src/HOL/Data_Structures/RBT_Map.thy

   EQ \<Rightarrow> R l (x,y) r)"
 
 definition update :: "'a::linorder \<Rightarrow> 'b \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "update x y t = paint Black (upd x y t)"
 
-fun del :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
-and delL :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
-and delR :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> 'a*'b \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt"
-where
+fun del :: "'a::linorder \<Rightarrow> ('a*'b)rbt \<Rightarrow> ('a*'b)rbt" where
 "del x Leaf = Leaf" |
-"del x (Node t1 (a,b) c t2) = (case cmp x a of
-  LT \<Rightarrow> delL x t1 (a,b) t2 |
-  GT \<Rightarrow> delR x t1 (a,b) t2 |
-  EQ \<Rightarrow> combine t1 t2)" |
-"delL x (B t1 a t2) b t3 = baldL (del x (B t1 a t2)) b t3" |
-"delL x t1 a t2 = R (del x t1) a t2" |
-"delR x t1 a (B t2 b t3) = baldR t1 a (del x (B t2 b t3))" | 
-"delR x t1 a t2 = R t1 a (del x t2)"
+"del x (Node l (a,b) c r) = (case cmp x a of
+     LT \<Rightarrow> if l \<noteq> Leaf \<and> color l = Black
+           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)"
 
 definition delete :: "'a::linorder \<Rightarrow> ('a*'b) rbt \<Rightarrow> ('a*'b) rbt" where
 "delete x t = paint Black (del x t)"
 
 

 lemma inorder_update:
   "sorted1(inorder t) \<Longrightarrow> inorder(update x y t) = upd_list x y (inorder t)"
 by(simp add: update_def inorder_upd inorder_paint)
 
 lemma inorder_del:
- "sorted1(inorder t1) \<Longrightarrow>  inorder(del x t1) = del_list x (inorder t1)" and
- "sorted1(inorder t1) \<Longrightarrow>  inorder(delL x t1 a t2) =
-    del_list x (inorder t1) @ a # inorder t2" and
- "sorted1(inorder t2) \<Longrightarrow>  inorder(delR x t1 a t2) =
-    inorder t1 @ a # del_list x (inorder t2)"
-by(induction x t1 and x t1 a t2 and x t1 a t2 rule: del_delL_delR.induct)
+ "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)
 
 lemma inorder_delete:
   "sorted1(inorder t) \<Longrightarrow> inorder(delete x t) = del_list x (inorder t)"
 by(simp add: delete_def inorder_del inorder_paint)
 
+
+subsection \<open>Structural invariants\<close>
+
+subsubsection \<open>Update\<close>
+
+lemma invc_upd: assumes "invc t"
+  shows "color t = Black \<Longrightarrow> invc (upd x y t)" "invc2 (upd x y t)"
+using assms
+by (induct x y t rule: upd.induct) (auto simp: invc_baliL invc_baliR invc2I)
+
+lemma invh_upd: assumes "invh t"
+  shows "invh (upd x y t)" "bheight (upd x y t) = bheight t"
+using assms
+by(induct x y t rule: upd.induct)
+  (auto simp: invh_baliL invh_baliR bheight_baliL bheight_baliR)
+
+theorem rbt_update: "rbt t \<Longrightarrow> rbt (update x y t)"
+by (simp add: invc_upd(2) invh_upd(1) color_paint_Black invc_paint_Black invh_paint
+  rbt_def update_def)
+
+
+subsubsection \<open>Deletion\<close>
+
+lemma del_invc_invh: "invh t \<Longrightarrow> invc t \<Longrightarrow> invh (del x t) \<and>
+   (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))"
+proof (induct x t rule: del.induct)
+case (2 x _ y _ c)
+  have "x = y \<or> x < y \<or> x > y" by auto
+  thus ?case proof (elim disjE)
+    assume "x = y"
+    with 2 show ?thesis
+    by (cases c) (simp_all add: invh_combine invc_combine)
+  next
+    assume "x < y"
+    with 2 show ?thesis
+      by(cases c)
+        (auto simp: invh_baldL_invc invc_baldL invc2_baldL dest: neq_LeafD)
+  next
+    assume "y < x"
+    with 2 show ?thesis
+      by(cases c)
+        (auto simp: invh_baldR_invc invc_baldR invc2_baldR dest: neq_LeafD)
+  qed
+qed auto
+
+theorem rbt_delete: "rbt t \<Longrightarrow> rbt (delete k t)"
+by (metis delete_def rbt_def color_paint_Black del_invc_invh invc_paint_Black invc2I invh_paint)
+
 interpretation Map_by_Ordered
 where empty = Leaf and lookup = lookup and update = update and delete = delete
-and inorder = inorder and inv = "\<lambda>_. True"
+and inorder = inorder and inv = rbt
 proof (standard, goal_cases)
   case 1 show ?case by simp
 next
   case 2 thus ?case by(simp add: lookup_map_of)
 next
   case 3 thus ?case by(simp add: inorder_update)
 next
   case 4 thus ?case by(simp add: inorder_delete)
-qed auto
+next
+  case 5 thus ?case by (simp add: rbt_Leaf) 
+next
+  case 6 thus ?case by (simp add: rbt_update) 
+next
+  case 7 thus ?case by (simp add: rbt_delete) 
+qed
 
 end