author | nipkow |
Mon, 11 Jun 2018 16:29:27 +0200 | |
changeset 68413 | b56ed5010e69 |
parent 68261 | 035c78bb0a66 |
child 68484 | 59793df7f853 |
permissions | -rw-r--r-- |
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(* Author: Tobias Nipkow *) |
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section "Join-Based Implementation of Sets" |
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theory Set2_Join |
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imports |
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Isin2 |
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begin |
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text \<open>This theory implements the set operations \<open>insert\<close>, \<open>delete\<close>, |
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\<open>union\<close>, \<open>inter\<close>section and \<open>diff\<close>erence. The implementation is based on binary search trees. |
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All operations are reduced to a single operation \<open>join l x r\<close> that joins two BSTs \<open>l\<close> and \<open>r\<close> |
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and an element \<open>x\<close> such that \<open>l < x < r\<close>. |
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The theory is based on theory @{theory Tree2} where nodes have an additional field. |
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This field is ignored here but it means that this theory can be instantiated |
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with red-black trees (see theory @{file "Set2_Join_RBT.thy"}) and other balanced trees. |
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This approach is very concrete and fixes the type of trees. |
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Alternatively, one could assume some abstract type @{typ 't} of trees with suitable decomposition |
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and recursion operators on it.\<close> |
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locale Set2_Join = |
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fixes join :: "('a::linorder,'b) tree \<Rightarrow> 'a \<Rightarrow> ('a,'b) tree \<Rightarrow> ('a,'b) tree" |
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fixes inv :: "('a,'b) tree \<Rightarrow> bool" |
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assumes set_join: "set_tree (join l a r) = set_tree l \<union> {a} \<union> set_tree r" |
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assumes bst_join: |
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"\<lbrakk> bst l; bst r; \<forall>x \<in> set_tree l. x < a; \<forall>y \<in> set_tree r. a < y \<rbrakk> |
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\<Longrightarrow> bst (join l a r)" |
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assumes inv_Leaf: "inv \<langle>\<rangle>" |
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assumes inv_join: "\<lbrakk> inv l; inv r \<rbrakk> \<Longrightarrow> inv (join l k r)" |
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assumes inv_Node: "\<lbrakk> inv (Node l x h r) \<rbrakk> \<Longrightarrow> inv l \<and> inv r" |
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begin |
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declare set_join [simp] |
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subsection "\<open>split_min\<close>" |
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fun split_min :: "('a,'b) tree \<Rightarrow> 'a \<times> ('a,'b) tree" where |
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"split_min (Node l x _ r) = |
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(if l = Leaf then (x,r) else let (m,l') = split_min l in (m, join l' x r))" |
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lemma split_min_set: |
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"\<lbrakk> split_min t = (x,t'); t \<noteq> Leaf \<rbrakk> \<Longrightarrow> x \<in> set_tree t \<and> set_tree t = Set.insert x (set_tree t')" |
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proof(induction t arbitrary: t') |
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case Node thus ?case by(auto split: prod.splits if_splits dest: inv_Node) |
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next |
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case Leaf thus ?case by simp |
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qed |
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lemma split_min_bst: |
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"\<lbrakk> split_min t = (x,t'); bst t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> bst t' \<and> (\<forall>x' \<in> set_tree t'. x < x')" |
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proof(induction t arbitrary: t') |
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case Node thus ?case by(fastforce simp: split_min_set bst_join split: prod.splits if_splits) |
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next |
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case Leaf thus ?case by simp |
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qed |
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lemma split_min_inv: |
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"\<lbrakk> split_min t = (x,t'); inv t; t \<noteq> Leaf \<rbrakk> \<Longrightarrow> inv t'" |
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proof(induction t arbitrary: t') |
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case Node thus ?case by(auto simp: inv_join split: prod.splits if_splits dest: inv_Node) |
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next |
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case Leaf thus ?case by simp |
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qed |
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subsection "\<open>join2\<close>" |
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definition join2 :: "('a,'b) tree \<Rightarrow> ('a,'b) tree \<Rightarrow> ('a,'b) tree" where |
