# HG changeset patch
# User nipkow
# Date 1523173593 -7200
# Node ID 08cc5ab18c84ee4726e67eb8293b445687389514
# Parent  9541f2c5ce8daa888c5eada4624b877de33d26d9
better name; added binary operations

diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/AVL_Set.thy
--- a/src/HOL/Data_Structures/AVL_Set.thy	Sat Apr 07 22:09:57 2018 +0200
+++ b/src/HOL/Data_Structures/AVL_Set.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -7,8 +7,8 @@
 
 theory AVL_Set
 imports
- Cmp
- Isin2
+  Cmp
+  Isin2
   "HOL-Number_Theory.Fib"
 begin
 
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Brother12_Set.thy
--- a/src/HOL/Data_Structures/Brother12_Set.thy	Sat Apr 07 22:09:57 2018 +0200
+++ b/src/HOL/Data_Structures/Brother12_Set.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -5,7 +5,7 @@
 theory Brother12_Set
 imports
   Cmp
-  Set_by_Ordered
+  Set_Interfaces
   "HOL-Number_Theory.Fib"
 begin
 
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Isin2.thy
--- a/src/HOL/Data_Structures/Isin2.thy	Sat Apr 07 22:09:57 2018 +0200
+++ b/src/HOL/Data_Structures/Isin2.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -6,7 +6,7 @@
 imports
   Tree2
   Cmp
-  Set_by_Ordered
+  Set_Interfaces
 begin
 
