6 |
6 |
7 theory Lift_RBT |
7 theory Lift_RBT |
8 imports Main "~~/src/HOL/Library/RBT_Impl" |
8 imports Main "~~/src/HOL/Library/RBT_Impl" |
9 begin |
9 begin |
10 |
10 |
11 (* TODO: Replace the ancient Library/RBT theory by this example of the lifting and transfer mechanism. *) |
11 (* Moved to ~~/HOL/Library/RBT" *) |
12 |
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13 subsection {* Type definition *} |
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14 |
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15 typedef (open) ('a, 'b) rbt = "{t :: ('a\<Colon>linorder, 'b) RBT_Impl.rbt. is_rbt t}" |
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16 morphisms impl_of RBT |
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17 proof - |
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18 have "RBT_Impl.Empty \<in> ?rbt" by simp |
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19 then show ?thesis .. |
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20 qed |
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21 |
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22 lemma rbt_eq_iff: |
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23 "t1 = t2 \<longleftrightarrow> impl_of t1 = impl_of t2" |
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24 by (simp add: impl_of_inject) |
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25 |
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26 lemma rbt_eqI: |
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27 "impl_of t1 = impl_of t2 \<Longrightarrow> t1 = t2" |
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28 by (simp add: rbt_eq_iff) |
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29 |
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30 lemma is_rbt_impl_of [simp, intro]: |
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31 "is_rbt (impl_of t)" |
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32 using impl_of [of t] by simp |
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33 |
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34 lemma RBT_impl_of [simp, code abstype]: |
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35 "RBT (impl_of t) = t" |
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36 by (simp add: impl_of_inverse) |
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37 |
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38 subsection {* Primitive operations *} |
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39 |
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40 setup_lifting type_definition_rbt |
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41 |
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42 lift_definition lookup :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a \<rightharpoonup> 'b" is "rbt_lookup" |
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43 by simp |
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44 |
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45 lift_definition empty :: "('a\<Colon>linorder, 'b) rbt" is RBT_Impl.Empty |
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46 by (simp add: empty_def) |
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47 |
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48 lift_definition insert :: "'a\<Colon>linorder \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_insert" |
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49 by simp |
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50 |
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51 lift_definition delete :: "'a\<Colon>linorder \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_delete" |
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52 by simp |
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53 |
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54 lift_definition entries :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a \<times> 'b) list" is RBT_Impl.entries |
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55 by simp |
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56 |
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57 lift_definition keys :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'a list" is RBT_Impl.keys |
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58 by simp |
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59 |
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60 lift_definition bulkload :: "('a\<Colon>linorder \<times> 'b) list \<Rightarrow> ('a, 'b) rbt" is "rbt_bulkload" |
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61 by simp |
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62 |
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63 lift_definition map_entry :: "'a \<Rightarrow> ('b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is rbt_map_entry |
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64 by simp |
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65 |
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66 lift_definition map :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is RBT_Impl.map |
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67 by simp |
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68 |
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69 lift_definition fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a\<Colon>linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" is RBT_Impl.fold |
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70 by simp |
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71 |
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72 lift_definition union :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt" is "rbt_union" |
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73 by (simp add: rbt_union_is_rbt) |
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74 |
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75 lift_definition foldi :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a :: linorder, 'b) rbt \<Rightarrow> 'c \<Rightarrow> 'c" |
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76 is RBT_Impl.foldi by simp |
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77 |
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78 export_code lookup empty insert delete entries keys bulkload map_entry map fold union foldi in SML |
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79 |
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80 subsection {* Derived operations *} |
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81 |
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82 definition is_empty :: "('a\<Colon>linorder, 'b) rbt \<Rightarrow> bool" where |
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83 [code]: "is_empty t = (case impl_of t of RBT_Impl.Empty \<Rightarrow> True | _ \<Rightarrow> False)" |
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84 |
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85 |
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86 subsection {* Abstract lookup properties *} |
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87 |
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88 lemma lookup_RBT: |
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89 "is_rbt t \<Longrightarrow> lookup (RBT t) = rbt_lookup t" |
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90 by (simp add: lookup_def RBT_inverse) |
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91 |
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92 lemma lookup_impl_of: |
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93 "rbt_lookup (impl_of t) = lookup t" |
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94 by transfer (rule refl) |
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95 |
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96 lemma entries_impl_of: |
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97 "RBT_Impl.entries (impl_of t) = entries t" |
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98 by transfer (rule refl) |