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"join2 l r = (if r = Leaf then l else let (x,r') = split_min r in join l x r')" |
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lemma set_join2[simp]: "set_tree (join2 l r) = set_tree l \<union> set_tree r" |
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by(simp add: join2_def split_min_set split: prod.split) |
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lemma bst_join2: "\<lbrakk> bst l; bst r; \<forall>x \<in> set_tree l. \<forall>y \<in> set_tree r. x < y \<rbrakk> |
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\<Longrightarrow> bst (join2 l r)" |
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by(simp add: join2_def bst_join split_min_set split_min_bst split: prod.split) |
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lemma inv_join2: "\<lbrakk> inv l; inv r \<rbrakk> \<Longrightarrow> inv (join2 l r)" |
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by(simp add: join2_def inv_join split_min_set split_min_inv split: prod.split) |
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subsection "\<open>split\<close>" |
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fun split :: "('a,'b)tree \<Rightarrow> 'a \<Rightarrow> ('a,'b)tree \<times> bool \<times> ('a,'b)tree" where |
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"split Leaf k = (Leaf, False, Leaf)" | |
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"split (Node l a _ r) k = |
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(if k < a then let (l1,b,l2) = split l k in (l1, b, join l2 a r) else |
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if a < k then let (r1,b,r2) = split r k in (join l a r1, b, r2) |
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else (l, True, r))" |
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lemma split: "split t k = (l,kin,r) \<Longrightarrow> bst t \<Longrightarrow> |
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set_tree l = {x \<in> set_tree t. x < k} \<and> set_tree r = {x \<in> set_tree t. k < x} |
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\<and> (kin = (k \<in> set_tree t)) \<and> bst l \<and> bst r" |
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proof(induction t arbitrary: l kin r) |
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case Leaf thus ?case by simp |
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next |
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case Node thus ?case by(force split!: prod.splits if_splits intro!: bst_join) |
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qed |
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lemma split_inv: "split t k = (l,kin,r) \<Longrightarrow> inv t \<Longrightarrow> inv l \<and> inv r" |
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proof(induction t arbitrary: l kin r) |
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case Leaf thus ?case by simp |
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next |
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case Node |
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thus ?case by(force simp: inv_join split!: prod.splits if_splits dest!: inv_Node) |
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qed |
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declare split.simps[simp del] |
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subsection "\<open>insert\<close>" |
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definition insert :: "'a \<Rightarrow> ('a,'b) tree \<Rightarrow> ('a,'b) tree" where |
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"insert k t = (let (l,_,r) = split t k in join l k r)" |
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lemma set_tree_insert: "bst t \<Longrightarrow> set_tree (insert x t) = Set.insert x (set_tree t)" |
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by(auto simp add: insert_def split split: prod.split) |
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lemma bst_insert: "bst t \<Longrightarrow> bst (insert x t)" |
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by(auto simp add: insert_def bst_join dest: split split: prod.split) |
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lemma inv_insert: "inv t \<Longrightarrow> inv (insert x t)" |
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by(force simp: insert_def inv_join dest: split_inv split: prod.split) |
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subsection "\<open>delete\<close>" |
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definition delete :: "'a \<Rightarrow> ('a,'b) tree \<Rightarrow> ('a,'b) tree" where |
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"delete k t = (let (l,_,r) = split t k in join2 l r)" |
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lemma set_tree_delete: "bst t \<Longrightarrow> set_tree (delete k t) = set_tree t - {k}" |
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by(auto simp: delete_def split split: prod.split) |
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lemma bst_delete: "bst t \<Longrightarrow> bst (delete x t)" |
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by(force simp add: delete_def intro: bst_join2 dest: split split: prod.split) |
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lemma inv_delete: "inv t \<Longrightarrow> inv (delete x t)" |
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by(force simp: delete_def inv_join2 dest: split_inv split: prod.split) |
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subsection "\<open>union\<close>" |
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diff
changeset
|
144 |
fun union :: "('a,'b)tree \<Rightarrow> ('a,'b)tree \<Rightarrow> ('a,'b)tree" where |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
145 |
"union t1 t2 = |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
146 |
(if t1 = Leaf then t2 else |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
147 |
if t2 = Leaf then t1 else |
68413 | 148 |
case t1 of Node l1 k _ r1 \<Rightarrow> |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
149 |
let (l2,_ ,r2) = split t2 k; |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
