--- a/src/HOL/Data_Structures/Trie_Map.thy Mon Jun 24 22:52:54 2024 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,127 +0,0 @@
-section "Tries via Search Trees"
-
-theory Trie_Map
-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 a binary search tree.
-Although this implementation uses maps implemented by red-black trees it works for any
-implementation of maps.
-
-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>In principle one should be able to given an implementation of tries
-once and for all for any map implementation and not just for a specific one (unbalanced trees) as done here.
-But because the map (@{type tree}) is used in a datatype, the HOL type system does not support this.
-
-However, the development below works verbatim for any map 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
--- /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
--- a/src/HOL/ROOT Mon Jun 24 22:52:54 2024 +0200
+++ b/src/HOL/ROOT Tue Jun 25 11:08:00 2024 +0200
@@ -307,8 +307,8 @@
Set2_Join_RBT
Array_Braun
Trie_Fun
- Trie_Map
Tries_Binary
+ Trie_Ternary
Queue_2Lists
Heaps
Leftist_Heap