# HG changeset patch
# User nipkow
# Date 1582303916 -3600
# Node ID a31a9da4369460e87007e79d42819de839e2fdcb
# Parent  ed8d50969995f4a9c6361f43555a1f881e7222cd
tuned deletion

diff -r ed8d50969995 -r a31a9da43694 src/HOL/Data_Structures/RBT.thy
--- a/src/HOL/Data_Structures/RBT.thy	Thu Feb 20 09:05:19 2020 +0100
+++ b/src/HOL/Data_Structures/RBT.thy	Fri Feb 21 17:51:56 2020 +0100
@@ -39,18 +39,18 @@
 "baldR (R t1 a (B t2 b t3)) c t4 = R (baliL (paint Red t1) a t2) b (B t3 c t4)" |
 "baldR t1 a t2 = R t1 a t2"
 
-fun combine :: "'a rbt \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
-"combine Leaf t = t" |
-"combine t Leaf = t" |
-"combine (R t1 a t2) (R t3 c t4) =
-  (case combine t2 t3 of
+fun app :: "'a rbt \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
+"app Leaf t = t" |
+"app t Leaf = t" |
+"app (R t1 a t2) (R t3 c t4) =
+  (case app t2 t3 of
      R u2 b u3 \<Rightarrow> (R (R t1 a u2) b (R u3 c t4)) |
      t23 \<Rightarrow> R t1 a (R t23 c t4))" |
-"combine (B t1 a t2) (B t3 c t4) =
-  (case combine t2 t3 of
+"app (B t1 a t2) (B t3 c t4) =
+  (case app t2 t3 of
      R u2 b u3 \<Rightarrow> R (B t1 a u2) b (B u3 c t4) |
      t23 \<Rightarrow> baldL t1 a (B t23 c t4))" |
-"combine t1 (R t2 a t3) = R (combine t1 t2) a t3" |
-"combine (R t1 a t2) t3 = R t1 a (combine t2 t3)" 
+"app t1 (R t2 a t3) = R (app t1 t2) a t3" |
+"app (R t1 a t2) t3 = R t1 a (app t2 t3)" 
 
 end
diff -r ed8d50969995 -r a31a9da43694 src/HOL/Data_Structures/RBT_Map.thy
--- a/src/HOL/Data_Structures/RBT_Map.thy	Thu Feb 20 09:05:19 2020 +0100
+++ b/src/HOL/Data_Structures/RBT_Map.thy	Fri Feb 21 17:51:56 2020 +0100
@@ -29,7 +29,7 @@
            then baldL (del x l) (a,b) r else R (del x l) (a,b) r |
      GT \<Rightarrow> if r \<noteq> Leaf\<and> color r = Black
            then baldR l (a,b) (del x r) else R l (a,b) (del x r) |
-  EQ \<Rightarrow> combine l r)"
+  EQ \<Rightarrow> app l r)"
 
 definition delete :: "'a::linorder \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "delete x t = paint Black (del x t)"
@@ -49,7 +49,7 @@
 lemma inorder_del:
  "sorted1(inorder t) \<Longrightarrow>  inorder(del x t) = del_list x (inorder t)"
 by(induction x t rule: del.induct)
-  (auto simp: del_list_simps inorder_combine inorder_baldL inorder_baldR)
+  (auto simp: del_list_simps inorder_app inorder_baldL inorder_baldR)
 
 lemma inorder_delete:
   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
@@ -86,7 +86,7 @@
   thus ?case proof (elim disjE)
     assume "x = y"
     with 2 show ?thesis
-    by (cases c) (simp_all add: invh_combine invc_combine)
+    by (cases c) (simp_all add: invh_app invc_app)
   next
     assume "x < y"
     with 2 show ?thesis
diff -r ed8d50969995 -r a31a9da43694 src/HOL/Data_Structures/RBT_Set.thy
--- a/src/HOL/Data_Structures/RBT_Set.thy	Thu Feb 20 09:05:19 2020 +0100
+++ b/src/HOL/Data_Structures/RBT_Set.thy	Fri Feb 21 17:51:56 2020 +0100
@@ -41,7 +41,7 @@
            then baldL (del x l) a r else R (del x l) a r |
      GT \<Rightarrow> if r \<noteq> Leaf\<and> color r = Black
            then baldR l a (del x r) else R l a (del x r) |
-     EQ \<Rightarrow> combine l r)"
+     EQ \<Rightarrow> app l r)"
 
 definition delete :: "'a::linorder \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
 "delete x t = paint Black (del x t)"
@@ -79,15 +79,15 @@
 by(cases "(l,a,r)" rule: baldR.cases)
   (auto simp:  inorder_baliL inorder_baliR inorder_paint)
 
-lemma inorder_combine:
-  "inorder(combine l r) = inorder l @ inorder r"
-by(induction l r rule: combine.induct)
+lemma inorder_app:
+  "inorder(app l r) = inorder l @ inorder r"
+by(induction l r rule: app.induct)
   (auto simp: inorder_baldL inorder_baldR split: tree.split color.split)
 
 lemma inorder_del:
  "sorted(inorder t) \<Longrightarrow>  inorder(del x t) = del_list x (inorder t)"
 by(induction x t rule: del.induct)
-  (auto simp: del_list_simps inorder_combine inorder_baldL inorder_baldR)
+  (auto simp: del_list_simps inorder_app inorder_baldL inorder_baldR)
 
 lemma inorder_delete:
   "sorted(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
@@ -225,16 +225,16 @@
 lemma invc2_baldR: "\<lbrakk> invc l; invc2 r \<rbrakk> \<Longrightarrow>invc2 (baldR l a r)"
 by (induct l a r rule: baldR.induct) (auto simp: invc_baliL paint2 invc2I)
 
