1 | include "basics/types.ma". |
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2 | include "basics/list.ma". |
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3 | include "common/Graphs.ma". |
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4 | include "common/Order.ma". |
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5 | include "common/Registers.ma". |
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6 | include "ERTL/ERTL.ma". |
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7 | include "ERTL/liveness.ma". |
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8 | |
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9 | include "utilities/adt/table_adt.ma". |
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10 | include "utilities/adt/priority_set_adt.ma". |
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11 | include "utilities/adt/set_adt.ma". |
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12 | include "utilities/adt/set_table_adt.ma". |
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13 | include "utilities/adt/register_table.ma". |
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14 | |
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15 | definition vertex ≝ register ⊎ Register. |
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16 | |
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17 | inductive decision: Type[0] ≝ |
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18 | | decision_spill: nat → decision |
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19 | | decision_colour: Register → decision. |
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20 | |
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21 | definition is_member ≝ |
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22 | λvertex. |
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23 | λregister_lattice. |
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24 | let 〈l, r〉 ≝ register_lattice in |
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25 | match vertex with |
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26 | [ inl v ⇒ set_member ? (eq_identifier RegisterTag) v l |
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27 | | inr v ⇒ set_member ? eq_Register v r |
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28 | ]. |
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29 | |
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30 | record coloured_graph (v: valuation): Type[1] ≝ |
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31 | { |
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32 | colouring: vertex → decision; |
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33 | prop_colouring: ∀l. ∀v1, v2: vertex. |
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34 | is_member v1 (v l) → is_member v2 (v l) → colouring v1 ≠ colouring v2 |
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35 | }. |
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36 | |
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37 | axiom build: ∀valuation. coloured_graph valuation. |
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38 | |
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39 | (* definition vertex_set ≝ set vertex. *) |
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40 | definition vertex_priority_set ≝ priority_set vertex. |
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41 | definition vertex_set_table ≝ set_table vertex (set vertex). |
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42 | definition vertex_set ≝ set vertex. |
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43 | definition Register_set_table ≝ set_table vertex (set Register). |
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44 | definition Register_set ≝ set Register. |
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45 | |
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46 | (* |
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47 | record graph: Type[0] ≝ |
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48 | { |
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49 | g_regmap : register_table; |
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50 | g_ivv : vertex_set_table; |
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51 | g_ivh : Register_set_table; |
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52 | g_pvv : vertex_set_table; |
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53 | g_pvh : Register_set_table; |
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54 | g_degree : vertex_priority_set; |
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55 | g_nmr : vertex_priority_set |
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56 | }. |
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57 | |
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58 | definition set_ivv ≝ |
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59 | λgraph. |
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60 | λivv: vertex_set_table. |
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61 | let regmap ≝ g_regmap graph in |
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62 | let ivh ≝ g_ivh graph in |
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63 | let pvv ≝ g_pvv graph in |
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64 | let pvh ≝ g_pvh graph in |
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65 | let degree ≝ g_degree graph in |
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66 | let nmr ≝ g_nmr graph in |
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67 | mk_graph |
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68 | regmap ivv ivh pvv pvh degree nmr. |
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69 | |
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70 | definition set_ivh ≝ |
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71 | λgraph. |
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72 | λivh: Register_set_table. |
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73 | let regmap ≝ g_regmap graph in |
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74 | let ivv ≝ g_ivv graph in |
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75 | let pvv ≝ g_pvv graph in |
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76 | let pvh ≝ g_pvh graph in |
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77 | let degree ≝ g_degree graph in |
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78 | let nmr ≝ g_nmr graph in |
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79 | mk_graph |
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80 | regmap ivv ivh pvv pvh degree nmr. |
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81 | |
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82 | definition set_degree ≝ |
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83 | λgraph. |
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84 | λdegree: vertex_priority_set. |
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85 | let regmap ≝ g_regmap graph in |
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86 | let ivv ≝ g_ivv graph in |
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87 | let ivh ≝ g_ivh graph in |
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88 | let pvv ≝ g_pvv graph in |
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89 | let pvh ≝ g_pvh graph in |
