split_conv]); |
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212 by (rtac Least_equality 1); |
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213 by Auto_tac; |
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214 qed "ndepth_K0"; |
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215 |
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216 Goal "nat_case (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k"; |
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217 by (induct_tac "k" 1); |
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218 by (ALLGOALS Simp_tac); |
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219 by (rtac impI 1); |
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220 by (etac Least_le 1); |
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221 val lemma = result(); |
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222 |
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223 Goalw [ndepth_def,Push_Node_def] |
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224 "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))"; |
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225 by (stac (Rep_Node RS Node_Push_I RS Abs_Node_inverse) 1); |
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226 by (cut_facts_tac [rewrite_rule [Node_def] Rep_Node] 1); |
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227 by Safe_tac; |
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228 by (etac ssubst 1); (*instantiates type variables!*) |
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229 by (Simp_tac 1); |
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230 by (rtac Least_equality 1); |
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231 by (rewtac Push_def); |
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232 by (auto_tac (claset(), simpset() addsimps [lemma])); |
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233 by (etac LeastI 1); |
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234 qed "ndepth_Push_Node"; |
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235 |
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236 |
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237 (*** ntrunc applied to the various node sets ***) |
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238 |
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239 Goalw [ntrunc_def] "ntrunc 0 M = {}"; |
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240 by (Blast_tac 1); |
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241 qed "ntrunc_0"; |
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242 |
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243 Goalw [Atom_def,ntrunc_def] "ntrunc (Suc k) (Atom a) = Atom(a)"; |
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244 by (fast_tac (claset() addss (simpset() addsimps [ndepth_K0])) 1); |
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245 qed "ntrunc_Atom"; |
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246 |
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247 Goalw [Leaf_def,o_def] "ntrunc (Suc k) (Leaf a) = Leaf(a)"; |
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248 by (rtac ntrunc_Atom 1); |
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249 qed "ntrunc_Leaf"; |
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250 |
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251 Goalw [Numb_def,o_def] "ntrunc (Suc k) (Numb i) = Numb(i)"; |
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252 by (rtac ntrunc_Atom 1); |
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253 qed "ntrunc_Numb"; |
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254 |
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255 Goalw [Scons_def,ntrunc_def,One_nat_def] |
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256 "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)"; |
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257 by (safe_tac (claset() addSIs [imageI])); |
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258 by (REPEAT (stac ndepth_Push_Node 3 THEN etac Suc_mono 3)); |
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259 by (REPEAT (rtac Suc_less_SucD 1 THEN |
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260 rtac (ndepth_Push_Node RS subst) 1 THEN |
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261 assume_tac 1)); |
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262 qed "ntrunc_Scons"; |
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263 |
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264 Addsimps [ntrunc_0, ntrunc_Atom, ntrunc_Leaf, ntrunc_Numb, ntrunc_Scons]; |
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265 |
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266 |
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267 (** Injection nodes **) |
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268 |
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269 Goalw [In0_def] "ntrunc (Suc 0) (In0 M) = {}"; |
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270 by (Simp_tac 1); |
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271 by (rewtac Scons_def); |
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272 by (Blast_tac 1); |
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273 qed "ntrunc_one_In0"; |
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274 |
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275 Goalw [In0_def] |
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276 "ntrunc (Suc (Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)"; |
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277 by (Simp_tac 1); |
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278 qed "ntrunc_In0"; |
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279 |
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280 Goalw [In1_def] "ntrunc (Suc 0) (In1 M) = {}"; |
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281 by (Simp_tac 1); |
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282 by (rewtac Scons_def); |
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283 by (Blast_tac 1); |
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284 qed "ntrunc_one_In1"; |
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285 |
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286 Goalw [In1_def] |
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287 "ntrunc (Suc (Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)"; |
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288 by (Simp_tac 1); |
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289 qed "ntrunc_In1"; |
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290 |
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291 Addsimps [ntrunc_one_In0, ntrunc_In0, ntrunc_one_In1, ntrunc_In1]; |
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292 |
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293 |
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294 (*** Cartesian Product ***) |
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295 |
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296 Goalw [uprod_def] "[| M:A; N:B |] ==> Scons M N : uprod A B"; |
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297 by (REPEAT (ares_tac [singletonI,UN_I] 1)); |