 fun isin :: "('a::linorder,'b) tree \<Rightarrow> 'a \<Rightarrow> bool" where
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Set_Interfaces.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Data_Structures/Set_Interfaces.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -0,0 +1,86 @@
+(* Author: Tobias Nipkow *)
+
+section \<open>Interfaces for Set ADT\<close>
+
+theory Set_Interfaces
+imports List_Ins_Del
+begin
+
+text \<open>The basic set interface with traditional specification (based on \<open>set\<close> and \<open>bst\<close>):\<close>
+
+locale Set =
+fixes empty :: "'s"
+fixes insert :: "'a \<Rightarrow> 's \<Rightarrow> 's"
+fixes delete :: "'a \<Rightarrow> 's \<Rightarrow> 's"
+fixes isin :: "'s \<Rightarrow> 'a \<Rightarrow> bool"
+fixes set :: "'s \<Rightarrow> 'a set"
+fixes invar :: "'s \<Rightarrow> bool"
+assumes set_empty:    "set empty = {}"
+assumes set_isin:     "invar s \<Longrightarrow> isin s x = (x \<in> set s)"
+assumes set_insert:   "invar s \<Longrightarrow> set(insert x s) = Set.insert x (set s)"
+assumes set_delete:   "invar s \<Longrightarrow> set(delete x s) = set s - {x}"
+assumes invar_empty:  "invar empty"
+assumes invar_insert: "invar s \<Longrightarrow> invar(insert x s)"
+assumes invar_delete: "invar s \<Longrightarrow> invar(delete x s)"
+
+
+text \<open>The basic set interface with \<open>inorder\<close>-based specification:\<close>
+
+locale Set_by_Ordered =
+fixes empty :: "'t"
+fixes insert :: "'a::linorder \<Rightarrow> 't \<Rightarrow> 't"
+fixes delete :: "'a \<Rightarrow> 't \<Rightarrow> 't"
+fixes isin :: "'t \<Rightarrow> 'a \<Rightarrow> bool"
+fixes inorder :: "'t \<Rightarrow> 'a list"
+fixes inv :: "'t \<Rightarrow> bool"
+assumes empty: "inorder empty = []"
+assumes isin: "inv t \<and> sorted(inorder t) \<Longrightarrow>
+  isin t x = (x \<in> set (inorder t))"
+assumes insert: "inv t \<and> sorted(inorder t) \<Longrightarrow>
+  inorder(insert x t) = ins_list x (inorder t)"
+assumes delete: "inv t \<and> sorted(inorder t) \<Longrightarrow>
+  inorder(delete x t) = del_list x (inorder t)"
+assumes inv_empty:  "inv empty"
+assumes inv_insert: "inv t \<and> sorted(inorder t) \<Longrightarrow> inv(insert x t)"
+assumes inv_delete: "inv t \<and> sorted(inorder t) \<Longrightarrow> inv(delete x t)"
+begin
+
+text \<open>It implements the traditional specification:\<close>
+
+sublocale Set
+  empty insert delete isin "set o inorder" "\<lambda>t. inv t \<and> sorted(inorder t)"
+proof(standard, goal_cases)
+  case 1 show ?case by (auto simp: empty)
+next
+  case 2 thus ?case by(simp add: isin)
+next
+  case 3 thus ?case by(simp add: insert set_ins_list)
+next
+  case (4 s x) thus ?case
+    using delete[OF 4, of x] by (auto simp: distinct_if_sorted set_del_list_eq)
+next
+  case 5 thus ?case by(simp add: empty inv_empty)
+next
+  case 6 thus ?case by(simp add: insert inv_insert sorted_ins_list)
+next
+  case 7 thus ?case by (auto simp: delete inv_delete sorted_del_list)
+qed
+
+end
+
+
+text \<open>Set2 = Set with binary operations:\<close>
+
+locale Set2 = Set
+  where insert = insert for insert :: "'a \<Rightarrow> 's \<Rightarrow> 's" (*for typing purposes only*) +
+fixes union :: "'s \<Rightarrow> 's \<Rightarrow> 's"
+fixes inter :: "'s \<Rightarrow> 's \<Rightarrow> 's"
+fixes diff  :: "'s \<Rightarrow> 's \<Rightarrow> 's"
+assumes set_union:   "\<lbrakk> invar s1; invar s2 \<rbrakk> \<Longrightarrow> set(union s1 s2) = set s1 \<union> set s2"
+assumes set_inter:   "\<lbrakk> invar s1; invar s2 \<rbrakk> \<Longrightarrow> set(inter s1 s2) = set s1 \<inter> set s2"
+assumes set_diff:   "\<lbrakk> invar s1; invar s2 \<rbrakk> \<Longrightarrow> set(diff s1 s2) = set s1 - set s2"
+assumes invar_union:   "\<lbrakk> invar s1; invar s2 \<rbrakk> \<Longrightarrow> invar(union s1 s2)"
+assumes invar_inter:   "\<lbrakk> invar s1; invar s2 \<rbrakk> \<Longrightarrow> invar(inter s1 s2)"
