--- a/src/HOL/Data_Structures/Balance.thy Mon Nov 28 15:09:23 2016 +0100
+++ b/src/HOL/Data_Structures/Balance.thy Tue Nov 29 10:53:52 2016 +0100
@@ -8,6 +8,63 @@
"~~/src/HOL/Library/Tree"
begin
+(* The following two lemmas should go into theory \<open>Tree\<close>, except that that
+theory would then depend on \<open>Complex_Main\<close>. *)
+
+lemma min_height_balanced: assumes "balanced t"
+shows "min_height t = nat(floor(log 2 (size1 t)))"
+proof cases
+ assume *: "complete t"
+ hence "size1 t = 2 ^ min_height t"
+ by (simp add: complete_iff_height size1_if_complete)
+ hence "size1 t = 2 powr min_height t"
+ using * by (simp add: powr_realpow)
+ hence "min_height t = log 2 (size1 t)"
+ by simp
+ thus ?thesis
+ by linarith
+next
+ assume *: "\<not> complete t"
+ hence "height t = min_height t + 1"
+ using assms min_hight_le_height[of t]
+ by(auto simp add: balanced_def complete_iff_height)
+ hence "2 ^ min_height t \<le> size1 t \<and> size1 t < 2 ^ (min_height t + 1)"
+ by (metis * min_height_size1 size1_height_if_incomplete)
+ hence "2 powr min_height t \<le> size1 t \<and> size1 t < 2 powr (min_height t + 1)"
+ by(simp only: powr_realpow)
+ (metis of_nat_less_iff of_nat_le_iff of_nat_numeral of_nat_power)
+ hence "min_height t \<le> log 2 (size1 t) \<and> log 2 (size1 t) < min_height t + 1"
+ by(simp add: log_less_iff le_log_iff)
+ thus ?thesis by linarith
+qed
+
+lemma height_balanced: assumes "balanced t"
+shows "height t = nat(ceiling(log 2 (size1 t)))"
+proof cases
+ assume *: "complete t"
+ hence "size1 t = 2 ^ height t"
+ by (simp add: size1_if_complete)
+ hence "size1 t = 2 powr height t"
+ using * by (simp add: powr_realpow)
+ hence "height t = log 2 (size1 t)"
+ by simp
+ thus ?thesis
+ by linarith
+next
+ assume *: "\<not> complete t"
+ hence **: "height t = min_height t + 1"
+ using assms min_hight_le_height[of t]
+ by(auto simp add: balanced_def complete_iff_height)
+ hence 0: "2 ^ min_height t < size1 t \<and> size1 t \<le> 2 ^ (min_height t + 1)"
+ by (metis "*" min_height_size1_if_incomplete size1_height)
+ hence "2 powr min_height t < size1 t \<and> size1 t \<le> 2 powr (min_height t + 1)"
+ by(simp only: powr_realpow)
+ (metis of_nat_less_iff of_nat_le_iff of_nat_numeral of_nat_power)
+ hence "min_height t < log 2 (size1 t) \<and> log 2 (size1 t) \<le> min_height t + 1"
+ by(simp add: log_le_iff less_log_iff)
+ thus ?thesis using ** by linarith
+qed
+
(* mv *)
text \<open>The lemmas about \<open>floor\<close> and \<open>ceiling\<close> of \<open>log 2\<close> should be generalized