-lemma invh_combine:
+lemma invh_app:
   "\<lbrakk> invh l; invh r; bheight l = bheight r \<rbrakk>
-  \<Longrightarrow> invh (combine l r) \<and> bheight (combine l r) = bheight l"
-by (induct l r rule: combine.induct) 
+  \<Longrightarrow> invh (app l r) \<and> bheight (app l r) = bheight l"
+by (induct l r rule: app.induct) 
    (auto simp: invh_baldL_Black split: tree.splits color.splits)
 
-lemma invc_combine: 
+lemma invc_app: 
   "\<lbrakk> invc l; invc r \<rbrakk> \<Longrightarrow>
-  (color l = Black \<and> color r = Black \<longrightarrow> invc (combine l r)) \<and> invc2 (combine l r)"
-by (induct l r rule: combine.induct)
+  (color l = Black \<and> color r = Black \<longrightarrow> invc (app l r)) \<and> invc2 (app l r)"
+by (induct l r rule: app.induct)
    (auto simp: invc_baldL invc2I split: tree.splits color.splits)
 
 text \<open>All in one:\<close>
@@ -253,11 +253,11 @@
 by (induct l a r rule: baldR.induct)
    (auto simp: inv_baliL invh_paint bheight_baliL bheight_paint_Red paint2 invc2I)
 
-lemma inv_combine:
+lemma inv_app:
   "\<lbrakk> invh l; invh r; bheight l = bheight r; invc l; invc r \<rbrakk>
-  \<Longrightarrow> invh (combine l r) \<and> bheight (combine l r) = bheight l
-  \<and> invc2 (combine l r) \<and> (color l = Black \<and> color r = Black \<longrightarrow> invc (combine l r))"
-by (induct l r rule: combine.induct) 
+  \<Longrightarrow> invh (app l r) \<and> bheight (app l r) = bheight l
+  \<and> invc2 (app l r) \<and> (color l = Black \<and> color r = Black \<longrightarrow> invc (app l r))"
+by (induct l r rule: app.induct) 
    (auto simp: invh_baldL_Black inv_baldL invc2I split: tree.splits color.splits)
 
 lemma neq_LeafD: "t \<noteq> Leaf \<Longrightarrow> \<exists>l x c r. t = Node l (x,c) r"
@@ -268,7 +268,7 @@
    (color t = Red \<and> bheight (del x t) = bheight t \<and> invc (del x t) \<or>
     color t = Black \<and> bheight (del x t) = bheight t - 1 \<and> invc2 (del x t))"
 by(induct x t rule: del.induct)
-  (auto simp: inv_baldL inv_baldR inv_combine dest!: neq_LeafD)
+  (auto simp: inv_baldL inv_baldR inv_app dest!: neq_LeafD)
 
 theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete x t)"
 by (metis delete_def rbt_def color_paint_Black inv_del invc2I invh_paint)
diff -r ed8d50969995 -r a31a9da43694 src/HOL/Data_Structures/Tree_Set.thy
--- a/src/HOL/Data_Structures/Tree_Set.thy	Thu Feb 20 09:05:19 2020 +0100
+++ b/src/HOL/Data_Structures/Tree_Set.thy	Fri Feb 21 17:51:56 2020 +0100
@@ -30,6 +30,8 @@
      EQ \<Rightarrow> Node l a r |
      GT \<Rightarrow> Node l a (insert x r))"
 
+text \<open>Deletion by replacing:\<close>
+
 fun split_min :: "'a tree \<Rightarrow> 'a * 'a tree" where
 "split_min (Node l a r) =
   (if l = Leaf then (a,r) else let (x,l') = split_min l in (x, Node l' a r))"
@@ -42,6 +44,24 @@
      GT \<Rightarrow>  Node l a (delete x r) |
      EQ \<Rightarrow> if r = Leaf then l else let (a',r') = split_min r in Node l a' r')"
 
+text \<open>Deletion by appending:\<close>
+
+fun app :: "('a::linorder)tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"app t Leaf = t" |
+"app Leaf t = t" |
+"app (Node t1 a t2) (Node t3 b t4) =
+  (case app t2 t3 of
+     Leaf \<Rightarrow> Node t1 a (Node Leaf b t4) |
+     Node u2 x u3 \<Rightarrow> Node (Node t1 a u2) x (Node u3 b t4))"
+
+fun delete2 :: "'a::linorder \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
+"delete2 x Leaf = Leaf" |
+"delete2 x (Node l a r) =
+  (case cmp x a of
+     LT \<Rightarrow>  Node (delete2 x l) a r |
+     GT \<Rightarrow>  Node l a (delete2 x r) |
+     EQ \<Rightarrow> app l r)"
+
 
 subsection "Functional Correctness Proofs"
 
@@ -75,4 +95,25 @@
   case 4 thus ?case by(simp add: inorder_delete)
 qed (rule TrueI)+
 
+lemma inorder_app:
+  "inorder(app l r) = inorder l @ inorder r"
+by(induction l r rule: app.induct) (auto split: tree.split)
+
+lemma inorder_delete2:
+  "sorted(inorder t) \<Longrightarrow> inorder(delete2 x t) = del_list x (inorder t)"
+by(induction t) (auto simp: inorder_app del_list_simps)
+
+interpretation S2: Set_by_Ordered
+where empty = empty and isin = isin and insert = insert and delete = delete2
+and inorder = inorder and inv = "\<lambda>_. True"
+proof (standard, goal_cases)
+  case 1 show ?case by (simp add: empty_def)
+next
+  case 2 thus ?case by(simp add: isin_set)
+next
+  case 3 thus ?case by(simp add: inorder_insert)
+next
+  case 4 thus ?case by(simp add: inorder_delete2)
+qed (rule TrueI)+
+
 end