--- a/src/HOL/Data_Structures/AVL_Bal_Set.thy Mon May 11 11:15:41 2020 +0100
+++ b/src/HOL/Data_Structures/AVL_Bal_Set.thy Mon May 11 19:41:31 2020 +0200
@@ -26,9 +26,11 @@
subsection \<open>Code\<close>
-datatype 'a tree_bal2 = Same "'a tree_bal" | Diff "'a tree_bal"
+datatype 'a alt = Same 'a | Diff 'a
-fun tree :: "'a tree_bal2 \<Rightarrow> 'a tree_bal" where
+type_synonym 'a tree_bal2 = "'a tree_bal alt"
+
+fun tree :: "'a alt \<Rightarrow> 'a" where
"tree(Same t) = t" |
"tree(Diff t) = t"
@@ -59,12 +61,15 @@
Node B (c,Rh) C \<Rightarrow> Same(Node (Node A (a,Bal) B) (c,Bal) C) |
Node B (c,Lh) C \<Rightarrow> Same(rot2 A a B c 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
+fun ins :: "'a::linorder \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal2" where
+"ins x Leaf = Diff(Node Leaf (x, Bal) Leaf)" |
+"ins 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))"
+ LT \<Rightarrow> balL (ins x l) a b r |
+ GT \<Rightarrow> balR l a b (ins x r))"
+
+definition insert :: "'a::linorder \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal" where
+"insert x t = tree(ins x t)"
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
@@ -92,51 +97,54 @@
"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) =
+fun del :: "'a::linorder \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal2" where
+"del _ Leaf = Same Leaf" |
+"del 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))"
+ LT \<Rightarrow> baldL (del x l) a ba r |
+ GT \<Rightarrow> baldR l a ba (del x r))"
+
+definition delete :: "'a::linorder \<Rightarrow> 'a tree_bal \<Rightarrow> 'a tree_bal" where
+"delete x t = tree(del x t)"
lemmas split_max_induct = split_max.induct[case_names Node Leaf]
-lemmas splits = if_splits tree.splits tree_bal2.splits bal.splits
+lemmas splits = if_splits tree.splits alt.splits bal.splits
subsection \<open>Proofs\<close>
-subsubsection "Proofs about insert"
+subsubsection "Proofs about insertion"
-lemma avl_insert_case: "avl t \<Longrightarrow> case insert x t of
+lemma avl_ins_case: "avl t \<Longrightarrow> case ins x t of
Same t' \<Rightarrow> avl t' \<and> height t' = height t |
Diff t' \<Rightarrow> avl t' \<and> height t' = height t + 1 \<and>
(\<forall>l a r. t' = Node l (a,Bal) r \<longrightarrow> a = x \<and> l = Leaf \<and> r = Leaf)"
-apply(induction x t rule: insert.induct)
+apply(induction x t rule: ins.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)
+corollary avl_insert: "avl t \<Longrightarrow> avl(insert x t)"
+using avl_ins_case[of t x] by (simp add: insert_def split: splits)
-(* The following aux lemma simplifies the inorder_insert proof *)
+(* The following aux lemma simplifies the inorder_ins proof *)
-lemma insert_Diff[simp]: "avl t \<Longrightarrow>
- insert x t \<noteq> Diff Leaf \<and>
- (insert x t = Diff (Node l (a,Bal) r) \<longleftrightarrow> t = Leaf \<and> a = x \<and> l=Leaf \<and> r=Leaf) \<and>
- insert x t \<noteq> Diff (Node l (a,Rh) Leaf) \<and>
- insert x t \<noteq> Diff (Node Leaf (a,Lh) r)"
-by(drule avl_insert_case[of _ x]) (auto split: splits)
+lemma ins_Diff[simp]: "avl t \<Longrightarrow>
+ ins x t \<noteq> Diff Leaf \<and>
+ (ins x t = Diff (Node l (a,Bal) r) \<longleftrightarrow> t = Leaf \<and> a = x \<and> l=Leaf \<and> r=Leaf) \<and>
+ ins x t \<noteq> Diff (Node l (a,Rh) Leaf) \<and>
+ ins x t \<noteq> Diff (Node Leaf (a,Lh) r)"
+by(drule avl_ins_case[of _ x]) (auto split: splits)
-theorem inorder_insert:
- "\<lbrakk> avl t; sorted(inorder t) \<rbrakk> \<Longrightarrow> inorder(tree(insert x t)) = ins_list x (inorder t)"
+theorem inorder_ins:
+ "\<lbrakk> avl t; sorted(inorder t) \<rbrakk> \<Longrightarrow> inorder(tree(ins x t)) = ins_list x (inorder t)"
apply(induction t)
apply (auto simp: ins_list_simps split!: splits)
done
-subsubsection "Proofs for delete"
+subsubsection "Proofs about deletion"
lemma inorder_baldL:
"\<lbrakk> ba = Rh \<longrightarrow> r \<noteq> Leaf; avl r \<rbrakk>
@@ -157,15 +165,15 @@
apply simp
done
-lemma avl_delete_case: "avl t \<Longrightarrow> case delete x t of
+lemma avl_del_case: "avl t \<Longrightarrow> case del 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(induction x t rule: del.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)
+corollary avl_delete: "avl t \<Longrightarrow> avl(delete x t)"
+using avl_del_case[of t x] by(simp add: delete_def split: splits)
lemma inorder_split_maxD:
"\<lbrakk> split_max t = (t',a); t \<noteq> Leaf; avl t \<rbrakk> \<Longrightarrow>
@@ -178,8 +186,8 @@
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)"
+theorem inorder_del:
+ "\<lbrakk> avl t; sorted(inorder t) \<rbrakk> \<Longrightarrow> inorder (tree(del 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)
@@ -190,19 +198,19 @@
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 insert = insert
+ and delete = delete
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)
+ case 3 thus ?case by(simp add: inorder_ins insert_def)
next
- case 4 thus ?case by(simp add: inorder_delete)
+ case 4 thus ?case by(simp add: inorder_del delete_def)
next
- case 5 thus ?case by (simp add: empty_def)
+ case 5 thus ?case by (simp)
next
case 6 thus ?case by (simp add: avl_insert)
next