--- a/src/HOL/Data_Structures/AVL_Set.thy Wed Nov 04 15:07:23 2015 +0100
+++ b/src/HOL/Data_Structures/AVL_Set.thy Thu Nov 05 08:27:14 2015 +0100
@@ -6,7 +6,7 @@
section "AVL Tree Implementation of Sets"
theory AVL_Set
-imports Isin2
+imports Cmp Isin2
begin
type_synonym 'a avl_tree = "('a,nat) tree"
@@ -26,8 +26,8 @@
definition node :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
"node l a r = Node (max (ht l) (ht r) + 1) l a r"
-definition node_bal_l :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
-"node_bal_l l a r = (
+definition balL :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"balL l a r = (
if ht l = ht r + 2 then (case l of
Node _ bl b br \<Rightarrow> (if ht bl < ht br
then case br of
@@ -35,8 +35,8 @@
else node bl b (node br a r)))
else node l a r)"
-definition node_bal_r :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
-"node_bal_r l a r = (
+definition balR :: "'a avl_tree \<Rightarrow> 'a \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+"balR l a r = (
if ht r = ht l + 2 then (case r of
Node _ bl b br \<Rightarrow> (if ht bl > ht br
then case bl of
@@ -44,19 +44,17 @@
else node (node l a bl) b br))
else node l a r)"
-fun insert :: "'a::order \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+fun insert :: "'a::cmp \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
"insert x Leaf = Node 1 Leaf x Leaf" |
-"insert x (Node h l a r) =
- (if x=a then Node h l a r
- else if x<a
- then node_bal_l (insert x l) a r
- else node_bal_r l a (insert x r))"
+"insert x (Node h l a r) = (case cmp x a of
+ EQ \<Rightarrow> Node h l a r |
+ LT \<Rightarrow> balL (insert x l) a r |
+ GT \<Rightarrow> balR l a (insert x r))"
fun delete_max :: "'a avl_tree \<Rightarrow> 'a avl_tree * 'a" where
"delete_max (Node _ l a Leaf) = (l,a)" |
-"delete_max (Node _ l a r) = (
- let (r',a') = delete_max r in
- (node_bal_l l a r', a'))"
+"delete_max (Node _ l a r) =
+ (let (r',a') = delete_max r in (balL l a r', a'))"
lemmas delete_max_induct = delete_max.induct[case_names Leaf Node]
@@ -64,16 +62,16 @@
"delete_root (Node h Leaf a r) = r" |
"delete_root (Node h l a Leaf) = l" |
"delete_root (Node h l a r) =
- (let (l', a') = delete_max l in node_bal_r l' a' r)"
+ (let (l', a') = delete_max l in balR l' a' r)"
lemmas delete_root_cases = delete_root.cases[case_names Leaf_t Node_Leaf Node_Node]
-fun delete :: "'a::order \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
+fun delete :: "'a::cmp \<Rightarrow> 'a avl_tree \<Rightarrow> 'a avl_tree" where
"delete _ Leaf = Leaf" |
-"delete x (Node h l a r) = (
- if x = a then delete_root (Node h l a r)
- else if x < a then node_bal_r (delete x l) a r
- else node_bal_l l a (delete x r))"
+"delete x (Node h l a r) = (case cmp x a of
+ EQ \<Rightarrow> delete_root (Node h l a r) |
+ LT \<Rightarrow> balR (delete x l) a r |
+ GT \<Rightarrow> balL l a (delete x r))"
subsection {* Functional Correctness Proofs *}
@@ -83,18 +81,18 @@
subsubsection "Proofs for insert"
-lemma inorder_node_bal_l:
- "inorder (node_bal_l l a r) = inorder l @ a # inorder r"
-by (auto simp: node_def node_bal_l_def split:tree.splits)
+lemma inorder_balL:
+ "inorder (balL l a r) = inorder l @ a # inorder r"
+by (auto simp: node_def balL_def split:tree.splits)
-lemma inorder_node_bal_r:
- "inorder (node_bal_r l a r) = inorder l @ a # inorder r"
-by (auto simp: node_def node_bal_r_def split:tree.splits)
+lemma inorder_balR:
+ "inorder (balR l a r) = inorder l @ a # inorder r"
+by (auto simp: node_def balR_def split:tree.splits)
theorem inorder_insert:
"sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"
by (induct t)
- (auto simp: ins_list_simps inorder_node_bal_l inorder_node_bal_r)
