Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
Size:
100K - 1M
License:
File size: 10,092 Bytes
4365a98 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 |
(*
File: Interval_Tree.thy
Author: Bohua Zhan
*)
section \<open>Interval tree\<close>
theory Interval_Tree
imports Lists_Ex Interval
begin
text \<open>
Functional version of interval tree. This is an augmented data
structure on top of regular binary search trees (see BST.thy).
See \cite[Section 14.3]{cormen2009introduction} for a reference.
\<close>
subsection \<open>Definition of an interval tree\<close>
datatype interval_tree =
Tip
| Node (lsub: interval_tree) (val: "nat idx_interval") (tmax: nat) (rsub: interval_tree)
where
"tmax Tip = 0"
setup \<open>add_resolve_prfstep @{thm interval_tree.distinct(1)}\<close>
setup \<open>fold add_rewrite_rule @{thms interval_tree.sel}\<close>
setup \<open>add_forward_prfstep @{thm interval_tree.collapse}\<close>
setup \<open>add_var_induct_rule @{thm interval_tree.induct}\<close>
subsection \<open>Inorder traversal, and set of elements of a tree\<close>
fun in_traverse :: "interval_tree \<Rightarrow> nat idx_interval list" where
"in_traverse Tip = []"
| "in_traverse (Node l it m r) = in_traverse l @ it # in_traverse r"
setup \<open>fold add_rewrite_rule @{thms in_traverse.simps}\<close>
fun tree_set :: "interval_tree \<Rightarrow> nat idx_interval set" where
"tree_set Tip = {}"
| "tree_set (Node l it m r) = {it} \<union> tree_set l \<union> tree_set r"
setup \<open>fold add_rewrite_rule @{thms tree_set.simps}\<close>
fun tree_sorted :: "interval_tree \<Rightarrow> bool" where
"tree_sorted Tip = True"
| "tree_sorted (Node l it m r) = ((\<forall>x\<in>tree_set l. x < it) \<and> (\<forall>x\<in>tree_set r. it < x)
\<and> tree_sorted l \<and> tree_sorted r)"
setup \<open>fold add_rewrite_rule @{thms tree_sorted.simps}\<close>
lemma tree_sorted_lr [forward]:
"tree_sorted (Node l it m r) \<Longrightarrow> tree_sorted l \<and> tree_sorted r" by auto2
lemma tree_sortedD1 [forward]:
"tree_sorted (Node l it m r) \<Longrightarrow> x \<in> tree_set l \<Longrightarrow> x < it" by auto2
lemma tree_sortedD2 [forward]:
"tree_sorted (Node l it m r) \<Longrightarrow> x \<in> tree_set r \<Longrightarrow> x > it" by auto2
lemma inorder_preserve_set [rewrite]:
"tree_set t = set (in_traverse t)"
@proof @induct t @qed
lemma inorder_sorted [rewrite]:
"tree_sorted t \<longleftrightarrow> strict_sorted (in_traverse t)"
@proof @induct t @qed
text \<open>Use definition in terms of in\_traverse from now on.\<close>
setup \<open>fold del_prfstep_thm (@{thms tree_set.simps} @ @{thms tree_sorted.simps})\<close>
subsection \<open>Invariant on the maximum\<close>
definition max3 :: "nat idx_interval \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" where [rewrite]:
"max3 it b c = max (high (int it)) (max b c)"
fun tree_max_inv :: "interval_tree \<Rightarrow> bool" where
"tree_max_inv Tip = True"
| "tree_max_inv (Node l it m r) \<longleftrightarrow> (tree_max_inv l \<and> tree_max_inv r \<and> m = max3 it (tmax l) (tmax r))"
setup \<open>fold add_rewrite_rule @{thms tree_max_inv.simps}\<close>
lemma tree_max_is_max [resolve]:
"tree_max_inv t \<Longrightarrow> it \<in> tree_set t \<Longrightarrow> high (int it) \<le> tmax t"
@proof @induct t @qed