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90 | let nmr ≝ g_nmr graph in |
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91 | mk_graph |
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92 | regmap ivv ivh pvv pvh degree nmr. |
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93 | |
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94 | definition set_nmr ≝ |
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95 | λgraph. |
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96 | λnmr: vertex_priority_set. |
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97 | let regmap ≝ g_regmap graph in |
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98 | let ivv ≝ g_ivv graph in |
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99 | let ivh ≝ g_ivh graph in |
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100 | let pvv ≝ g_pvv graph in |
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101 | let pvh ≝ g_pvh graph in |
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102 | let degree ≝ g_degree graph in |
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103 | mk_graph |
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104 | regmap ivv ivh pvv pvh degree nmr. |
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105 | |
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106 | definition sg_neighboursv ≝ |
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107 | λgraph: graph. |
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108 | λv: vertex. |
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109 | set_tbl_find … v (g_ivv graph). |
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110 | |
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111 | definition sg_existsvv ≝ |
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112 | λgraph. |
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113 | λv1. |
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114 | λv2. |
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115 | match sg_neighboursv graph v2 with |
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116 | [ None ⇒ false (* XXX: ok? *) |
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117 | | Some neigh ⇒ set_member ? eq_nat v1 neigh |
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118 | ]. |
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119 | |
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120 | definition sg_neighboursh ≝ |
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121 | λgraph. |
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122 | λv. |
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123 | set_tbl_find ? ? v (g_ivh graph). |
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124 | |
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125 | definition sg_existsvh ≝ |
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126 | λgraph. |
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127 | λv. |
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128 | λh. |
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129 | match sg_neighboursh graph v with |
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130 | [ None ⇒ false (* XXX: ok? *) |
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131 | | Some neigh ⇒ set_member ? eq_Register h neigh |
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132 | ]. |
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133 | |
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134 | definition sg_degree ≝ |
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135 | λgraph. |
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136 | λv. |
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137 | match sg_neighboursv graph v with |
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138 | [ None ⇒ None ? |
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139 | | Some neigh ⇒ |
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140 | match sg_neighboursh graph v with |
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141 | [ None ⇒ None ? |
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142 | | Some neigh' ⇒ Some ? ((set_size … neigh) + (set_size … neigh')) |
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143 | ] |
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144 | ]. |
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145 | |
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146 | definition sg_hwregs ≝ |
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147 | λgraph: graph. |
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148 | let union ≝ λkey: vertex. set_union ? in |
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149 | set_tbl_fold vertex ? ? union (g_ivh graph) (set_empty Register). |
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150 | |
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151 | axiom sg_iter: Type[0]. (* XXX: todo when i can be bothered *) |
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152 | |
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153 | definition sg_mkvvi ≝ |
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154 | λgraph. |
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155 | λv1. |
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156 | λv2. |
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157 | set_ivv graph (set_tbl_homo_mkbiedge … v1 v2 (g_ivv graph)). |
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158 | |
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159 | definition sg_mkvv ≝ |
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160 | λgraph. |
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161 | λv1. |
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162 | λv2. |
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163 | if eq_nat v1 v2 then |
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164 | graph |
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165 | else if sg_existsvv graph v1 v2 then |
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166 | graph |
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167 | else |
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168 | sg_mkvvi graph v1 v2. |
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169 | |
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170 | definition sg_rmvv ≝ |
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171 | λgraph. |
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172 | λv1. |
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173 | λv2. |
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174 | set_ivv graph (set_tbl_homo_rmbiedge … v1 v2 (g_ivv graph)). |
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175 | |
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176 | definition sg_rmvvifx ≝ |
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177 | λgraph. |
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178 | λv1. |
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179 | λv2. |
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180 | if sg_existsvv graph v1 v2 then |
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181 | sg_rmvv graph v1 v2 |
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182 | else |
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183 | graph. |
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184 | |
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185 | definition sg_mkvhi ≝ |