+ (auto simp: ins_list_simps inorder_balL inorder_balR)
subsubsection "Proofs for delete"
@@ -103,17 +101,17 @@
"\<lbrakk> delete_max t = (t',a); t \<noteq> Leaf \<rbrakk> \<Longrightarrow>
inorder t' @ [a] = inorder t"
by(induction t arbitrary: t' rule: delete_max.induct)
- (auto simp: inorder_node_bal_l split: prod.splits tree.split)
+ (auto simp: inorder_balL split: prod.splits tree.split)
lemma inorder_delete_root:
"inorder (delete_root (Node h l a r)) = inorder l @ inorder r"
by(induction "Node h l a r" arbitrary: l a r h rule: delete_root.induct)
- (auto simp: inorder_node_bal_r inorder_delete_maxD split: prod.splits)
+ (auto simp: inorder_balR inorder_delete_maxD split: prod.splits)
theorem inorder_delete:
"sorted(inorder t) \<Longrightarrow> inorder (delete x t) = del_list x (inorder t)"
by(induction t)
- (auto simp: del_list_simps inorder_node_bal_l inorder_node_bal_r
+ (auto simp: del_list_simps inorder_balL inorder_balR
inorder_delete_root inorder_delete_maxD split: prod.splits)
@@ -145,17 +143,17 @@
lemma [simp]: "avl t \<Longrightarrow> ht t = height t"
by (induct t) simp_all
-lemma height_node_bal_l:
+lemma height_balL:
"\<lbrakk> height l = height r + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
- height (node_bal_l l a r) = height r + 2 \<or>
- height (node_bal_l l a r) = height r + 3"
-by (cases l) (auto simp:node_def node_bal_l_def split:tree.split)
+ height (balL l a r) = height r + 2 \<or>
+ height (balL l a r) = height r + 3"
+by (cases l) (auto simp:node_def balL_def split:tree.split)
-lemma height_node_bal_r:
+lemma height_balR:
"\<lbrakk> height r = height l + 2; avl l; avl r \<rbrakk> \<Longrightarrow>
- height (node_bal_r l a r) = height l + 2 \<or>
- height (node_bal_r l a r) = height l + 3"
-by (cases r) (auto simp add:node_def node_bal_r_def split:tree.split)
+ height (balR l a r) = height l + 2 \<or>
+ height (balR l a r) = height l + 3"
+by (cases r) (auto simp add:node_def balR_def split:tree.split)
lemma [simp]: "height(node l a r) = max (height l) (height r) + 1"
by (simp add: node_def)
@@ -166,53 +164,53 @@
\<rbrakk> \<Longrightarrow> avl(node l a r)"
by (auto simp add:max_def node_def)
-lemma height_node_bal_l2:
+lemma height_balL2:
"\<lbrakk> avl l; avl r; height l \<noteq> height r + 2 \<rbrakk> \<Longrightarrow>
- height (node_bal_l l a r) = (1 + max (height l) (height r))"
-by (cases l, cases r) (simp_all add: node_bal_l_def)
+ height (balL l a r) = (1 + max (height l) (height r))"
+by (cases l, cases r) (simp_all add: balL_def)
-lemma height_node_bal_r2:
+lemma height_balR2:
"\<lbrakk> avl l; avl r; height r \<noteq> height l + 2 \<rbrakk> \<Longrightarrow>
- height (node_bal_r l a r) = (1 + max (height l) (height r))"
-by (cases l, cases r) (simp_all add: node_bal_r_def)
+ height (balR l a r) = (1 + max (height l) (height r))"
+by (cases l, cases r) (simp_all add: balR_def)
-lemma avl_node_bal_l:
+lemma avl_balL:
assumes "avl l" "avl r" and "height l = height r \<or> height l = height r + 1
\<or> height r = height l + 1 \<or> height l = height r + 2"
- shows "avl(node_bal_l l a r)"
+ shows "avl(balL l a r)"
proof(cases l)
case Leaf
- with assms show ?thesis by (simp add: node_def node_bal_l_def)
+ with assms show ?thesis by (simp add: node_def balL_def)
next
case (Node ln ll lr lh)
with assms show ?thesis
proof(cases "height l = height r + 2")
case True
from True Node assms show ?thesis
- by (auto simp: node_bal_l_def intro!: avl_node split: tree.split) arith+
+ by (auto simp: balL_def intro!: avl_node split: tree.split) arith+
next
case False
- with assms show ?thesis by (simp add: avl_node node_bal_l_def)
+ with assms show ?thesis by (simp add: avl_node balL_def)