150 |
l' = union l1 l2; r' = union r1 r2 |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
151 |
in join l' k r')" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
152 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
153 |
declare union.simps [simp del] |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
154 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
155 |
lemma set_tree_union: "bst t2 \<Longrightarrow> set_tree (union t1 t2) = set_tree t1 \<union> set_tree t2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
156 |
proof(induction t1 t2 rule: union.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
157 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
158 |
then show ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
159 |
by (auto simp: union.simps[of t1 t2] split split: tree.split prod.split) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
160 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
161 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
162 |
lemma bst_union: "\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> bst (union t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
163 |
proof(induction t1 t2 rule: union.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
164 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
165 |
thus ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
166 |
by(fastforce simp: union.simps[of t1 t2] set_tree_union split intro!: bst_join |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
167 |
split: tree.split prod.split) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
168 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
169 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
170 |
lemma inv_union: "\<lbrakk> inv t1; inv t2 \<rbrakk> \<Longrightarrow> inv (union t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
171 |
proof(induction t1 t2 rule: union.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
172 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
173 |
thus ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
174 |
by(auto simp:union.simps[of t1 t2] inv_join split_inv |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
175 |
split!: tree.split prod.split dest: inv_Node) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
176 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
177 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
178 |
subsection "\<open>inter\<close>" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
179 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
180 |
fun inter :: "('a,'b)tree \<Rightarrow> ('a,'b)tree \<Rightarrow> ('a,'b)tree" where |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
181 |
"inter t1 t2 = |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
182 |
(if t1 = Leaf then Leaf else |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
183 |
if t2 = Leaf then Leaf else |
68413 | 184 |
case t1 of Node l1 k _ r1 \<Rightarrow> |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
185 |
let (l2,kin,r2) = split t2 k; |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
186 |
l' = inter l1 l2; r' = inter r1 r2 |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
187 |
in if kin then join l' k r' else join2 l' r')" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
188 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
189 |
declare inter.simps [simp del] |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
190 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
191 |
lemma set_tree_inter: |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
192 |
"\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> set_tree (inter t1 t2) = set_tree t1 \<inter> set_tree t2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
193 |
proof(induction t1 t2 rule: inter.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
194 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
195 |
show ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
196 |
proof (cases t1) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
197 |
case Leaf thus ?thesis by (simp add: inter.simps) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
198 |
next |
68413 | 199 |
case [simp]: (Node l1 k _ r1) |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
200 |
show ?thesis |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
201 |
proof (cases "t2 = Leaf") |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
202 |
case True thus ?thesis by (simp add: inter.simps) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
203 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
204 |
case False |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
205 |
let ?L1 = "set_tree l1" let ?R1 = "set_tree r1" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
206 |
have *: "k \<notin> ?L1 \<union> ?R1" using \<open>bst t1\<close> by (fastforce) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
207 |
obtain l2 kin r2 where sp: "split t2 k = (l2,kin,r2)" using prod_cases3 by blast |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
208 |
let ?L2 = "set_tree l2" let ?R2 = "set_tree r2" let ?K = "if kin then {k} else {}" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
209 |
have t2: "set_tree t2 = ?L2 \<union> ?R2 \<union> ?K" and |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
210 |
**: "?L2 \<inter> ?R2 = {}" "k \<notin> ?L2 \<union> ?R2" "?L1 \<inter> ?R2 = {}" "?L2 \<inter> ?R1 = {}" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
211 |
using split[OF sp] \<open>bst t1\<close> \<open>bst t2\<close> by (force, force, force, force, force) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
212 |
have IHl: "set_tree (inter l1 l2) = set_tree l1 \<inter> set_tree l2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
213 |
using "1.IH"(1)[OF _ False _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
214 |
have IHr: "set_tree (inter r1 r2) = set_tree r1 \<inter> set_tree r2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
215 |
using "1.IH"(2)[OF _ False _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
216 |