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186 | λgraph. |
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187 | λv. |
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188 | λh. |
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189 | set_ivh graph (set_tbl_update … v (set_insert … h) (g_ivh graph)). |
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190 | |
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191 | definition sg_mkvh ≝ |
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192 | λgraph. |
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193 | λv. |
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194 | λh. |
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195 | if sg_existsvh graph v h then |
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196 | graph |
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197 | else |
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198 | sg_mkvhi graph v h. |
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199 | |
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200 | definition sg_rmvh ≝ |
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201 | λgraph. |
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202 | λv. |
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203 | λh. |
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204 | set_ivh graph (set_tbl_update … v (set_remove … h) (g_ivh graph)). |
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205 | |
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206 | definition sg_rmvhifx ≝ |
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207 | λgraph. |
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208 | λv. |
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209 | λh. |
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210 | if sg_existsvh graph v h then |
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211 | sg_rmvh graph v h |
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212 | else |
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213 | graph. |
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214 | |
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215 | definition sg_coalesce ≝ |
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216 | λg. |
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217 | λx. |
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218 | λy. |
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219 | match sg_neighboursv g x with |
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220 | [ None ⇒ None ? |
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221 | | Some neigh ⇒ |
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222 | let graph ≝ set_fold ? graph (λw. λg. |
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223 | sg_mkvv (sg_rmvv g x w) y w) neigh g |
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224 | in |
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225 | match sg_neighboursh g x with |
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226 | [ None ⇒ None ? |
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227 | | Some neigh ⇒ |
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228 | let graph ≝ set_fold ? ? (λh. λg. |
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229 | sg_mkvh (sg_rmvh g x h) y h) neigh g |
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230 | in |
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231 | Some … graph |
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232 | ] |
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233 | ]. |
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234 | |
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235 | definition sg_coalesceh ≝ |
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236 | λg. |
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237 | λx. |
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238 | λh. |
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239 | match sg_neighboursv g x with |
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240 | [ None ⇒ None ? |
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241 | | Some neigh ⇒ |
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242 | let graph ≝ set_fold ? graph (λw. λg. |
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243 | sg_mkvh (sg_rmvv g x w) w h) neigh g |
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244 | in |
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245 | match sg_neighboursh g x with |
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246 | [ None ⇒ None ? |
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247 | | Some neigh ⇒ |
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248 | let graph ≝ set_fold ? ? (λk. λg. |
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249 | sg_rmvh graph x k) neigh g |
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250 | in |
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251 | Some … graph |
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252 | ] |
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253 | ]. |
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254 | |
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255 | definition sg_remove ≝ |
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256 | λg. |
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257 | λx. |
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258 | match sg_neighboursv g x with |
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259 | [ None ⇒ None ? |
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260 | | Some neigh ⇒ |
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261 | let graph ≝ |
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262 | set_fold … (λw. λgraph. |
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263 | sg_rmvv graph x w) neigh g |
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264 | in |
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265 | match sg_neighboursh graph x with |
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266 | [ None ⇒ None ? |
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267 | | Some neigh ⇒ |
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268 | let graph ≝ set_fold … (λh. λg. |
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269 | sg_rmvh g x h) neigh graph |
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270 | in |
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271 | Some ? graph |
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272 | ] |
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273 | ]. |
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274 | |
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275 | definition ig_mkvvi ≝ |
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276 | λgraph. |
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277 | λv1. |
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278 | λv2. |
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279 | let graph ≝ sg_mkvvi graph v1 v2 in |
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280 | let graph ≝ sg_rmvvifx graph v1 v2 in |
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281 | let degree' ≝ pset_increment ? v1 (repr 1) (pset_increment ? v2 (repr 1) (g_degree graph)) in |
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282 | let nmr' ≝ pset_incrementifx ? v1 (repr 1) (pset_incrementifx ? v2 (repr 1) (g_nmr graph)) in |
