(* Tobias Nipkow *)
section "AVL Tree with Balance Factors"
theory AVL_Bal_Set
imports
Cmp
Isin2
begin
datatype bal = Lh | Bal | Rh
type_synonym 'a tree_bal = "('a * bal) tree"
text \<open>Invariant:\<close>
fun avl :: "'a tree_bal \<Rightarrow> bool" where
"avl Leaf = True" |
"avl (Node l (a,b) r) =
((case b of
Bal \<Rightarrow> height r = height l |
Lh \<Rightarrow> height l = height r + 1 |
Rh \<Rightarrow> height r = height l + 1)
\<and> avl l \<and> avl r)"
subsection \<open>Code\<close>
datatype 'a tree_bal2 = Same "'a tree_bal" | Diff "'a tree_bal"
fun tree :: "'a tree_bal2 \<Rightarrow> 'a tree_bal" where
"tree(Same t) = t" |
"tree(Diff t) = t"
fun rot21 :: "bal \<Rightarrow> bal" where
"rot21 b = (if b=Rh then Lh else Bal)"
fun rot22 :: "bal \<Rightarrow> bal" where
"rot22 b = (if b=Lh then Rh else Bal)"
fun balL :: "'a tree_bal2 \<Rightarrow> 'a \<Rightarrow> bal \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal2" where
"balL AB' c bc C = (case AB' of
Same AB \<Rightarrow> Same (Node AB (c,bc) C) |
Diff AB \<Rightarrow> (case bc of
Bal \<Rightarrow> Diff (Node AB (c,Lh) C) |
Rh \<Rightarrow> Same (Node AB (c,Bal) C) |
Lh \<Rightarrow> Same(case AB of
Node A (a,Lh) B \<Rightarrow> Node A (a,Bal) (Node B (c,Bal) C) |
Node A (a,Rh) B \<Rightarrow> (case B of
Node B\<^sub>1 (b, bb) B\<^sub>2 \<Rightarrow>
Node (Node A (a,rot21 bb) B\<^sub>1) (b,Bal) (Node B\<^sub>2 (c,rot22 bb) C)))))"
fun balR :: "'a tree_bal \<Rightarrow> 'a \<Rightarrow> bal \<Rightarrow> 'a tree_bal2 \<Rightarrow> 'a tree_bal2" where
"balR A a ba BC' = (case BC' of
Same BC \<Rightarrow> Same (Node A (a,ba) BC) |
Diff BC \<Rightarrow> (case ba of
Bal \<Rightarrow> Diff (Node A (a,Rh) BC) |
Lh \<Rightarrow> Same (Node A (a,Bal) BC) |
Rh \<Rightarrow> Same(case BC of
Node B (c,Rh) C \<Rightarrow> Node (Node A (a,Bal) B) (c,Bal) C |
Node B (c,Lh) C \<Rightarrow> (case B of
Node B\<^sub>1 (b, bb) B\<^sub>2 \<Rightarrow>
Node (Node A (a,rot21 bb) B\<^sub>1) (b,Bal) (Node B\<^sub>2 (c,rot22 bb) C)))))"
fun insert :: "'a::linorder \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal2" where
"insert x Leaf = Diff(Node Leaf (x, Bal) Leaf)" |
"insert x (Node l (a, b) r) = (case cmp x a of
EQ \<Rightarrow> Same(Node l (a, b) r) |
LT \<Rightarrow> balL (insert x l) a b r |
GT \<Rightarrow> balR l a b (insert x r))"
fun baldR :: "'a tree_bal \<Rightarrow> 'a \<Rightarrow> bal \<Rightarrow> 'a tree_bal2 \<Rightarrow> 'a tree_bal2" where
"baldR AB c bc C' = (case C' of
Same C \<Rightarrow> Same (Node AB (c,bc) C) |
Diff C \<Rightarrow> (case bc of
Bal \<Rightarrow> Same (Node AB (c,Lh) C) |
Rh \<Rightarrow> Diff (Node AB (c,Bal) C) |
Lh \<Rightarrow> (case AB of
Node A (a,Lh) B \<Rightarrow> Diff(Node A (a,Bal) (Node B (c,Bal) C)) |
Node A (a,Bal) B \<Rightarrow> Same(Node A (a,Rh) (Node B (c,Lh) C)) |
Node A (a,Rh) B \<Rightarrow> (case B of
Node B\<^sub>1 (b, bb) B\<^sub>2 \<Rightarrow>
Diff(Node (Node A (a,rot21 bb) B\<^sub>1) (b,Bal) (Node B\<^sub>2 (c,rot22 bb) C))))))"
fun baldL :: "'a tree_bal2 \<Rightarrow> 'a \<Rightarrow> bal \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal2" where
"baldL A' a ba BC = (case A' of
Same A \<Rightarrow> Same (Node A (a,ba) BC) |
Diff A \<Rightarrow> (case ba of
Bal \<Rightarrow> Same (Node A (a,Rh) BC) |
Lh \<Rightarrow> Diff (Node A (a,Bal) BC) |
Rh \<Rightarrow> (case BC of
Node B (c,Rh) C \<Rightarrow> Diff(Node (Node A (a,Bal) B) (c,Bal) C) |
Node B (c,Bal) C \<Rightarrow> Same(Node (Node A (a,Rh) B) (c,Lh) C) |
Node B (c,Lh) C \<Rightarrow> (case B of
Node B\<^sub>1 (b, bb) B\<^sub>2 \<Rightarrow>
Diff(Node (Node A (a,rot21 bb) B\<^sub>1) (b,Bal) (Node B\<^sub>2 (c,rot22 bb) C))))))"
fun split_max :: "'a tree_bal \<Rightarrow> 'a tree_bal2 * 'a" where
"split_max (Node l (a, ba) r) =
(if r = Leaf then (Diff l,a) else let (r',a') = split_max r in (baldR l a ba r', a'))"
fun delete :: "'a::linorder \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal2" where
"delete _ Leaf = Same Leaf" |
"delete x (Node l (a, ba) r) =
(case cmp x a of
