src/HOL/Data_Structures/Trie_Map.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Trie_Map.thy	Thu May 09 12:32:47 2019 +0200
@@ -0,0 +1,112 @@
+section "Tries via Search Trees"
+
+theory Trie_Map
+imports
+  RBT_Map
+  Trie_Fun
+begin
+
+text \<open>An implementation of tries based on maps implemented by red-black trees.
+Works for any kind of search tree.\<close>
+
+text \<open>Implementation of map:\<close>
+
+type_synonym 'a mapi = "'a rbt"
+
+datatype 'a trie_map = Lf | Nd bool "('a * 'a trie_map) mapi"
+
+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 (RBT) as done here.
+But because the map (@{typ "'a rbt"}) 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>Tree_Map\<close>,
+and not just \<open>RBT_Map\<close>, except for the termination lemma \<open>lookup_size\<close>.\<close>
+
+lemma lookup_size[termination_simp]:
+  fixes t :: "('a::linorder * 'a trie_map) rbt"
+  shows "lookup t a = Some b \<Longrightarrow> size b < Suc (size_tree (\<lambda>ab. Suc (size (snd ab))) (\<lambda>x. 0) t)"
+apply(induction t a rule: lookup.induct)
+apply(auto split: if_splits)
+done
+
+fun isin :: "('a::linorder) trie_map \<Rightarrow> 'a list \<Rightarrow> bool" where
+"isin Lf xs = False" |
+"isin (Nd b m) [] = b" |
+"isin (Nd b m) (x # xs) = (case lookup m x of None \<Rightarrow> False | Some t \<Rightarrow> isin t xs)"
+
+fun insert :: "('a::linorder) list \<Rightarrow> 'a trie_map \<Rightarrow> 'a trie_map" where
+"insert [] Lf = Nd True empty" |
+"insert [] (Nd b m) = Nd True m" |
+"insert (x#xs) Lf = Nd False (update x (insert xs Lf) empty)" |
+"insert (x#xs) (Nd b m) =
+  Nd b (update x (insert xs (case lookup m x of None \<Rightarrow> Lf | Some t \<Rightarrow> t)) m)"
+
+fun delete :: "('a::linorder) list \<Rightarrow> 'a trie_map \<Rightarrow> 'a trie_map" where
+"delete xs Lf = Lf" |
+"delete [] (Nd b m) = Nd False m" |
+"delete (x#xs) (Nd b m) = Nd b
+   (case lookup m x of
+      None \<Rightarrow> m |
+      Some t \<Rightarrow> update x (delete xs t) m)"
+
+
+subsection "Correctness"
+
+text \<open>Proof by stepwise refinement. First abstract to type @{typ "'a trie"}.\<close>
+
+fun abs :: "'a::linorder trie_map \<Rightarrow> 'a trie" where
+"abs Lf = Trie_Fun.Lf" |
+"abs (Nd b t) = Trie_Fun.Nd b (\<lambda>a. map_option abs (lookup t a))"
+
+fun invar :: "('a::linorder)trie_map \<Rightarrow> bool" where
+"invar Lf = True" |
+"invar (Nd b m) = (M.invar m \<and> (\<forall>a t. lookup m a = Some t \<longrightarrow> invar t))"
+
+lemma isin_abs: "isin t xs = Trie_Fun.isin (abs t) xs"
+apply(induction t xs rule: isin.induct)
+apply(auto split: option.split)
+done
+
+lemma abs_insert: "invar t \<Longrightarrow> abs(insert xs t) = Trie_Fun.insert xs (abs t)"
+apply(induction xs t rule: insert.induct)
+apply(auto simp: M.map_specs split: option.split)
+done
+
+lemma abs_delete: "invar t \<Longrightarrow> abs(delete xs t) = Trie_Fun.delete xs (abs t)"
+apply(induction xs t rule: delete.induct)
+apply(auto simp: M.map_specs split: option.split)
+done
+
+lemma invar_insert: "invar t \<Longrightarrow> invar (insert xs t)"
+apply(induction xs t rule: insert.induct)
+apply(auto simp: M.map_specs split: option.split)
+done
+
+lemma invar_delete: "invar t \<Longrightarrow> invar (delete xs t)"
+apply(induction xs t rule: delete.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 = Lf and isin = isin and insert = insert and delete = delete
+and set = "set o abs" and invar = invar
+proof (standard, goal_cases)
+  case 1 show ?case by (simp)
+next
+  case 2 thus ?case by (simp add: isin_set isin_abs)
+next
+  case 3 thus ?case by (simp add: set_insert abs_insert)
+next
+  case 4 thus ?case by (simp add: set_delete abs_delete)
+next
+  case 5 thus ?case by (simp)
+next
+  case 6 thus ?case by (simp add: invar_insert)
+next
+  case 7 thus ?case by (simp add: invar_delete)
+qed
+
+
+end