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(* Author: Tobias Nipkow *)
theory Tree_Real
imports
Complex_Main
Tree
begin
text \<open>
This theory is separate from \<^theory>\<open>HOL-Library.Tree\<close> because the former is discrete and
builds on \<^theory>\<open>Main\<close> whereas this theory builds on \<^theory>\<open>Complex_Main\<close>.
\<close>
lemma size1_height_log: "log 2 (size1 t) \ height t"
by (simp add: log2_of_power_le size1_height)
lemma min_height_size1_log: "min_height t \ log 2 (size1 t)"
by (simp add: le_log2_of_power min_height_size1)
lemma size1_log_if_complete: "complete t \ height t = log 2 (size1 t)"
by (simp add: size1_if_complete)
lemma min_height_size1_log_if_incomplete:
"\ complete t \ min_height t < log 2 (size1 t)"
by (simp add: less_log2_of_power min_height_size1_if_incomplete)
lemma min_height_acomplete: assumes "acomplete 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)
from log2_of_power_eq[OF this] show ?thesis by linarith
next
assume *: "\ complete t"
hence "height t = min_height t + 1"
using assms min_height_le_height[of t]
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_acomplete: assumes "acomplete 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)
from log2_of_power_eq[OF this] show ?thesis by linarith
next
assume *: "\ complete t"
hence **: "height t = min_height t + 1"
using assms min_height_le_height[of t]
by(auto simp add: acomplete_def complete_iff_height)
hence "size1 t \ 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 acomplete_Node_if_wbal1:
assumes "acomplete l" "acomplete r" "size l = size r + 1"
shows "acomplete \l, x, r\"
proof -
from assms(3) have [simp]: "size1 l = size1 r + 1" by(simp add: size1_size)
have "nat \log 2 (1 + size1 r)\ \ nat \log 2 (size1 r)\"
by(rule nat_mono[OF ceiling_mono]) simp
hence 1: "height(Node l x r) = nat \log 2 (1 + size1 r)\ + 1"
using height_acomplete[OF assms(1)] height_acomplete[OF assms(2)]
by (simp del: nat_ceiling_le_eq add: max_def)
have "nat \log 2 (1 + size1 r)\ \ nat \log 2 (size1 r)\"
by(rule nat_mono[OF floor_mono]) simp
hence 2: "min_height(Node l x r) = nat \log 2 (size1 r)\ + 1"
using min_height_acomplete[OF assms(1)] min_height_acomplete[OF assms(2)]
by (simp)
have "size1 r \ 1" by(simp add: size1_size)
then obtain i where i: "2 ^ i \ 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 \ 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:acomplete_def)
qed
lemma acomplete_sym: "acomplete \l, x, r\ \ acomplete \r, y, l\"
by(auto simp: acomplete_def)
lemma acomplete_Node_if_wbal2:
assumes "acomplete l" "acomplete r" "abs(int(size l) - int(size r)) \ 1"
shows "acomplete \l, x, r\"
proof -
have "size l = size r \ (size l = size r + 1 \ size r = size l + 1)" (is "?A \ ?B")
using assms(3) by linarith
thus ?thesis
proof
assume "?A"
thus ?thesis using assms(1,2)
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) acomplete_sym acomplete_Node_if_wbal1)
qed
qed
lemma acomplete_if_wbalanced: "wbalanced t \ acomplete t"
proof(induction t)
case Leaf show ?case by (simp add: acomplete_def)
next
case (Node l x r)
thus ?case by(simp add: acomplete_Node_if_wbal2)
qed
end
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