qed
qed
-lemma avl_node_bal_r:
+lemma avl_balR:
assumes "avl l" and "avl r" and "height l = height r \<or> height l = height r + 1
\<or> height r = height l + 1 \<or> height r = height l + 2"
- shows "avl(node_bal_r l a r)"
+ shows "avl(balR l a r)"
proof(cases r)
case Leaf
- with assms show ?thesis by (simp add: node_def node_bal_r_def)
+ with assms show ?thesis by (simp add: node_def balR_def)
next
case (Node rn rl rr rh)
with assms show ?thesis
proof(cases "height r = height l + 2")
case True
from True Node assms show ?thesis
- by (auto simp: node_bal_r_def intro!: avl_node split: tree.split) arith+
+ by (auto simp: balR_def intro!: avl_node split: tree.split) arith+
next
case False
- with assms show ?thesis by (simp add: node_bal_r_def avl_node)
+ with assms show ?thesis by (simp add: balR_def avl_node)
qed
qed
@@ -237,10 +235,10 @@
with Node 1 show ?thesis
proof(cases "x<a")
case True
- with Node 1 show ?thesis by (auto simp add:avl_node_bal_l)
+ with Node 1 show ?thesis by (auto simp add:avl_balL)
next
case False
- with Node 1 `x\<noteq>a` show ?thesis by (auto simp add:avl_node_bal_r)
+ with Node 1 `x\<noteq>a` show ?thesis by (auto simp add:avl_balR)
qed
qed
case 2
@@ -255,12 +253,12 @@
case True
with Node 2 show ?thesis
proof(cases "height (insert x l) = height r + 2")
- case False with Node 2 `x < a` show ?thesis by (auto simp: height_node_bal_l2)
+ case False with Node 2 `x < a` show ?thesis by (auto simp: height_balL2)
next
case True
- hence "(height (node_bal_l (insert x l) a r) = height r + 2) \<or>
- (height (node_bal_l (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
- using Node 2 by (intro height_node_bal_l) simp_all
+ hence "(height (balL (insert x l) a r) = height r + 2) \<or>
+ (height (balL (insert x l) a r) = height r + 3)" (is "?A \<or> ?B")
+ using Node 2 by (intro height_balL) simp_all
thus ?thesis
proof
assume ?A
@@ -275,12 +273,12 @@
with Node 2 show ?thesis
proof(cases "height (insert x r) = height l + 2")
case False
- with Node 2 `\<not>x < a` show ?thesis by (auto simp: height_node_bal_r2)
+ with Node 2 `\<not>x < a` show ?thesis by (auto simp: height_balR2)
next
case True
- hence "(height (node_bal_r l a (insert x r)) = height l + 2) \<or>
- (height (node_bal_r l a (insert x r)) = height l + 3)" (is "?A \<or> ?B")
- using Node 2 by (intro height_node_bal_r) simp_all
+ hence "(height (balR l a (insert x r)) = height l + 2) \<or>
+ (height (balR l a (insert x r)) = height l + 3)" (is "?A \<or> ?B")
+ using Node 2 by (intro height_balR) simp_all
thus ?thesis
proof
assume ?A
@@ -306,10 +304,10 @@
case (Node h l a rh rl b rr)
case 1
with Node have "avl l" "avl (fst (delete_max (Node rh rl b rr)))" by auto
- with 1 Node have "avl (node_bal_l l a (fst (delete_max (Node rh rl b rr))))"
- by (intro avl_node_bal_l) fastforce+
+ with 1 Node have "avl (balL l a (fst (delete_max (Node rh rl b rr))))"
+ by (intro avl_balL) fastforce+
thus ?case
- by (auto simp: height_node_bal_l height_node_bal_l2
+ by (auto simp: height_balL height_balL2
linorder_class.max.absorb1 linorder_class.max.absorb2
split:prod.split)
next
@@ -318,7 +316,7 @@
let ?r = "Node rh rl b rr"
let ?r' = "fst (delete_max ?r)"
from `avl x` Node 2 have "avl l" and "avl ?r" by simp_all
- thus ?case using Node 2 height_node_bal_l[of l ?r' a] height_node_bal_l2[of l ?r' a]
+ thus ?case using Node 2 height_balL[of l ?r' a] height_balL2[of l ?r' a]
apply (auto split:prod.splits simp del:avl.simps) by arith+
qed auto
@@ -337,8 +335,8 @@
height ?l = height(?l') + 1" by (rule avl_delete_max,simp)+
with `avl t` Node_Node have "height ?l' = height ?r \<or> height ?l' = height ?r + 1
\<or> height ?r = height ?l' + 1 \<or> height ?r = height ?l' + 2" by fastforce
