added AA trees

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/AA_Set.thy	Sun Dec 06 17:27:42 2015 +0100
@@ -0,0 +1,138 @@
+(*
+Author: Tobias Nipkow
+Invariants are under development
+*)
+
+section \<open>An AA Tree Implementation of Sets\<close>
+
+theory AA_Set
+imports
+  Isin2
+  Cmp
+begin
+
+type_synonym 'a aa_tree = "('a,nat) tree"
+
+fun lvl :: "'a aa_tree \<Rightarrow> nat" where
+"lvl Leaf = 0" |
+"lvl (Node lv _ _ _) = lv"
+
+fun invar :: "'a aa_tree \<Rightarrow> bool" where
+"invar Leaf = True" |
+"invar (Node h l a r) =
+ (invar l \<and> invar r \<and>
+  h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = Node h lr b rr \<and> h = lvl rr + 1)))"
+
+fun skew :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
+"skew (Node lva (Node lvb t1 b t2) a t3) =
+  (if lva = lvb then Node lva t1 b (Node lva t2 a t3) else Node lva (Node lvb t1 b t2) a t3)" |
+"skew t = t"
+
+fun split :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
+"split (Node lva t1 a (Node lvb t2 b (Node lvc t3 c t4))) =
+   (if lva = lvb \<and> lvb = lvc (* lva = lvc suffices *)
+    then Node (lva+1) (Node lva t1 a t2) b (Node lva t3 c t4)
+    else Node lva t1 a (Node lvb t2 b (Node lvc t3 c t4)))" |
+"split t = t"
+
+hide_const (open) insert
+
+fun insert :: "'a::cmp \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
+"insert x Leaf = Node 1 Leaf x Leaf" |
+"insert x (Node lv t1 a t2) =
+  (case cmp x a of
+     LT \<Rightarrow> split (skew (Node lv (insert x t1) a t2)) |
+     GT \<Rightarrow> split (skew (Node lv t1 a (insert x t2))) |
+     EQ \<Rightarrow> Node lv t1 x t2)"
+
+(* wrong in paper! *)
+fun del_max :: "'a aa_tree \<Rightarrow> 'a aa_tree * 'a" where
+"del_max (Node lv l a Leaf) = (l,a)" |
+"del_max (Node lv l a r) = (let (r',b) = del_max r in (Node lv l a r', b))"
+
+fun sngl :: "'a aa_tree \<Rightarrow> bool" where
+"sngl Leaf = False" |
+"sngl (Node _ _ _ Leaf) = True" |
+"sngl (Node lva _ _ (Node lvb _ _ _)) = (lva > lvb)"
+
+definition adjust :: "'a aa_tree \<Rightarrow> 'a aa_tree" where
+"adjust t =
+ (case t of
+  Node lv l x r \<Rightarrow>
+   (if lvl l >= lv-1 \<and> lvl r >= lv-1 then t else
+    if lvl r < lv-1 \<and> sngl l then skew (Node (lv-1) l x r) else
+    if lvl r < lv-1
+    then case l of
+           Node lva t1 a (Node lvb t2 b t3)
+             \<Rightarrow> Node (lvb+1) (Node lva t1 a t2) b (Node (lv-1) t3 x r) |
+           _ \<Rightarrow> t (* unreachable *)
+    else
+    if lvl r < lv then split (Node (lv-1) l x r)
+    else
+      case r of
+        Leaf \<Rightarrow> Leaf (* unreachable *) |
+        Node _ t1 b t4 \<Rightarrow>
+          (case t1 of
+             Node lva t2 a t3
+               \<Rightarrow> Node (lva+1) (Node (lv-1) l x t2) a
+                    (split (Node (if sngl t1 then lva-1 else lva) t3 b t4))
+           | _ \<Rightarrow> t (* unreachable *))))"
+
+fun delete :: "'a::cmp \<Rightarrow> 'a aa_tree \<Rightarrow> 'a aa_tree" where
+"delete _ Leaf = Leaf" |
+"delete x (Node lv l a r) =
+  (case cmp x a of
+     LT \<Rightarrow> adjust (Node lv (delete x l) a r) |
+     GT \<Rightarrow> adjust (Node lv l a (delete x r)) |
+     EQ \<Rightarrow> (if l = Leaf then r
+            else let (l',b) = del_max l in adjust (Node lv l' b r)))"
+
+
+subsection "Functional Correctness"
+
+subsubsection "Proofs for insert"
+
+lemma inorder_split: "inorder(split t) = inorder t"
+by(cases t rule: split.cases) (auto)
+
+lemma inorder_skew: "inorder(skew t) = inorder t"
+by(cases t rule: skew.cases) (auto)
+
+lemma inorder_insert:
+  "sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
+by(induction t) (auto simp: ins_list_simps inorder_split inorder_skew)
+
+subsubsection "Proofs for delete"
+
+lemma del_maxD:
+  "\<lbrakk> del_max t = (t',x); t \<noteq> Leaf; sorted(inorder t) \<rbrakk> \<Longrightarrow>
+   inorder t' @ [x] = inorder t"
+by(induction t arbitrary: t' rule: del_max.induct)
+  (auto simp: sorted_lems split: prod.splits)
+
+lemma inorder_adjust: "t \<noteq> Leaf \<Longrightarrow> inorder(adjust t) = inorder t"
+by(induction t)
+  (auto simp: adjust_def inorder_skew inorder_split split: tree.splits)
+
+lemma inorder_delete:
+  "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
+by(induction t)
+  (auto simp: del_list_simps inorder_adjust del_maxD split: prod.splits)
+
+
+subsection "Overall correctness"
+
+interpretation Set_by_Ordered
+where empty = Leaf and isin = isin and insert = insert and delete = delete
+and inorder = inorder and inv = "\<lambda>_. True"
+proof (standard, goal_cases)
+  case 1 show ?case by simp
+next
+  case 2 thus ?case by(simp add: isin_set)
+next
+  case 3 thus ?case by(simp add: inorder_insert)
+next
+  case 4 thus ?case by(simp add: inorder_delete)
+qed auto
+
+end
\ No newline at end of file

--- a/src/HOL/ROOT	Sun Dec 06 11:26:38 2015 +0100
+++ b/src/HOL/ROOT	Sun Dec 06 17:27:42 2015 +0100
@@ -179,6 +179,7 @@
     Tree23_Map
     Tree234_Map
     Brother12_Map
+    AA_Set
     Splay_Map
   document_files "root.tex" "root.bib"