src/HOL/Data_Structures/Trie_Ternary.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Trie_Ternary.thy	Tue Jun 25 11:08:00 2024 +0200
@@ -0,0 +1,123 @@
+section "Ternary Tries"
+
+theory Trie_Ternary
+imports
+  Tree_Map
+  Trie_Fun
+begin
+
+text \<open>An implementation of tries for an arbitrary alphabet \<open>'a\<close> where the mapping
+from an element of type \<open>'a\<close> to the sub-trie is implemented by an (unbalanced) binary search tree.
+In principle, other search trees (e.g. red-black trees) work just as well,
+with some small adjustments (Exercise!).
+
+This is an implementation of the ``ternary search trees'' by Bentley and Sedgewick
+[SODA 1997, Dr. Dobbs 1998]. The name derives from the fact that a node in the BST can now
+be drawn to have 3 children, where the middle child is the sub-trie that the node maps
+its key to. Hence the name \<open>trie3\<close>.
+
+Example from @{url "https://en.wikipedia.org/wiki/Ternary_search_tree#Description"}:
+
+          c
+        / | \
+       a  u  h
+       |  |  | \
+       t. t  e. u
+     /  / |   / |
+    s. p. e. i. s.
+
+Characters with a dot are final.
+Thus the tree represents the set of strings "cute","cup","at","as","he","us" and "i".
+\<close>
+
+datatype 'a trie3 = Nd3 bool "('a * 'a trie3) tree"
+
+text \<open>The development below works almost verbatim for any search tree implementation, eg \<open>RBT_Map\<close>,
+and not just \<open>Tree_Map\<close>, except for the termination lemma \<open>lookup_size\<close>.\<close>
+
+term size_tree
+lemma lookup_size[termination_simp]:
+  fixes t :: "('a::linorder * 'a trie3) tree"
+  shows "lookup t a = Some b \<Longrightarrow> size b < Suc (size_tree (\<lambda>ab. Suc (size (snd( ab)))) t)"
+apply(induction t a rule: lookup.induct)
+apply(auto split: if_splits)
+done
+
+
+definition empty3 :: "'a trie3" where
+[simp]: "empty3 = Nd3 False Leaf"
+
+fun isin3 :: "('a::linorder) trie3 \<Rightarrow> 'a list \<Rightarrow> bool" where
+"isin3 (Nd3 b m) [] = b" |
+"isin3 (Nd3 b m) (x # xs) = (case lookup m x of None \<Rightarrow> False | Some t \<Rightarrow> isin3 t xs)"
+
+fun insert3 :: "('a::linorder) list \<Rightarrow> 'a trie3 \<Rightarrow> 'a trie3" where
+"insert3 [] (Nd3 b m) = Nd3 True m" |
+"insert3 (x#xs) (Nd3 b m) =
+  Nd3 b (update x (insert3 xs (case lookup m x of None \<Rightarrow> empty3 | Some t \<Rightarrow> t)) m)"
+
+fun delete3 :: "('a::linorder) list \<Rightarrow> 'a trie3 \<Rightarrow> 'a trie3" where
+"delete3 [] (Nd3 b m) = Nd3 False m" |
+"delete3 (x#xs) (Nd3 b m) = Nd3 b
+   (case lookup m x of
+      None \<Rightarrow> m |
+      Some t \<Rightarrow> update x (delete3 xs t) m)"
+
+
+subsection "Correctness"
+
+text \<open>Proof by stepwise refinement. First abs3tract to type @{typ "'a trie"}.\<close>
+
+fun abs3 :: "'a::linorder trie3 \<Rightarrow> 'a trie" where
+"abs3 (Nd3 b t) = Nd b (\<lambda>a. map_option abs3 (lookup t a))"
+
+fun invar3 :: "('a::linorder)trie3 \<Rightarrow> bool" where
+"invar3 (Nd3 b m) = (M.invar m \<and> (\<forall>a t. lookup m a = Some t \<longrightarrow> invar3 t))"
+
+lemma isin_abs3: "isin3 t xs = isin (abs3 t) xs"
+apply(induction t xs rule: isin3.induct)
+apply(auto split: option.split)
+done
+
+lemma abs3_insert3: "invar3 t \<Longrightarrow> abs3(insert3 xs t) = insert xs (abs3 t)"
+apply(induction xs t rule: insert3.induct)
+apply(auto simp: M.map_specs Tree_Set.empty_def[symmetric] split: option.split)
+done
+
+lemma abs3_delete3: "invar3 t \<Longrightarrow> abs3(delete3 xs t) = delete xs (abs3 t)"
+apply(induction xs t rule: delete3.induct)
+apply(auto simp: M.map_specs split: option.split)
+done
+
+lemma invar3_insert3: "invar3 t \<Longrightarrow> invar3 (insert3 xs t)"
+apply(induction xs t rule: insert3.induct)
+apply(auto simp: M.map_specs Tree_Set.empty_def[symmetric] split: option.split)
+done
+
+lemma invar3_delete3: "invar3 t \<Longrightarrow> invar3 (delete3 xs t)"
+apply(induction xs t rule: delete3.induct)
+apply(auto simp: M.map_specs split: option.split)
+done
+
+text \<open>Overall correctness w.r.t. the \<open>Set\<close> ADT:\<close>
+
+interpretation S2: Set
+where empty = empty3 and isin = isin3 and insert = insert3 and delete = delete3
+and set = "set o abs3" and invar = invar3
+proof (standard, goal_cases)
+  case 1 show ?case by (simp add: isin_case split: list.split)
+next
+  case 2 thus ?case by (simp add: isin_abs3)
+next
+  case 3 thus ?case by (simp add: set_insert abs3_insert3 del: set_def)
+next
+  case 4 thus ?case by (simp add: set_delete abs3_delete3 del: set_def)
+next
+  case 5 thus ?case by (simp add: M.map_specs Tree_Set.empty_def[symmetric])
+next
+  case 6 thus ?case by (simp add: invar3_insert3)
+next
+  case 7 thus ?case by (simp add: invar3_delete3)
+qed
+
+end