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283 | set_degree (set_nmr graph nmr') degree'. |
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284 | |
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285 | definition ig_rmvv ≝ |
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286 | λgraph. |
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287 | λv1. |
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288 | λv2. |
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289 | let graph ≝ sg_rmvv graph v1 v2 in |
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290 | let degree' ≝ pset_increment ? v1 (neg (repr 1)) (pset_increment ? v2 (neg (repr 1)) (g_degree graph)) in |
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291 | let nmr' ≝ pset_incrementifx ? v1 (neg (repr 1)) (pset_incrementifx ? v2 (neg (repr 1)) (g_nmr graph)) in |
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292 | set_degree (set_nmr graph nmr') degree'. |
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293 | |
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294 | definition ig_mkvhi ≝ |
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295 | λgraph. |
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296 | λv. |
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297 | λh. |
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298 | let graph ≝ sg_mkvhi graph v h in |
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299 | let graph ≝ sg_rmvhifx graph v h in |
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300 | let degree ≝ pset_increment ? v (repr 1) (g_degree graph) in |
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301 | let nmr ≝ pset_incrementifx ? v (repr 1) (g_nmr graph) in |
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302 | set_degree (set_nmr graph nmr) degree. |
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303 | |
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304 | definition ig_rmvh ≝ |
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305 | λgraph. |
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306 | λv. |
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307 | λh. |
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308 | let graph ≝ sg_rmvh graph v h in |
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309 | let degree ≝ pset_increment ? v (neg (repr 1)) (g_degree graph) in |
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310 | let nmr ≝ pset_incrementifx ? v (neg (repr 1)) (g_nmr graph) in |
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311 | set_degree (set_nmr graph nmr) degree. |
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312 | |
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313 | definition pref_nmr ≝ |
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314 | λgraph. |
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315 | λv. |
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316 | match sg_neighboursv graph v with |
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317 | [ None ⇒ false (* XXX: ok? *) |
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318 | | Some neigh ⇒ |
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319 | match sg_neighboursh graph v with |
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320 | [ None ⇒ false |
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321 | | Some neigh' ⇒ |
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322 | andb (set_is_empty ? neigh) (set_is_empty ? neigh') |
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323 | ] |
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324 | ]. |
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325 | |
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326 | definition pref_mkcheck ≝ |
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327 | λgraph. |
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328 | λv. |
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329 | if pref_nmr graph v then |
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330 | let nmr' ≝ pset_remove ? v (g_nmr graph) in |
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331 | set_nmr graph nmr' |
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332 | else |
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333 | graph. |
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334 | |
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335 | definition pref_mkvvi ≝ |
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336 | λgraph. |
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337 | λv1. |
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338 | λv2. |
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339 | if sg_existsvv graph v1 v2 then |
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340 | graph |
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341 | else |
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342 | let graph ≝ pref_mkcheck graph v1 in |
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343 | let graph ≝ pref_mkcheck graph v2 in |
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344 | sg_mkvvi graph v1 v2. |
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345 | |
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346 | definition pref_mkvhi ≝ |
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347 | λgraph. |
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348 | λv. |
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349 | λh. |
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350 | if sg_existsvh graph v h then |
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351 | graph |
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352 | else |
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353 | let graph ≝ pref_mkcheck graph v in |
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354 | sg_mkvhi graph v h. |
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355 | |
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356 | (* XXX: look at this carefully *) |
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357 | definition pref_rmcheck ≝ |
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358 | λgraph. |
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359 | λv. |
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360 | if pref_nmr graph v then |
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361 | match pset_lookup ? v (g_degree graph) with |
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362 | [ None ⇒ graph (* XXX: ok? *) |
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363 | | Some pref ⇒ |
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364 | let nmr ≝ pset_insert ? v pref (g_nmr graph) in |
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365 | set_nmr graph nmr |
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366 | ] |
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367 | else |
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368 | graph. |
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369 | |
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370 | definition pref_rmvv ≝ |
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371 | λgraph. |
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372 | λv1. |
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373 | λv2. |
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374 | let graph ≝ sg_rmvv graph v1 v2 in |
