diff -r ed5b907bbf50 -r 831f17da1aab src/HOL/Library/Tree_Real.thy
--- a/src/HOL/Library/Tree_Real.thy	Sun Nov 08 21:27:08 2020 +0100
+++ b/src/HOL/Library/Tree_Real.thy	Tue Nov 10 17:42:41 2020 +0100
@@ -26,7 +26,7 @@
 by (simp add: less_log2_of_power min_height_size1_if_incomplete)
 
 
-lemma min_height_balanced: assumes "balanced t"
+lemma min_height_acomplete: assumes "acomplete t"
 shows "min_height t = nat(floor(log 2 (size1 t)))"
 proof cases
   assume *: "complete t"
@@ -37,12 +37,12 @@
   assume *: "\<not> complete t"
   hence "height t = min_height t + 1"
     using assms min_height_le_height[of t]
-    by(auto simp: balanced_def complete_iff_height)
+    by(auto simp: acomplete_def complete_iff_height)
   hence "size1 t < 2 ^ (min_height t + 1)" by (metis * size1_height_if_incomplete)
   from floor_log_nat_eq_if[OF min_height_size1 this] show ?thesis by simp
 qed
 
-lemma height_balanced: assumes "balanced t"
+lemma height_acomplete: assumes "acomplete t"
 shows "height t = nat(ceiling(log 2 (size1 t)))"
 proof cases
   assume *: "complete t"
@@ -52,41 +52,41 @@
   assume *: "\<not> complete t"
   hence **: "height t = min_height t + 1"
     using assms min_height_le_height[of t]
-    by(auto simp add: balanced_def complete_iff_height)
+    by(auto simp add: acomplete_def complete_iff_height)
   hence "size1 t \<le> 2 ^ (min_height t + 1)" by (metis size1_height)
   from log2_of_power_le[OF this size1_ge0] min_height_size1_log_if_incomplete[OF *] **
   show ?thesis by linarith
 qed
 
-lemma balanced_Node_if_wbal1:
-assumes "balanced l" "balanced r" "size l = size r + 1"
-shows "balanced \<langle>l, x, r\<rangle>"
+lemma acomplete_Node_if_wbal1:
+assumes "acomplete l" "acomplete r" "size l = size r + 1"
+shows "acomplete \<langle>l, x, r\<rangle>"
 proof -
   from assms(3) have [simp]: "size1 l = size1 r + 1" by(simp add: size1_size)
   have "nat \<lceil>log 2 (1 + size1 r)\<rceil> \<ge> nat \<lceil>log 2 (size1 r)\<rceil>"
     by(rule nat_mono[OF ceiling_mono]) simp
   hence 1: "height(Node l x r) = nat \<lceil>log 2 (1 + size1 r)\<rceil> + 1"
-    using height_balanced[OF assms(1)] height_balanced[OF assms(2)]
+    using height_acomplete[OF assms(1)] height_acomplete[OF assms(2)]
     by (simp del: nat_ceiling_le_eq add: max_def)
   have "nat \<lfloor>log 2 (1 + size1 r)\<rfloor> \<ge> nat \<lfloor>log 2 (size1 r)\<rfloor>"
     by(rule nat_mono[OF floor_mono]) simp
   hence 2: "min_height(Node l x r) = nat \<lfloor>log 2 (size1 r)\<rfloor> + 1"
-    using min_height_balanced[OF assms(1)] min_height_balanced[OF assms(2)]
+    using min_height_acomplete[OF assms(1)] min_height_acomplete[OF assms(2)]
     by (simp)
   have "size1 r \<ge> 1" by(simp add: size1_size)
   then obtain i where i: "2 ^ i \<le> size1 r" "size1 r < 2 ^ (i + 1)"
     using ex_power_ivl1[of 2 "size1 r"] by auto
   hence i1: "2 ^ i < size1 r + 1" "size1 r + 1 \<le> 2 ^ (i + 1)" by auto
   from 1 2 floor_log_nat_eq_if[OF i] ceiling_log_nat_eq_if[OF i1]
-  show ?thesis by(simp add:balanced_def)
+  show ?thesis by(simp add:acomplete_def)
 qed
 
-lemma balanced_sym: "balanced \<langle>l, x, r\<rangle> \<Longrightarrow> balanced \<langle>r, y, l\<rangle>"
-by(auto simp: balanced_def)
+lemma acomplete_sym: "acomplete \<langle>l, x, r\<rangle> \<Longrightarrow> acomplete \<langle>r, y, l\<rangle>"
+by(auto simp: acomplete_def)
 
-lemma balanced_Node_if_wbal2:
-assumes "balanced l" "balanced r" "abs(int(size l) - int(size r)) \<le> 1"
-shows "balanced \<langle>l, x, r\<rangle>"
+lemma acomplete_Node_if_wbal2:
+assumes "acomplete l" "acomplete r" "abs(int(size l) - int(size r)) \<le> 1"
+shows "acomplete \<langle>l, x, r\<rangle>"
 proof -
   have "size l = size r \<or> (size l = size r + 1 \<or> size r = size l + 1)" (is "?A \<or> ?B")
     using assms(3) by linarith
@@ -94,21 +94,21 @@
   proof
     assume "?A"
     thus ?thesis using assms(1,2)
-      apply(simp add: balanced_def min_def max_def)
-      by (metis assms(1,2) balanced_optimal le_antisym le_less)
+      apply(simp add: acomplete_def min_def max_def)
+      by (metis assms(1,2) acomplete_optimal le_antisym le_less)
   next
     assume "?B"
     thus ?thesis
-      by (meson assms(1,2) balanced_sym balanced_Node_if_wbal1)
+      by (meson assms(1,2) acomplete_sym acomplete_Node_if_wbal1)
   qed
 qed
 
-lemma balanced_if_wbalanced: "wbalanced t \<Longrightarrow> balanced t"
+lemma acomplete_if_wbalanced: "wbalanced t \<Longrightarrow> acomplete t"
 proof(induction t)
-  case Leaf show ?case by (simp add: balanced_def)
+  case Leaf show ?case by (simp add: acomplete_def)
 next
   case (Node l x r)
-  thus ?case by(simp add: balanced_Node_if_wbal2)
+  thus ?case by(simp add: acomplete_Node_if_wbal2)
 qed
 
 end