lemma tmax_exists [backward]:
"tree_max_inv t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> \<exists>p\<in>tree_set t. high (int p) = tmax t"
@proof @induct t @with
@subgoal "t = Node l it m r"
@case "l = Tip" @with @case "r = Tip" @end
@case "r = Tip"
@endgoal @end
@qed
text \<open>For insertion\<close>
lemma max3_insert [rewrite]: "max3 it 0 0 = high (int it)" by auto2
setup \<open>del_prfstep_thm @{thm max3_def}\<close>
subsection \<open>Condition on the values\<close>
definition tree_interval_inv :: "interval_tree \<Rightarrow> bool" where [rewrite]:
"tree_interval_inv t \<longleftrightarrow> (\<forall>p\<in>tree_set t. is_interval (int p))"
definition is_interval_tree :: "interval_tree \<Rightarrow> bool" where [rewrite]:
"is_interval_tree t \<longleftrightarrow> (tree_sorted t \<and> tree_max_inv t \<and> tree_interval_inv t)"
lemma is_interval_tree_lr [forward]:
"is_interval_tree (Node l x m r) \<Longrightarrow> is_interval_tree l \<and> is_interval_tree r" by auto2
subsection \<open>Insertion on trees\<close>
fun insert :: "nat idx_interval \<Rightarrow> interval_tree \<Rightarrow> interval_tree" where
"insert x Tip = Node Tip x (high (int x)) Tip"
| "insert x (Node l y m r) =
(if x = y then Node l y m r
else if x < y then
let l' = insert x l in
Node l' y (max3 y (tmax l') (tmax r)) r
else
let r' = insert x r in
Node l y (max3 y (tmax l) (tmax r')) r')"
setup \<open>fold add_rewrite_rule @{thms insert.simps}\<close>
lemma tree_insert_in_traverse [rewrite]:
"tree_sorted t \<Longrightarrow> in_traverse (insert x t) = ordered_insert x (in_traverse t)"
@proof @induct t @qed
lemma tree_insert_max_inv [forward]:
"tree_max_inv t \<Longrightarrow> tree_max_inv (insert x t)"
@proof @induct t @qed
text \<open>Correctness of insertion.\<close>
theorem tree_insert_all_inv [forward]:
"is_interval_tree t \<Longrightarrow> is_interval (int it) \<Longrightarrow> is_interval_tree (insert it t)" by auto2
theorem tree_insert_on_set [rewrite]:
"tree_sorted t \<Longrightarrow> tree_set (insert it t) = {it} \<union> tree_set t" by auto2
subsection \<open>Deletion on trees\<close>
fun del_min :: "interval_tree \<Rightarrow> nat idx_interval \<times> interval_tree" where
"del_min Tip = undefined"
| "del_min (Node lt v m rt) =
(if lt = Tip then (v, rt) else
let lt' = snd (del_min lt) in
(fst (del_min lt), Node lt' v (max3 v (tmax lt') (tmax rt)) rt))"
setup \<open>add_rewrite_rule @{thm del_min.simps(2)}\<close>
setup \<open>register_wellform_data ("del_min t", ["t \<noteq> Tip"])\<close>
lemma delete_min_del_hd:
"t \<noteq> Tip \<Longrightarrow> fst (del_min t) # in_traverse (snd (del_min t)) = in_traverse t"
@proof @induct t @qed
setup \<open>add_forward_prfstep_cond @{thm delete_min_del_hd} [with_term "in_traverse (snd (del_min ?t))"]\<close>
lemma delete_min_max_inv [forward_arg]:
"tree_max_inv t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> tree_max_inv (snd (del_min t))"
@proof @induct t @qed
lemma delete_min_on_set:
"t \<noteq> Tip \<Longrightarrow> {fst (del_min t)} \<union> tree_set (snd (del_min t)) = tree_set t" by auto2
setup \<open>add_forward_prfstep_cond @{thm delete_min_on_set} [with_term "tree_set (snd (del_min ?t))"]\<close>
lemma delete_min_interval_inv [forward_arg]:
"tree_interval_inv t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> tree_interval_inv (snd (del_min t))" by auto2
lemma delete_min_all_inv [forward_arg]:
"is_interval_tree t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> is_interval_tree (snd (del_min t))" by auto2
fun delete_elt_tree :: "interval_tree \<Rightarrow> interval_tree" where
"delete_elt_tree Tip = undefined"
| "delete_elt_tree (Node lt x m rt) =
(if lt = Tip then rt else if rt = Tip then lt else
let x' = fst (del_min rt);
rt' = snd (del_min rt);
m' = max3 x' (tmax lt) (tmax rt') in
Node lt (fst (del_min rt)) m' rt')"
setup \<open>add_rewrite_rule @{thm delete_elt_tree.simps(2)}\<close>
lemma delete_elt_in_traverse [rewrite]:
"in_traverse (delete_elt_tree (Node lt x m rt)) = in_traverse lt @ in_traverse rt" by auto2
lemma delete_elt_max_inv [forward_arg]:
"tree_max_inv t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> tree_max_inv (delete_elt_tree t)" by auto2
lemma delete_elt_on_set [rewrite]:
"t \<noteq> Tip \<Longrightarrow> tree_set (delete_elt_tree (Node lt x m rt)) = tree_set lt \<union> tree_set rt" by auto2
lemma delete_elt_interval_inv [forward_arg]:
"tree_interval_inv t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> tree_interval_inv (delete_elt_tree t)" by auto2
lemma delete_elt_all_inv [forward_arg]:
"is_interval_tree t \<Longrightarrow> t \<noteq> Tip \<Longrightarrow> is_interval_tree (delete_elt_tree t)" by auto2
fun delete :: "nat idx_interval \<Rightarrow> interval_tree \<Rightarrow> interval_tree" where
"delete x Tip = Tip"
| "delete x (Node l y m r) =
(if x = y then delete_elt_tree (Node l y m r)
else if x < y then
let l' = delete x l;
m' = max3 y (tmax l') (tmax r) in Node l' y m' r
else
let r' = delete x r;
m' = max3 y (tmax l) (tmax r') in Node l y m' r')"
setup \<open>fold add_rewrite_rule @{thms delete.simps}\<close>
lemma tree_delete_in_traverse [rewrite]:
"tree_sorted t \<Longrightarrow> in_traverse (delete x t) = remove_elt_list x (in_traverse t)"
@proof @induct t @qed
lemma tree_delete_max_inv [forward]:
"tree_max_inv t \<Longrightarrow> tree_max_inv (delete x t)"
@proof @induct t @qed
text \<open>Correctness of deletion.\<close>
theorem tree_delete_all_inv [forward]:
"is_interval_tree t \<Longrightarrow> is_interval_tree (delete x t)"
@proof @have "tree_set (delete x t) \<subseteq> tree_set t" @qed
theorem tree_delete_on_set [rewrite]:
"tree_sorted t \<Longrightarrow> tree_set (delete x t) = tree_set t - {x}" by auto2
subsection \<open>Search on interval trees\<close>
fun search :: "interval_tree \<Rightarrow> nat interval \<Rightarrow> bool" where
"search Tip x = False"
| "search (Node l y m r) x =
(if is_overlap (int y) x then True
else if l \<noteq> Tip \<and> tmax l \<ge> low x then search l x
else search r x)"
setup \<open>fold add_rewrite_rule @{thms search.simps}\<close>
text \<open>Correctness of search\<close>
theorem search_correct [rewrite]:
"is_interval_tree t \<Longrightarrow> is_interval x \<Longrightarrow> search t x \<longleftrightarrow> has_overlap (tree_set t) x"
@proof
@induct t @with
@subgoal "t = Node l y m r"
@let "t = Node l y m r"
@case "is_overlap (int y) x"
@case "l \<noteq> Tip \<and> tmax l \<ge> low x" @with
@obtain "p\<in>tree_set l" where "high (int p) = tmax l"
@case "is_overlap (int p) x"
@end
@case "l = Tip"
@endgoal
@end
@qed
end
|