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375 | let graph ≝ pref_rmcheck graph v1 in |
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376 | let graph ≝ pref_rmcheck graph v2 in |
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377 | graph. |
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378 | |
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379 | definition pref_rmvh ≝ |
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380 | λgraph. |
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381 | λv. |
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382 | λh. |
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383 | let graph ≝ sg_rmvh graph v h in |
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384 | let graph ≝ pref_rmcheck graph v in |
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385 | graph. |
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386 | |
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387 | definition ig_ipp ≝ sg_neighboursv. |
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388 | definition ig_iph ≝ sg_neighboursh. |
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389 | definition ig_ppp ≝ sg_neighboursv. |
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390 | definition ig_pph ≝ sg_neighboursh. |
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391 | definition ig_degree ≝ λgraph. λv. pset_lookup ? v (g_degree graph). |
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392 | definition ig_lowest ≝ λgraph. pset_lowest ? (g_degree graph). |
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393 | definition ig_lowest_non_move_related ≝ λgraph. pset_lowest ? (g_nmr graph). |
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394 | definition ig_fold ≝ λA: Type[0]. λf: vertex → A → A. λgraph. λaccu. |
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395 | rt_fold … (λv. λ_. λaccu. f v accu) (g_regmap graph) accu. |
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396 | |
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397 | definition ig_minimum: ∀a: Type[0]. (a → a → order) → (vertex → a) → graph → option vertex ≝ |
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398 | λa: Type[0]. |
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399 | λcompare: a → a → order. |
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400 | λf: vertex → a. |
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401 | λgraph. |
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402 | let folded ≝ ig_fold … (λw. λaccu. |
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403 | let dw ≝ f w in |
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404 | match accu with |
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405 | [ None ⇒ Some … 〈dw, w〉 |
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406 | | Some dv_v ⇒ |
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407 | let 〈dv, v〉 ≝ dv_v in |
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408 | match compare dw dv with |
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409 | [ order_lt ⇒ Some … 〈dw, w〉 |
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410 | | _ ⇒ accu |
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411 | ] |
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412 | ]) graph (None …) |
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413 | in |
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414 | match folded with |
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415 | [ None ⇒ None … |
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416 | | Some ignore_v ⇒ |
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417 | let 〈ignore, v〉 ≝ ignore_v in |
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418 | Some … v |
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419 | ]. |
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420 | |
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421 | definition ig_ppedge ≝ vertex × vertex. |
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422 | |
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423 | definition ig_pppick ≝ λgraph. λp. set_tbl_pick … (g_pvv graph) p. |
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424 | |
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425 | definition ig_phedge ≝ vertex × Register. |
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426 | |
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427 | definition ig_phpick ≝ λgraph. λp. set_tbl_pick … (g_pvh graph) p. |
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428 | |
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429 | definition ig_create ≝ |
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430 | λregs. |
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431 | let 〈ignore_int, table'', priority''〉 ≝ |
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432 | foldr … (λr. λv_table_priority'. |
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433 | let 〈v, table', priority'〉 ≝ v_table_priority' in |
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434 | let table'' ≝ rt_add r v table' in |
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435 | let priority'' ≝ pset_insert ? v 0 priority' in |
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436 | 〈v + 1, table'', priority''〉) 〈0, rt_empty …, pset_empty …〉 regs |
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437 | in |
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438 | mk_graph table'' (set_tbl_empty …) (set_tbl_empty …) (set_tbl_empty …) |
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439 | (set_tbl_empty …) priority'' priority''. |
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440 | definition ig_lookup ≝ λgraph. λr. rt_backward r (g_regmap graph). |
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441 | definition ig_registers ≝ λgraph. λv. rt_forward v (g_regmap graph). |
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442 | definition ig_mkipp ≝ |
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443 | λgraph. |
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444 | λregs1. |
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445 | λregs2. |
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446 | set_fold … (λr1. λgraph. |
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447 | let v1 ≝ ig_lookup graph r1 in |
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448 | set_fold … (λr2. λgraph. |
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449 | sg_mkvv graph v1 (ig_lookup graph r2) |
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450 | ) regs2 graph |
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451 | ) regs1 graph. |
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452 | definition ig_mkiph ≝ |
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453 | λgraph. |
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454 | λregs. |
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455 | λhwregs. |
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456 | set_fold … (λr. λgraph. |
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457 | let v ≝ ig_lookup graph r in |
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458 | set_fold … (λh. λgraph. |
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459 | sg_mkvh graph v h |
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460 | ) hwregs graph |
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461 | ) regs graph. |