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298 qed "uprodI"; |
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299 |
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300 (*The general elimination rule*) |
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301 val major::prems = Goalw [uprod_def] |
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302 "[| c : uprod A B; \ |
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303 \ !!x y. [| x:A; y:B; c = Scons x y |] ==> P \ |
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304 \ |] ==> P"; |
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305 by (cut_facts_tac [major] 1); |
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306 by (REPEAT (eresolve_tac [asm_rl,singletonE,UN_E] 1 |
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307 ORELSE resolve_tac prems 1)); |
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308 qed "uprodE"; |
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309 |
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310 (*Elimination of a pair -- introduces no eigenvariables*) |
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311 val prems = Goal |
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312 "[| Scons M N : uprod A B; [| M:A; N:B |] ==> P \ |
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313 \ |] ==> P"; |
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314 by (rtac uprodE 1); |
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315 by (REPEAT (ares_tac prems 1 ORELSE eresolve_tac [Scons_inject,ssubst] 1)); |
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316 qed "uprodE2"; |
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317 |
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318 |
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319 (*** Disjoint Sum ***) |
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320 |
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321 Goalw [usum_def] "M:A ==> In0(M) : usum A B"; |
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322 by (Blast_tac 1); |
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323 qed "usum_In0I"; |
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324 |
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325 Goalw [usum_def] "N:B ==> In1(N) : usum A B"; |
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326 by (Blast_tac 1); |
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327 qed "usum_In1I"; |
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328 |
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329 val major::prems = Goalw [usum_def] |
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330 "[| u : usum A B; \ |
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331 \ !!x. [| x:A; u=In0(x) |] ==> P; \ |
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332 \ !!y. [| y:B; u=In1(y) |] ==> P \ |
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333 \ |] ==> P"; |
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334 by (rtac (major RS UnE) 1); |
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335 by (REPEAT (rtac refl 1 |
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336 ORELSE eresolve_tac (prems@[imageE,ssubst]) 1)); |
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337 qed "usumE"; |
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338 |
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339 |
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340 (** Injection **) |
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341 |
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342 Goalw [In0_def,In1_def,One_nat_def] "In0(M) ~= In1(N)"; |
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343 by (rtac notI 1); |
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344 by (etac (Scons_inject1 RS Numb_inject RS Zero_neq_Suc) 1); |
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345 qed "In0_not_In1"; |
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346 |
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347 bind_thm ("In1_not_In0", In0_not_In1 RS not_sym); |
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348 |
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349 AddIffs [In0_not_In1, In1_not_In0]; |
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350 |
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351 Goalw [In0_def] "In0(M) = In0(N) ==> M=N"; |
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352 by (etac (Scons_inject2) 1); |
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353 qed "In0_inject"; |
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354 |
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355 Goalw [In1_def] "In1(M) = In1(N) ==> M=N"; |
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356 by (etac (Scons_inject2) 1); |
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357 qed "In1_inject"; |
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358 |
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359 Goal "(In0 M = In0 N) = (M=N)"; |
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360 by (blast_tac (claset() addSDs [In0_inject]) 1); |
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361 qed "In0_eq"; |
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362 |
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363 Goal "(In1 M = In1 N) = (M=N)"; |
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364 by (blast_tac (claset() addSDs [In1_inject]) 1); |
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365 qed "In1_eq"; |
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366 |
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367 AddIffs [In0_eq, In1_eq]; |
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368 |
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369 Goal "inj In0"; |
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370 by (blast_tac (claset() addSIs [injI]) 1); |
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371 qed "inj_In0"; |
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372 |
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373 Goal "inj In1"; |
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374 by (blast_tac (claset() addSIs [injI]) 1); |
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375 qed "inj_In1"; |
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376 |
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377 |
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378 (*** Function spaces ***) |
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379 |
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380 Goalw [Lim_def] "Lim f = Lim g ==> f = g"; |
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381 by (rtac ext 1); |
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382 by (blast_tac (claset() addSEs [Push_Node_inject]) 1); |
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383 qed "Lim_inject"; |
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384 |
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385 |
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386 (*** proving equality of sets and functions using ntrunc ***) |
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387 |