have "set_tree t1 \<inter> set_tree t2 = (?L1 \<union> ?R1 \<union> {k}) \<inter> (?L2 \<union> ?R2 \<union> ?K)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
217 |
by(simp add: t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
218 |
also have "\<dots> = (?L1 \<inter> ?L2) \<union> (?R1 \<inter> ?R2) \<union> ?K" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
219 |
using * ** by auto |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
220 |
also have "\<dots> = set_tree (inter t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
221 |
using IHl IHr sp inter.simps[of t1 t2] False by(simp) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
222 |
finally show ?thesis by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
223 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
224 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
225 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
226 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
227 |
lemma bst_inter: "\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> bst (inter t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
228 |
proof(induction t1 t2 rule: inter.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
229 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
230 |
thus ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
231 |
by(fastforce simp: inter.simps[of t1 t2] set_tree_inter split Let_def |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
232 |
intro!: bst_join bst_join2 split: tree.split prod.split) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
233 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
234 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
235 |
lemma inv_inter: "\<lbrakk> inv t1; inv t2 \<rbrakk> \<Longrightarrow> inv (inter t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
236 |
proof(induction t1 t2 rule: inter.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
237 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
238 |
thus ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
239 |
by(auto simp: inter.simps[of t1 t2] inv_join inv_join2 split_inv Let_def |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
240 |
split!: tree.split prod.split dest: inv_Node) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
241 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
242 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
243 |
subsection "\<open>diff\<close>" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
244 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
245 |
fun diff :: "('a,'b)tree \<Rightarrow> ('a,'b)tree \<Rightarrow> ('a,'b)tree" where |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
246 |
"diff t1 t2 = |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
247 |
(if t1 = Leaf then Leaf else |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
248 |
if t2 = Leaf then t1 else |
68413 | 249 |
case t2 of Node l2 k _ r2 \<Rightarrow> |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
250 |
let (l1,_,r1) = split t1 k; |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
251 |
l' = diff l1 l2; r' = diff r1 r2 |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
252 |
in join2 l' r')" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
253 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
254 |
declare diff.simps [simp del] |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
255 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
256 |
lemma set_tree_diff: |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
257 |
"\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> set_tree (diff t1 t2) = set_tree t1 - set_tree t2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
258 |
proof(induction t1 t2 rule: diff.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
259 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
260 |
show ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
261 |
proof (cases t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
262 |
case Leaf thus ?thesis by (simp add: diff.simps) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
263 |
next |
68413 | 264 |
case [simp]: (Node l2 k _ r2) |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
265 |
show ?thesis |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
266 |
proof (cases "t1 = Leaf") |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
267 |
case True thus ?thesis by (simp add: diff.simps) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
268 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
269 |
case False |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
270 |
let ?L2 = "set_tree l2" let ?R2 = "set_tree r2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
271 |
obtain l1 kin r1 where sp: "split t1 k = (l1,kin,r1)" using prod_cases3 by blast |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
272 |
let ?L1 = "set_tree l1" let ?R1 = "set_tree r1" let ?K = "if kin then {k} else {}" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
273 |
have t1: "set_tree t1 = ?L1 \<union> ?R1 \<union> ?K" and |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
274 |
**: "k \<notin> ?L1 \<union> ?R1" "?L1 \<inter> ?R2 = {}" "?L2 \<inter> ?R1 = {}" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
275 |
using split[OF sp] \<open>bst t1\<close> \<open>bst t2\<close> by (force, force, force, force) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
276 |
have IHl: "set_tree (diff l1 l2) = set_tree l1 - set_tree l2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
277 |
using "1.IH"(1)[OF False _ _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
278 |
have IHr: "set_tree (diff r1 r2) = set_tree r1 - set_tree r2" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
279 |
using "1.IH"(2)[OF False _ _ sp[symmetric]] "1.prems"(1,2) split[OF sp] by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
280 |
have "set_tree t1 - set_tree t2 = (?L1 \<union> ?R1) - (?L2 \<union> ?R2 \<union> {k})" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
281 |
by(simp add: t1) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
282 |
also have "\<dots> = (?L1 - ?L2) \<union> (?R1 - ?R2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