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99 |
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100 lemma keys_impl_of: |
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101 "RBT_Impl.keys (impl_of t) = keys t" |
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102 by transfer (rule refl) |
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103 |
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104 lemma lookup_empty [simp]: |
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105 "lookup empty = Map.empty" |
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106 by (simp add: empty_def lookup_RBT fun_eq_iff) |
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107 |
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108 lemma lookup_insert [simp]: |
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109 "lookup (insert k v t) = (lookup t)(k \<mapsto> v)" |
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110 by transfer (rule rbt_lookup_rbt_insert) |
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111 |
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112 lemma lookup_delete [simp]: |
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113 "lookup (delete k t) = (lookup t)(k := None)" |
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114 by transfer (simp add: rbt_lookup_rbt_delete restrict_complement_singleton_eq) |
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115 |
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116 lemma map_of_entries [simp]: |
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117 "map_of (entries t) = lookup t" |
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118 by transfer (simp add: map_of_entries) |
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119 |
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120 lemma entries_lookup: |
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121 "entries t1 = entries t2 \<longleftrightarrow> lookup t1 = lookup t2" |
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122 by transfer (simp add: entries_rbt_lookup) |
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123 |
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124 lemma lookup_bulkload [simp]: |
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125 "lookup (bulkload xs) = map_of xs" |
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126 by transfer (rule rbt_lookup_rbt_bulkload) |
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127 |
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128 lemma lookup_map_entry [simp]: |
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129 "lookup (map_entry k f t) = (lookup t)(k := Option.map f (lookup t k))" |
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130 by transfer (rule rbt_lookup_rbt_map_entry) |
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131 |
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132 lemma lookup_map [simp]: |
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133 "lookup (map f t) k = Option.map (f k) (lookup t k)" |
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134 by transfer (rule rbt_lookup_map) |
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135 |
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136 lemma fold_fold: |
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137 "fold f t = List.fold (prod_case f) (entries t)" |
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138 by transfer (rule RBT_Impl.fold_def) |
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139 |
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140 lemma impl_of_empty: |
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141 "impl_of empty = RBT_Impl.Empty" |
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142 by transfer (rule refl) |
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143 |
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144 lemma is_empty_empty [simp]: |
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145 "is_empty t \<longleftrightarrow> t = empty" |
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146 unfolding is_empty_def by transfer (simp split: rbt.split) |
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147 |
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148 lemma RBT_lookup_empty [simp]: (*FIXME*) |
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149 "rbt_lookup t = Map.empty \<longleftrightarrow> t = RBT_Impl.Empty" |
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150 by (cases t) (auto simp add: fun_eq_iff) |
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151 |
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152 lemma lookup_empty_empty [simp]: |
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153 "lookup t = Map.empty \<longleftrightarrow> t = empty" |
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154 by transfer (rule RBT_lookup_empty) |
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155 |
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156 lemma sorted_keys [iff]: |
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157 "sorted (keys t)" |
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158 by transfer (simp add: RBT_Impl.keys_def rbt_sorted_entries) |
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159 |
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160 lemma distinct_keys [iff]: |
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161 "distinct (keys t)" |
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162 by transfer (simp add: RBT_Impl.keys_def distinct_entries) |
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163 |
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164 lemma finite_dom_lookup [simp, intro!]: "finite (dom (lookup t))" |
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165 by transfer simp |
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166 |
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167 lemma lookup_union: "lookup (union s t) = lookup s ++ lookup t" |
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168 by transfer (simp add: rbt_lookup_rbt_union) |
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169 |
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170 lemma lookup_in_tree: "(lookup t k = Some v) = ((k, v) \<in> set (entries t))" |
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171 by transfer (simp add: rbt_lookup_in_tree) |
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172 |
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173 lemma keys_entries: "(k \<in> set (keys t)) = (\<exists>v. (k, v) \<in> set (entries t))" |
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174 by transfer (simp add: keys_entries) |
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175 |
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176 lemma fold_def_alt: |
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177 "fold f t = List.fold (prod_case f) (entries t)" |
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178 by transfer (auto simp: RBT_Impl.fold_def) |
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179 |
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180 lemma distinct_entries: "distinct (List.map fst (entries t))" |
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181 by transfer (simp add: distinct_entries) |
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182 |
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183 lemma non_empty_keys: "t \<noteq> Lift_RBT.empty \<Longrightarrow> keys t \<noteq> []" |
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184 by transfer (simp add: non_empty_rbt_keys) |
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185 |
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186 lemma keys_def_alt: |
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187 "keys t = List.map fst (entries t)" |
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188 by transfer (simp add: RBT_Impl.keys_def) |
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189 |
12 |
190 end |
13 end |