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388 Goalw [ntrunc_def] "ntrunc k M <= M"; |
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389 by (Blast_tac 1); |
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390 qed "ntrunc_subsetI"; |
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391 |
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392 val [major] = Goalw [ntrunc_def] "(!!k. ntrunc k M <= N) ==> M<=N"; |
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393 by (blast_tac (claset() addIs [less_add_Suc1, less_add_Suc2, |
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394 major RS subsetD]) 1); |
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395 qed "ntrunc_subsetD"; |
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396 |
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397 (*A generalized form of the take-lemma*) |
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398 val [major] = Goal "(!!k. ntrunc k M = ntrunc k N) ==> M=N"; |
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399 by (rtac equalityI 1); |
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400 by (ALLGOALS (rtac ntrunc_subsetD)); |
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401 by (ALLGOALS (rtac (ntrunc_subsetI RSN (2, subset_trans)))); |
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402 by (rtac (major RS equalityD1) 1); |
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403 by (rtac (major RS equalityD2) 1); |
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404 qed "ntrunc_equality"; |
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405 |
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406 val [major] = Goalw [o_def] |
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407 "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2"; |
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408 by (rtac (ntrunc_equality RS ext) 1); |
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409 by (rtac (major RS fun_cong) 1); |
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410 qed "ntrunc_o_equality"; |
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411 |
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412 (*** Monotonicity ***) |
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413 |
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414 Goalw [uprod_def] "[| A<=A'; B<=B' |] ==> uprod A B <= uprod A' B'"; |
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415 by (Blast_tac 1); |
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416 qed "uprod_mono"; |
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417 |
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418 Goalw [usum_def] "[| A<=A'; B<=B' |] ==> usum A B <= usum A' B'"; |
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419 by (Blast_tac 1); |
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420 qed "usum_mono"; |
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421 |
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422 Goalw [Scons_def] "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'"; |
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423 by (Blast_tac 1); |
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424 qed "Scons_mono"; |
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425 |
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426 Goalw [In0_def] "M<=N ==> In0(M) <= In0(N)"; |
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427 by (REPEAT (ares_tac [subset_refl,Scons_mono] 1)); |
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428 qed "In0_mono"; |
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429 |
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430 Goalw [In1_def] "M<=N ==> In1(M) <= In1(N)"; |
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431 by (REPEAT (ares_tac [subset_refl,Scons_mono] 1)); |
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432 qed "In1_mono"; |
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433 |
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434 |
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435 (*** Split and Case ***) |
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436 |
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437 Goalw [Split_def] "Split c (Scons M N) = c M N"; |
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438 by (Blast_tac 1); |
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439 qed "Split"; |
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440 |
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441 Goalw [Case_def] "Case c d (In0 M) = c(M)"; |
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442 by (Blast_tac 1); |
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443 qed "Case_In0"; |
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444 |
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445 Goalw [Case_def] "Case c d (In1 N) = d(N)"; |
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446 by (Blast_tac 1); |
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447 qed "Case_In1"; |
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448 |
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449 Addsimps [Split, Case_In0, Case_In1]; |
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450 |
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451 |
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452 (**** UN x. B(x) rules ****) |
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453 |
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454 Goalw [ntrunc_def] "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))"; |
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455 by (Blast_tac 1); |
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456 qed "ntrunc_UN1"; |
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457 |
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458 Goalw [Scons_def] "Scons (UN x. f x) M = (UN x. Scons (f x) M)"; |
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459 by (Blast_tac 1); |
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460 qed "Scons_UN1_x"; |
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461 |
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462 Goalw [Scons_def] "Scons M (UN x. f x) = (UN x. Scons M (f x))"; |
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463 by (Blast_tac 1); |
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464 qed "Scons_UN1_y"; |
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465 |
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466 Goalw [In0_def] "In0(UN x. f(x)) = (UN x. In0(f(x)))"; |
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467 by (rtac Scons_UN1_y 1); |
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468 qed "In0_UN1"; |
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469 |
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470 Goalw [In1_def] "In1(UN x. f(x)) = (UN x. In1(f(x)))"; |
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471 by (rtac Scons_UN1_y 1); |
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472 qed "In1_UN1"; |
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473 |
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474 |
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475 (*** Equality for Cartesian Product ***) |