+assumes invar_diff:   "\<lbrakk> invar s1; invar s2 \<rbrakk> \<Longrightarrow> invar(diff s1 s2)"
+
+end
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Set_by_Ordered.thy
--- a/src/HOL/Data_Structures/Set_by_Ordered.thy	Sat Apr 07 22:09:57 2018 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,64 +0,0 @@
-(* Author: Tobias Nipkow *)
-
-section \<open>Implementing Ordered Sets\<close>
-
-theory Set_by_Ordered
-imports List_Ins_Del
-begin
-
-locale Set =
-fixes empty :: "'s"
-fixes insert :: "'a \<Rightarrow> 's \<Rightarrow> 's"
-fixes delete :: "'a \<Rightarrow> 's \<Rightarrow> 's"
-fixes isin :: "'s \<Rightarrow> 'a \<Rightarrow> bool"
-fixes set :: "'s \<Rightarrow> 'a set"
-fixes invar :: "'s \<Rightarrow> bool"
-assumes set_empty:    "set empty = {}"
-assumes set_isin:     "invar s \<Longrightarrow> isin s x = (x \<in> set s)"
-assumes set_insert:   "invar s \<Longrightarrow> set(insert x s) = Set.insert x (set s)"
-assumes set_delete:   "invar s \<Longrightarrow> set(delete x s) = set s - {x}"
-assumes invar_empty:  "invar empty"
-assumes invar_insert: "invar s \<Longrightarrow> invar(insert x s)"
-assumes invar_delete: "invar s \<Longrightarrow> invar(delete x s)"
-
-locale Set_by_Ordered =
-fixes empty :: "'t"
-fixes insert :: "'a::linorder \<Rightarrow> 't \<Rightarrow> 't"
-fixes delete :: "'a \<Rightarrow> 't \<Rightarrow> 't"
-fixes isin :: "'t \<Rightarrow> 'a \<Rightarrow> bool"
-fixes inorder :: "'t \<Rightarrow> 'a list"
-fixes inv :: "'t \<Rightarrow> bool"
-assumes empty: "inorder empty = []"
-assumes isin: "inv t \<and> sorted(inorder t) \<Longrightarrow>
-  isin t x = (x \<in> set (inorder t))"
-assumes insert: "inv t \<and> sorted(inorder t) \<Longrightarrow>
-  inorder(insert x t) = ins_list x (inorder t)"
-assumes delete: "inv t \<and> sorted(inorder t) \<Longrightarrow>
-  inorder(delete x t) = del_list x (inorder t)"
-assumes inv_empty:  "inv empty"
-assumes inv_insert: "inv t \<and> sorted(inorder t) \<Longrightarrow> inv(insert x t)"
-assumes inv_delete: "inv t \<and> sorted(inorder t) \<Longrightarrow> inv(delete x t)"
-begin
-
-sublocale Set
-  empty insert delete isin "set o inorder" "\<lambda>t. inv t \<and> sorted(inorder t)"
-proof(standard, goal_cases)
-  case 1 show ?case by (auto simp: empty)
-next
-  case 2 thus ?case by(simp add: isin)
-next
-  case 3 thus ?case by(simp add: insert set_ins_list)
-next
-  case (4 s x) thus ?case
-    using delete[OF 4, of x] by (auto simp: distinct_if_sorted set_del_list_eq)
-next
-  case 5 thus ?case by(simp add: empty inv_empty)
-next
-  case 6 thus ?case by(simp add: insert inv_insert sorted_ins_list)
-next
-  case 7 thus ?case by (auto simp: delete inv_delete sorted_del_list)
-qed
-
-end
-
-end
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Tree234_Set.thy
--- a/src/HOL/Data_Structures/Tree234_Set.thy	Sat Apr 07 22:09:57 2018 +0200
+++ b/src/HOL/Data_Structures/Tree234_Set.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -6,7 +6,7 @@
 imports
   Tree234
   Cmp
-  "../Data_Structures/Set_by_Ordered"
+  "Set_Interfaces"
 begin
 
 subsection \<open>Set operations on 2-3-4 trees\<close>
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Tree23_Set.thy
--- a/src/HOL/Data_Structures/Tree23_Set.thy	Sat Apr 07 22:09:57 2018 +0200
+++ b/src/HOL/Data_Structures/Tree23_Set.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -6,7 +6,7 @@
 imports
   Tree23
   Cmp
-  Set_by_Ordered
+  Set_Interfaces
 begin
 
 fun isin :: "'a::linorder tree23 \<Rightarrow> 'a \<Rightarrow> bool" where
diff -r 9541f2c5ce8d -r 08cc5ab18c84 src/HOL/Data_Structures/Tree_Set.thy
--- a/src/HOL/Data_Structures/Tree_Set.thy	Sat Apr 07 22:09:57 2018 +0200
+++ b/src/HOL/Data_Structures/Tree_Set.thy	Sun Apr 08 09:46:33 2018 +0200
@@ -6,7 +6,7 @@
 imports
   "HOL-Library.Tree"
   Cmp
-  Set_by_Ordered
+  Set_Interfaces
 begin
 
 fun isin :: "'a::linorder tree \<Rightarrow> 'a \<Rightarrow> bool" where