src/HOL/Data_Structures/Trie.thy

+(* Author: Tobias Nipkow *)
+
+section \<open>Trie and Patricia Trie Implementations of \mbox{\<open>bool list set\<close>}\<close>
+
+theory Trie
+imports Set_Specs
+begin
+
+hide_const (open) insert
+
+declare Let_def[simp]
+
+
+subsection "Trie"
+
+datatype trie = Leaf | Node bool "trie * trie"
+
+text \<open>The pairing allows things like \<open>Node b (if \<dots> then (l,r) else (r,l))\<close>.\<close>
+
+fun isin :: "trie \<Rightarrow> bool list \<Rightarrow> bool" where
+"isin Leaf ks = False" |
+"isin (Node b (l,r)) ks =
+   (case ks of
+      [] \<Rightarrow> b |
+      k#ks \<Rightarrow> isin (if k then r else l) ks)"
+
+fun insert :: "bool list \<Rightarrow> trie \<Rightarrow> trie" where
+"insert [] Leaf = Node True (Leaf,Leaf)" |
+"insert [] (Node b lr) = Node True lr" |
+"insert (k#ks) Leaf =
+  Node False (if k then (Leaf, insert ks Leaf)
+                   else (insert ks Leaf, Leaf))" |
+"insert (k#ks) (Node b (l,r)) =
+  Node b (if k then (l, insert ks r)
+               else (insert ks l, r))"
+
+fun shrink_Node :: "bool \<Rightarrow> trie * trie \<Rightarrow> trie" where
+"shrink_Node b lr = (if \<not> b \<and> lr = (Leaf,Leaf) then Leaf else Node b lr)"
+
+fun delete :: "bool list \<Rightarrow> trie \<Rightarrow> trie" where
+"delete ks Leaf = Leaf" |
+"delete ks (Node b (l,r)) =
+   (case ks of
+      [] \<Rightarrow> shrink_Node False (l,r) |
+      k#ks' \<Rightarrow> shrink_Node b (if k then (l, delete ks' r) else (delete ks' l, r)))"
+
+fun invar :: "trie \<Rightarrow> bool" where
+"invar Leaf = True" |
+"invar (Node b (l,r)) = ((\<not> b \<longrightarrow> l \<noteq> Leaf \<or> r \<noteq> Leaf) \<and> invar l \<and> invar r)"
+
+
+subsubsection \<open>Functional Correctness\<close>
+
+lemma isin_insert: "isin (insert as t) bs = (as = bs \<or> isin t bs)"
+apply(induction as t arbitrary: bs rule: insert.induct)
+apply(auto split: list.splits)
+done
+
+lemma isin_delete: "isin (delete as t) bs = (as \<noteq> bs \<and> isin t bs)"
+apply(induction as t arbitrary: bs rule: delete.induct)
+ apply simp
+apply (auto split: list.splits; fastforce)
+done
+
+lemma insert_not_Leaf: "insert ks t \<noteq> Leaf"
+by(cases "(ks,t)" rule: insert.cases) auto
+
+lemma invar_insert: "invar t \<Longrightarrow> invar (insert ks t)"
+by(induction ks t rule: insert.induct)(auto simp: insert_not_Leaf)
+
+lemma invar_delete: "invar t \<Longrightarrow> invar (delete ks t)"
+by(induction ks t rule: delete.induct)(auto split: list.split)
+
+interpretation T: Set
+where empty = Leaf and insert = insert and delete = delete and isin = isin
+and set = "\<lambda>t. {x. isin t x}" and invar = invar
+proof (standard, goal_cases)
+  case 1 thus ?case by simp
+next
+  case 2 thus ?case by simp
+next
+  case 3 thus ?case by (auto simp add: isin_insert)
+next
+  case 4 thus ?case by (auto simp add: isin_delete)
+next
+  case 5 thus ?case by simp
+next
+  case 6 thus ?case by (auto simp add: invar_insert)
+next
+  case 7 thus ?case by (auto simp add: invar_delete)
+qed
+
+
+subsection "Patricia Trie"
+
+datatype trieP = LeafP | NodeP "bool list" bool "trieP * trieP"
+
+fun isinP :: "trieP \<Rightarrow> bool list \<Rightarrow> bool" where
+"isinP LeafP ks = False" |
+"isinP (NodeP ps b (l,r)) ks =
+  (let n = length ps in
+   if ps = take n ks
+   then case drop n ks of [] \<Rightarrow> b | k#ks' \<Rightarrow> isinP (if k then r else l) ks'