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476 |
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477 Goalw [dprod_def] |
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478 "[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s"; |
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479 by (Blast_tac 1); |
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480 qed "dprodI"; |
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481 |
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482 (*The general elimination rule*) |
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483 val major::prems = Goalw [dprod_def] |
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484 "[| c : dprod r s; \ |
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485 \ !!x y x' y'. [| (x,x') : r; (y,y') : s; c = (Scons x y, Scons x' y') |] ==> P \ |
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486 \ |] ==> P"; |
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487 by (cut_facts_tac [major] 1); |
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488 by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, mem_splitE, singletonE])); |
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489 by (REPEAT (ares_tac prems 1 ORELSE hyp_subst_tac 1)); |
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490 qed "dprodE"; |
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491 |
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492 |
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493 (*** Equality for Disjoint Sum ***) |
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494 |
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495 Goalw [dsum_def] "(M,M'):r ==> (In0(M), In0(M')) : dsum r s"; |
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496 by (Blast_tac 1); |
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497 qed "dsum_In0I"; |
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498 |
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499 Goalw [dsum_def] "(N,N'):s ==> (In1(N), In1(N')) : dsum r s"; |
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500 by (Blast_tac 1); |
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501 qed "dsum_In1I"; |
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502 |
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503 val major::prems = Goalw [dsum_def] |
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504 "[| w : dsum r s; \ |
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505 \ !!x x'. [| (x,x') : r; w = (In0(x), In0(x')) |] ==> P; \ |
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506 \ !!y y'. [| (y,y') : s; w = (In1(y), In1(y')) |] ==> P \ |
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507 \ |] ==> P"; |
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508 by (cut_facts_tac [major] 1); |
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509 by (REPEAT_FIRST (eresolve_tac [asm_rl, UN_E, UnE, mem_splitE, singletonE])); |
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510 by (DEPTH_SOLVE (ares_tac prems 1 ORELSE hyp_subst_tac 1)); |
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511 qed "dsumE"; |
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512 |
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513 AddSIs [uprodI, dprodI]; |
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514 AddIs [usum_In0I, usum_In1I, dsum_In0I, dsum_In1I]; |
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515 AddSEs [uprodE, dprodE, usumE, dsumE]; |
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516 |
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517 |
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518 (*** Monotonicity ***) |
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519 |
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520 Goal "[| r<=r'; s<=s' |] ==> dprod r s <= dprod r' s'"; |
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521 by (Blast_tac 1); |
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522 qed "dprod_mono"; |
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523 |
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524 Goal "[| r<=r'; s<=s' |] ==> dsum r s <= dsum r' s'"; |
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525 by (Blast_tac 1); |
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526 qed "dsum_mono"; |
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527 |
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528 |
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529 (*** Bounding theorems ***) |
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530 |
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531 Goal "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)"; |
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532 by (Blast_tac 1); |
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533 qed "dprod_Sigma"; |
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534 |
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535 bind_thm ("dprod_subset_Sigma", [dprod_mono, dprod_Sigma] MRS subset_trans |> standard); |
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536 |
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537 (*Dependent version*) |
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538 Goal "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))"; |
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539 by Safe_tac; |
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540 by (stac Split 1); |
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541 by (Blast_tac 1); |
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542 qed "dprod_subset_Sigma2"; |
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543 |
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544 Goal "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)"; |
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545 by (Blast_tac 1); |
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546 qed "dsum_Sigma"; |
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547 |
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548 bind_thm ("dsum_subset_Sigma", [dsum_mono, dsum_Sigma] MRS subset_trans |> standard); |
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549 |
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550 |
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551 (*** Domain ***) |
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552 |
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553 Goal "Domain (dprod r s) = uprod (Domain r) (Domain s)"; |
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554 by Auto_tac; |
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555 qed "Domain_dprod"; |
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556 |
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557 Goal "Domain (dsum r s) = usum (Domain r) (Domain s)"; |
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558 by Auto_tac; |
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559 qed "Domain_dsum"; |
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560 |
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561 Addsimps [Domain_dprod, Domain_dsum]; |
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