- with `avl ?l'` `avl ?r` have "avl(node_bal_r ?l' (snd(delete_max ?l)) ?r)"
- by (rule avl_node_bal_r)
+ with `avl ?l'` `avl ?r` have "avl(balR ?l' (snd(delete_max ?l)) ?r)"
+ by (rule avl_balR)
with Node_Node show ?thesis by (auto split:prod.splits)
qed simp_all
@@ -351,7 +349,7 @@
let ?l = "Node lh ll ln lr"
let ?r = "Node rh rl rn rr"
let ?l' = "fst (delete_max ?l)"
- let ?t' = "node_bal_r ?l' (snd(delete_max ?l)) ?r"
+ let ?t' = "balR ?l' (snd(delete_max ?l)) ?r"
from `avl t` and Node_Node have "avl ?r" by simp
from `avl t` and Node_Node have "avl ?l" by simp
hence "avl(?l')" by (rule avl_delete_max,simp)
@@ -360,11 +358,11 @@
have "height t = height ?t' \<or> height t = height ?t' + 1" using `avl t` Node_Node
proof(cases "height ?r = height ?l' + 2")
case False
- show ?thesis using l'_height t_height False by (subst height_node_bal_r2[OF `avl ?l'` `avl ?r` False])+ arith
+ show ?thesis using l'_height t_height False by (subst height_balR2[OF `avl ?l'` `avl ?r` False])+ arith
next
case True
show ?thesis
- proof(cases rule: disjE[OF height_node_bal_r[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]])
+ proof(cases rule: disjE[OF height_balR[OF True `avl ?l'` `avl ?r`, of "snd (delete_max ?l)"]])
case 1
thus ?thesis using l'_height t_height True by arith
next
@@ -393,10 +391,10 @@
with Node 1 show ?thesis
proof(cases "x<n")
case True
- with Node 1 show ?thesis by (auto simp add:avl_node_bal_r)
+ with Node 1 show ?thesis by (auto simp add:avl_balR)
next
case False
- with Node 1 `x\<noteq>n` show ?thesis by (auto simp add:avl_node_bal_l)
+ with Node 1 `x\<noteq>n` show ?thesis by (auto simp add:avl_balL)
qed
qed
case 2
@@ -414,38 +412,38 @@
case True
show ?thesis
proof(cases "height r = height (delete x l) + 2")
- case False with Node 1 `x < n` show ?thesis by(auto simp: node_bal_r_def)
+ case False with Node 1 `x < n` show ?thesis by(auto simp: balR_def)
next
case True
- hence "(height (node_bal_r (delete x l) n r) = height (delete x l) + 2) \<or>
- height (node_bal_r (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
- using Node 2 by (intro height_node_bal_r) auto
+ hence "(height (balR (delete x l) n r) = height (delete x l) + 2) \<or>
+ height (balR (delete x l) n r) = height (delete x l) + 3" (is "?A \<or> ?B")
+ using Node 2 by (intro height_balR) auto
thus ?thesis
proof
assume ?A
- with `x < n` Node 2 show ?thesis by(auto simp: node_bal_r_def)
+ with `x < n` Node 2 show ?thesis by(auto simp: balR_def)
next
assume ?B
- with `x < n` Node 2 show ?thesis by(auto simp: node_bal_r_def)
+ with `x < n` Node 2 show ?thesis by(auto simp: balR_def)
qed
qed
next
case False
show ?thesis
proof(cases "height l = height (delete x r) + 2")
- case False with Node 1 `\<not>x < n` `x \<noteq> n` show ?thesis by(auto simp: node_bal_l_def)
+ case False with Node 1 `\<not>x < n` `x \<noteq> n` show ?thesis by(auto simp: balL_def)
next
case True
- hence "(height (node_bal_l l n (delete x r)) = height (delete x r) + 2) \<or>
- height (node_bal_l l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
- using Node 2 by (intro height_node_bal_l) auto
+ hence "(height (balL l n (delete x r)) = height (delete x r) + 2) \<or>
+ height (balL l n (delete x r)) = height (delete x r) + 3" (is "?A \<or> ?B")
+ using Node 2 by (intro height_balL) auto
thus ?thesis
proof
assume ?A
- with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: node_bal_l_def)
+ with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: balL_def)
next
assume ?B
- with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: node_bal_l_def)
+ with `\<not>x < n` `x \<noteq> n` Node 2 show ?thesis by(auto simp: balL_def)
qed
qed
qed