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462 | definition ig_mki ≝ |
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463 | λgraph. |
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464 | λregs1_hwregs1. |
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465 | λregs2_hwregs2. |
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466 | let 〈regs1, hwregs1〉 ≝ regs1_hwregs1 in |
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467 | let 〈regs2, hwregs2〉 ≝ regs2_hwregs2 in |
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468 | let graph ≝ ig_mkipp graph regs1 regs2 in |
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469 | let graph ≝ ig_mkiph graph regs1 hwregs2 in |
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470 | let graph ≝ ig_mkiph graph regs2 hwregs1 in |
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471 | graph. |
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472 | definition ig_mkppp ≝ |
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473 | λgraph. |
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474 | λr1. |
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475 | λr2. |
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476 | let v1 ≝ ig_lookup graph r1 in |
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477 | let v2 ≝ ig_lookup graph r2 in |
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478 | let graph ≝ sg_mkvv graph v1 v2 in |
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479 | graph. |
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480 | definition ig_mkpph ≝ |
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481 | λgraph. |
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482 | λr. |
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483 | λh. |
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484 | let v ≝ ig_lookup graph r in |
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485 | let graph ≝ sg_mkvh graph v h in |
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486 | graph. |
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487 | (* |
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488 | (* XXX: precondition: |
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489 | x \not\eq y |
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490 | existsvv graph x y == false i.e. coalesce interfering edges *) |
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491 | definition ig_coalesce ≝ |
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492 | λgraph. |
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493 | λx. |
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494 | λy. |
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495 | let graph ≝ sg_coalesce graph x y in |
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496 | |
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497 | let coalesce graph x y = |
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498 | |
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499 | assert (x <> y); (* attempt to coalesce one vertex with itself *) |
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500 | assert (not (interference#existsvv graph x y)); (* attempt to coalesce two interfering vertices *) |
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501 | |
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502 | (* Perform coalescing in the two subgraphs. *) |
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503 | |
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504 | let graph = interference#coalesce graph x y in |
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505 | let graph = preference#coalesce graph x y in |
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506 | |
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507 | (* Remove [x] from all tables. *) |
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508 | |
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509 | { |
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510 | graph with |
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511 | regmap = RegMap.coalesce x y graph.regmap; |
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512 | ivh = Vertex.Map.remove x graph.ivh; |
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513 | pvh = Vertex.Map.remove x graph.pvh; |
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514 | degree = PrioritySet.remove x graph.degree; |
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515 | nmr = PrioritySet.remove x graph.nmr; |
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516 | } |
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517 | |
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518 | axiom ig_mkppp: graph → register → register → graph. |
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519 | axiom ig_mkpph: graph → register → Register → graph. |
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520 | axiom ig_coalesce: graph → vertex → vertex → graph. |
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521 | axiom ig_coalesceh: graph → vertex → Register → graph. |
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522 | axiom ig_remove: graph → vertex → graph. |
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523 | axiom ig_freeze: graph → vertex → graph. |
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524 | axiom ig_restrict: graph → (vertex → bool) → graph. |
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525 | axiom ig_droph: graph → graph. |
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526 | axiom ig_lookup: graph → register → vertex. |
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527 | axiom ig_registers: graph → vertex → list register. |
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528 | axiom ig_degree: graph → vertex → nat. |
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529 | axiom ig_lowest: graph → option (vertex × nat). |
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530 | axiom ig_lowest_non_move_related: graph → option (vertex × nat). |
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531 | axiom ig_minimum: ∀A: Type[0]. ∀ord: A → A → order. (vertex → A) → |
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532 | graph → option vertex. |
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533 | axiom ig_fold: ∀A: Type[0]. (vertex → A → A) → graph → A → A. |
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534 | axiom ig_ipp: graph → vertex → vertex_set. |
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535 | axiom ig_iph: graph → vertex → list Register. |
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536 | axiom ig_ppp: graph → vertex → vertex_set. |
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537 | axiom ig_pph: graph → vertex → list Register. |
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538 | definition ig_ppedge ≝ vertex × vertex. |
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539 | axiom ig_pppick: graph → (ig_ppedge → bool) → option ig_ppedge. |
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540 | definition ig_phedge ≝ vertex × Register. |
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541 | axiom ig_phpick: graph → (ig_phedge → bool) → option ig_phedge. |
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542 | *) |
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543 | *) |
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