+   else False)"
+
+text \<open>\<open>split xs ys = (zs, xs', ys')\<close> iff
+  \<open>zs\<close> is the longest common prefix of \<open>xs\<close> and \<open>ys\<close> and
+  \<open>xs = zs @ xs'\<close> and \<open>ys = zs @ ys'\<close>\<close>
+fun split where
+"split [] ys = ([],[],ys)" |
+"split xs [] = ([],xs,[])" |
+"split (x#xs) (y#ys) =
+  (if x\<noteq>y then ([],x#xs,y#ys)
+   else let (ps,xs',ys') = split xs ys in (x#ps,xs',ys'))"
+
+fun insertP :: "bool list \<Rightarrow> trieP \<Rightarrow> trieP" where
+"insertP ks LeafP  = NodeP ks True (LeafP,LeafP)" |
+"insertP ks (NodeP ps b (l,r)) =
+  (case split ks ps of
+     (qs,k#ks',p#ps') \<Rightarrow>
+       let tp = NodeP ps' b (l,r); tk = NodeP ks' True (LeafP,LeafP)
+       in NodeP qs False (if k then (tp,tk) else (tk,tp)) |
+     (qs,k#ks',[]) \<Rightarrow>
+       NodeP ps b (if k then (l,insertP ks' r) else (insertP ks' l, r)) |
+     (qs,[],p#ps') \<Rightarrow>
+       let t = NodeP ps' b (l,r)
+       in NodeP qs True (if p then (LeafP, t) else (t, LeafP)) |
+     (qs,[],[]) \<Rightarrow> NodeP ps True (l,r))"
+
+fun shrink_NodeP where
+"shrink_NodeP ps b lr = (if b then NodeP ps b lr else
+  case lr of
+     (LeafP, LeafP) \<Rightarrow> LeafP |
+     (LeafP, NodeP ps' b' lr') \<Rightarrow> NodeP (ps @ True # ps') b' lr' |
+     (NodeP ps' b' lr', LeafP) \<Rightarrow> NodeP (ps @ False # ps') b' lr' |
+     _ \<Rightarrow> NodeP ps b lr)"
+
+fun deleteP :: "bool list \<Rightarrow> trieP \<Rightarrow> trieP" where
+"deleteP ks LeafP  = LeafP" |
+"deleteP ks (NodeP ps b (l,r)) =
+  (case split ks ps of
+     (qs,_,p#ps') \<Rightarrow> NodeP ps b (l,r) |
+     (qs,k#ks',[]) \<Rightarrow>
+       shrink_NodeP ps b (if k then (l, deleteP ks' r) else (deleteP ks' l, r)) |
+     (qs,[],[]) \<Rightarrow> shrink_NodeP ps False (l,r))"
+
+fun invarP :: "trieP \<Rightarrow> bool" where
+"invarP LeafP = True" |
+"invarP (NodeP ps b (l,r)) = ((\<not>b \<longrightarrow> l \<noteq> LeafP \<or> r \<noteq> LeafP) \<and> invarP l \<and> invarP r)"
+
+text \<open>The abstraction function(s):\<close>
+
+fun prefix_trie :: "bool list \<Rightarrow> trie \<Rightarrow> trie" where
+"prefix_trie [] t = t" |
+"prefix_trie (k#ks) t =
+  (let t' = prefix_trie ks t in shrink_Node False (if k then (Leaf,t') else (t',Leaf)))"
+
+fun abs_trieP :: "trieP \<Rightarrow> trie" where
+"abs_trieP LeafP = Leaf" |
+"abs_trieP (NodeP ps b (l,r)) = prefix_trie ps (Node b (abs_trieP l, abs_trieP r))"
+
+
+subsubsection \<open>Functional Correctness\<close>
+
+text \<open>IsinP:\<close>
+
+lemma isin_prefix_trie: "isin (prefix_trie ps t) ks =
+  (length ks \<ge> length ps \<and>
+  (let n = length ps in ps = take n ks \<and> isin t (drop n ks)))"
+by(induction ps arbitrary: ks)(auto split: list.split)
+
+lemma isinP: "isinP t ks = isin (abs_trieP t) ks"
+apply(induction t arbitrary: ks rule: abs_trieP.induct)
+ apply(auto simp: isin_prefix_trie split: list.split)
+ using nat_le_linear apply force
+using nat_le_linear apply force
+done
+
+text \<open>Insert:\<close>
+
+lemma prefix_trie_Leaf_iff: "prefix_trie ps t = Leaf \<longleftrightarrow> t = Leaf"
+by (induction ps) auto
+
+lemma prefix_trie_Leafs: "prefix_trie ks (Node True (Leaf,Leaf)) = insert ks Leaf"
+by(induction ks) (auto simp: prefix_trie_Leaf_iff)
+
+lemma prefix_trie_Leaf: "prefix_trie ps Leaf = Leaf"