EQ \<Rightarrow> if l = Leaf then Diff r
else let (l', a') = split_max l in baldL l' a' ba r |
LT \<Rightarrow> baldL (delete x l) a ba r |
GT \<Rightarrow> baldR l a ba (delete x r))"
lemmas split_max_induct = split_max.induct[case_names Node Leaf]
lemmas splits = if_splits tree.splits tree_bal2.splits bal.splits
subsection \<open>Proofs\<close>
lemma insert_Diff1[simp]: "insert x t \<noteq> Diff Leaf"
by (cases t)(auto split!: splits)
lemma insert_Diff2[simp]: "insert x t = Diff (Node l (a,Bal) r) \<longleftrightarrow> t = Leaf \<and> a = x \<and> l=Leaf \<and> r=Leaf"
by (cases t)(auto split!: splits)
lemma insert_Diff3[simp]: "insert x t \<noteq> Diff (Node l (a,Rh) Leaf)"
by (cases t)(auto split!: splits)
lemma insert_Diff4[simp]: "insert x t \<noteq> Diff (Node Leaf (a,Lh) r)"
by (cases t)(auto split!: splits)
subsubsection "Proofs for insert"
theorem inorder_insert:
"\<lbrakk> avl t; sorted(inorder t) \<rbrakk> \<Longrightarrow> inorder(tree(insert x t)) = ins_list x (inorder t)"
by(induction t) (auto simp: ins_list_simps split!: splits)
lemma avl_insert_case: "avl t \<Longrightarrow> case insert x t of
Same t' \<Rightarrow> avl t' \<and> height t' = height t |
Diff t' \<Rightarrow> avl t' \<and> height t' = height t + 1"
apply(induction x t rule: insert.induct)
apply(auto simp: max_absorb1 split!: splits)
done
corollary avl_insert: "avl t \<Longrightarrow> avl(tree(insert x t))"
using avl_insert_case[of t x] by (simp split: splits)
subsubsection "Proofs for delete"
lemma inorder_baldL:
"\<lbrakk> ba = Rh \<longrightarrow> r \<noteq> Leaf; avl r \<rbrakk>
\<Longrightarrow> inorder (tree(baldL l a ba r)) = inorder (tree l) @ a # inorder r"
by (auto split: splits)
lemma inorder_baldR:
"\<lbrakk> ba = Lh \<longrightarrow> l \<noteq> Leaf; avl l \<rbrakk>
\<Longrightarrow> inorder (tree(baldR l a ba r)) = inorder l @ a # inorder (tree r)"
by (auto split: splits)
lemma avl_split_max:
"\<lbrakk> split_max t = (t',a); avl t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> case t' of
Same t' \<Rightarrow> avl t' \<and> height t = height t' |
Diff t' \<Rightarrow> avl t' \<and> height t = height t' + 1"
apply(induction t arbitrary: t' a rule: split_max_induct)
apply(fastforce simp: max_absorb1 max_absorb2 split!: splits prod.splits)
apply simp
done
lemma avl_delete_case: "avl t \<Longrightarrow> case delete x t of
Same t' \<Rightarrow> avl t' \<and> height t = height t' |
Diff t' \<Rightarrow> avl t' \<and> height t = height t' + 1"
apply(induction x t rule: delete.induct)
apply(auto simp: max_absorb1 max_absorb2 dest: avl_split_max split!: splits prod.splits)
done
corollary avl_delete: "avl t \<Longrightarrow> avl(tree(delete x t))"
using avl_delete_case[of t x] by(simp split: splits)
lemma inorder_split_maxD:
"\<lbrakk> split_max t = (t',a); t \<noteq> Leaf; avl t \<rbrakk> \<Longrightarrow>
inorder (tree t') @ [a] = inorder t"
apply(induction t arbitrary: t' rule: split_max.induct)
apply(fastforce split!: splits prod.splits)
apply simp
done
lemma neq_Leaf_if_height_neq_0[simp]: "height t \<noteq> 0 \<Longrightarrow> t \<noteq> Leaf"
by auto
theorem inorder_delete:
"\<lbrakk> avl t; sorted(inorder t) \<rbrakk> \<Longrightarrow> inorder (tree(delete x t)) = del_list x (inorder t)"
apply(induction t rule: tree2_induct)
apply(auto simp: del_list_simps inorder_baldL inorder_baldR avl_delete inorder_split_maxD
simp del: baldR.simps baldL.simps split!: splits prod.splits)
done
subsubsection \<open>Set Implementation\<close>
interpretation S: Set_by_Ordered
where empty = Leaf and isin = isin
and insert = "\<lambda>x t. tree(insert x t)"
and delete = "\<lambda>x t. tree(delete x t)"
and inorder = inorder and inv = avl
proof (standard, goal_cases)
case 1 show ?case by (simp)
next
case 2 thus ?case by(simp add: isin_set_inorder)
next
case 3 thus ?case by(simp add: inorder_insert)
next
case 4 thus ?case by(simp add: inorder_delete)
next
case 5 thus ?case by (simp add: empty_def)
next
case 6 thus ?case by (simp add: avl_insert)
next
case 7 thus ?case by (simp add: avl_delete)
qed
end