"app" -> "join" for RBTs

--- a/src/HOL/Data_Structures/RBT.thy	Tue May 12 10:24:53 2020 +0200
+++ b/src/HOL/Data_Structures/RBT.thy	Tue May 12 10:59:59 2020 +0200
@@ -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 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
+fun join :: "'a rbt \<Rightarrow> 'a rbt \<Rightarrow> 'a rbt" where
+"join Leaf t = t" |
+"join t Leaf = t" |
+"join (R t1 a t2) (R t3 c t4) =
+  (case join 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))" |
-"app (B t1 a t2) (B t3 c t4) =
-  (case app t2 t3 of
+"join (B t1 a t2) (B t3 c t4) =
+  (case join 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))" |
-"app t1 (R t2 a t3) = R (app t1 t2) a t3" |
-"app (R t1 a t2) t3 = R t1 a (app t2 t3)" 
+"join t1 (R t2 a t3) = R (join t1 t2) a t3" |
+"join (R t1 a t2) t3 = R t1 a (join t2 t3)" 
 
 end

--- a/src/HOL/Data_Structures/RBT_Map.thy	Tue May 12 10:24:53 2020 +0200
+++ b/src/HOL/Data_Structures/RBT_Map.thy	Tue May 12 10:59:59 2020 +0200
@@ -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> app l r)"
+  EQ \<Rightarrow> join 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_app inorder_baldL inorder_baldR)
+  (auto simp: del_list_simps inorder_join 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_app invc_app)
+    by (cases c) (simp_all add: invh_join invc_join)
   next
     assume "x < y"
     with 2 show ?thesis

--- a/src/HOL/Data_Structures/RBT_Set.thy	Tue May 12 10:24:53 2020 +0200
+++ b/src/HOL/Data_Structures/RBT_Set.thy	Tue May 12 10:59:59 2020 +0200
@@ -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> app l r)"
+     EQ \<Rightarrow> join 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_app:
-  "inorder(app l r) = inorder l @ inorder r"
-by(induction l r rule: app.induct)
+lemma inorder_join:
+  "inorder(join l r) = inorder l @ inorder r"
+by(induction l r rule: join.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_app inorder_baldL inorder_baldR)
+  (auto simp: del_list_simps inorder_join 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_app:
+lemma invh_join:
   "\<lbrakk> invh l; invh r; bheight l = bheight r \<rbrakk>
-  \<Longrightarrow> invh (app l r) \<and> bheight (app l r) = bheight l"
-by (induct l r rule: app.induct) 
+  \<Longrightarrow> invh (join l r) \<and> bheight (join l r) = bheight l"
+by (induct l r rule: join.induct) 
    (auto simp: invh_baldL_Black split: tree.splits color.splits)
 
-lemma invc_app: 
+lemma invc_join: 
   "\<lbrakk> invc l; invc r \<rbrakk> \<Longrightarrow>
-  (color l = Black \<and> color r = Black \<longrightarrow> invc (app l r)) \<and> invc2 (app l r)"
-by (induct l r rule: app.induct)
+  (color l = Black \<and> color r = Black \<longrightarrow> invc (join l r)) \<and> invc2 (join l r)"
+by (induct l r rule: join.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_app:
+lemma inv_join:
   "\<lbrakk> invh l; invh r; bheight l = bheight r; invc l; invc r \<rbrakk>
-  \<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) 
+  \<Longrightarrow> invh (join l r) \<and> bheight (join l r) = bheight l
+  \<and> invc2 (join l r) \<and> (color l = Black \<and> color r = Black \<longrightarrow> invc (join l r))"
+by (induct l r rule: join.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_app dest!: neq_LeafD)
+  (auto simp: inv_baldL inv_baldR inv_join 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)

--- a/src/HOL/Data_Structures/RBT_Set2.thy	Tue May 12 10:24:53 2020 +0200
+++ b/src/HOL/Data_Structures/RBT_Set2.thy	Tue May 12 10:59:59 2020 +0200
@@ -6,7 +6,7 @@
 imports RBT_Set
 begin
 
-text \<open>This is a conceptually simpler version of deletion. Instead of the tricky \<open>combine\<close>
+text \<open>This is a conceptually simpler version of deletion. Instead of the tricky \<open>join\<close>
 function this version follows the standard approach of replacing the deleted element
 (in function \<open>del\<close>) by the minimal element in its right subtree.\<close>