+by(induction ps) auto
+
+lemma insert_append: "insert (ks @ ks') (prefix_trie ks t) = prefix_trie ks (insert ks' t)"
+by(induction ks) (auto simp: prefix_trie_Leaf_iff insert_not_Leaf prefix_trie_Leaf)
+
+lemma prefix_trie_append: "prefix_trie (ps @ qs) t = prefix_trie ps (prefix_trie qs t)"
+by(induction ps) auto
+
+lemma split_iff: "split xs ys = (zs, xs', ys') \<longleftrightarrow>
+  xs = zs @ xs' \<and> ys = zs @ ys' \<and> (xs' \<noteq> [] \<and> ys' \<noteq> [] \<longrightarrow> hd xs' \<noteq> hd ys')"
+proof(induction xs ys arbitrary: zs xs' ys' rule: split.induct)
+  case 1 thus ?case by auto
+next
+  case 2 thus ?case by auto
+next
+  case 3 thus ?case by(clarsimp simp: Cons_eq_append_conv split: prod.splits if_splits) auto
+qed
+
+lemma abs_trieP_insertP:
+  "abs_trieP (insertP ks t) = insert ks (abs_trieP t)"
+apply(induction t arbitrary: ks)
+apply(auto simp: prefix_trie_Leafs insert_append prefix_trie_append
+        prefix_trie_Leaf_iff split_iff split: list.split prod.split)
+done
+
+corollary isinP_insertP: "isinP (insertP ks t) ks' = (ks=ks' \<or> isinP t ks')"
+by (simp add: isin_insert isinP abs_trieP_insertP)
+
+lemma insertP_not_LeafP: "insertP ks t \<noteq> LeafP"
+apply(induction ks t rule: insertP.induct)
+apply(auto split: prod.split list.split)
+done
+
+lemma invarP_insertP: "invarP t \<Longrightarrow> invarP (insertP ks t)"
+apply(induction ks t rule: insertP.induct)
+apply(auto simp: insertP_not_LeafP split: prod.split list.split)
+done
+
+text \<open>Delete:\<close>
+
+lemma invar_shrink_NodeP: "\<lbrakk> invarP l; invarP r \<rbrakk> \<Longrightarrow> invarP (shrink_NodeP ps b (l,r))"
+by(auto split: trieP.split)
+
+lemma invarP_deleteP: "invarP t \<Longrightarrow> invarP (deleteP ks t)"
+apply(induction t arbitrary: ks)
+apply(auto simp: invar_shrink_NodeP split_iff simp del: shrink_NodeP.simps
+           split!: list.splits prod.split if_splits)
+done
+
+lemma delete_append:
+  "delete (ks @ ks') (prefix_trie ks t) = prefix_trie ks (delete ks' t)"
+by(induction ks) auto
+
+lemma abs_trieP_shrink_NodeP:
+  "abs_trieP (shrink_NodeP ps b (l,r)) = prefix_trie ps (shrink_Node b (abs_trieP l, abs_trieP r))"
+apply(induction ps arbitrary: b l r)
+apply (auto simp: prefix_trie_Leaf prefix_trie_Leaf_iff prefix_trie_append
+            split!: trieP.splits if_splits)
+done
+
+lemma abs_trieP_deleteP:
+  "abs_trieP (deleteP ks t) = delete ks (abs_trieP t)"
+apply(induction t arbitrary: ks)
+apply(auto simp: prefix_trie_Leafs delete_append prefix_trie_Leaf
+                 abs_trieP_shrink_NodeP prefix_trie_append split_iff
+           simp del: shrink_NodeP.simps split!: list.split prod.split if_splits)
+done
+
+corollary isinP_deleteP: "isinP (deleteP ks t) ks' = (ks\<noteq>ks' \<and> isinP t ks')"
+by (simp add: isin_delete isinP abs_trieP_deleteP)
+
+interpretation PT: Set
+where empty = LeafP and insert = insertP and delete = deleteP
+and isin = isinP and set = "\<lambda>t. {x. isinP t x}" and invar = invarP
+proof (standard, goal_cases)
+  case 1 thus ?case by (simp)
+next
+  case 2 thus ?case by (simp)
+next
+  case 3 thus ?case by (auto simp add: isinP_insertP)
+next
+  case 4 thus ?case by (auto simp add: isinP_deleteP)
+next
+  case 5 thus ?case by (simp)
+next
+  case 6 thus ?case by (simp add: invarP_insertP)
+next
+  case 7 thus ?case by (auto simp add: invarP_deleteP)
+qed
+
+end