283 |
using ** by auto |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
284 |
also have "\<dots> = set_tree (diff t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
285 |
using IHl IHr sp diff.simps[of t1 t2] False by(simp) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
286 |
finally show ?thesis by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
287 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
288 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
289 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
290 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
291 |
lemma bst_diff: "\<lbrakk> bst t1; bst t2 \<rbrakk> \<Longrightarrow> bst (diff t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
292 |
proof(induction t1 t2 rule: diff.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
293 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
294 |
thus ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
295 |
by(fastforce simp: diff.simps[of t1 t2] set_tree_diff split Let_def |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
296 |
intro!: bst_join bst_join2 split: tree.split prod.split) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
297 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
298 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
299 |
lemma inv_diff: "\<lbrakk> inv t1; inv t2 \<rbrakk> \<Longrightarrow> inv (diff t1 t2)" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
300 |
proof(induction t1 t2 rule: diff.induct) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
301 |
case (1 t1 t2) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
302 |
thus ?case |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
303 |
by(auto simp: diff.simps[of t1 t2] inv_join inv_join2 split_inv Let_def |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
304 |
split!: tree.split prod.split dest: inv_Node) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
305 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
306 |
|
68261 | 307 |
text \<open>Locale @{locale Set2_Join} implements locale @{locale Set2}:\<close> |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
308 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
309 |
sublocale Set2 |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
310 |
where empty = Leaf and insert = insert and delete = delete and isin = isin |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
311 |
and union = union and inter = inter and diff = diff |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
312 |
and set = set_tree and invar = "\<lambda>t. inv t \<and> bst t" |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
313 |
proof (standard, goal_cases) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
314 |
case 1 show ?case by (simp) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
315 |
next |
67967 | 316 |
case 2 thus ?case by(simp add: isin_set_tree) |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
317 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
318 |
case 3 thus ?case by (simp add: set_tree_insert) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
319 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
320 |
case 4 thus ?case by (simp add: set_tree_delete) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
321 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
322 |
case 5 thus ?case by (simp add: inv_Leaf) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
323 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
324 |
case 6 thus ?case by (simp add: bst_insert inv_insert) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
325 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
326 |
case 7 thus ?case by (simp add: bst_delete inv_delete) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
327 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
328 |
case 8 thus ?case by(simp add: set_tree_union) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
329 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
330 |
case 9 thus ?case by(simp add: set_tree_inter) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
331 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
332 |
case 10 thus ?case by(simp add: set_tree_diff) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
333 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
334 |
case 11 thus ?case by (simp add: bst_union inv_union) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
335 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
336 |
case 12 thus ?case by (simp add: bst_inter inv_inter) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
337 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
338 |
case 13 thus ?case by (simp add: bst_diff inv_diff) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
339 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
340 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
341 |
end |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
342 |
|
68261 | 343 |
interpretation unbal: Set2_Join |
68413 | 344 |
where join = "\<lambda>l x r. Node l x () r" and inv = "\<lambda>t. True" |
67966
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
345 |
proof (standard, goal_cases) |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
346 |
case 1 show ?case by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
347 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
348 |
case 2 thus ?case by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
349 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
350 |
case 3 thus ?case by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
351 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
352 |
case 4 thus ?case by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
353 |
next |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
354 |
case 5 thus ?case by simp |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
355 |
qed |
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
356 |
|
f13796496e82
Added binary set operations with join-based implementation
nipkow
parents:
diff
changeset
|
357 |
end |