# Theory Tree_Real

```(* Author: Tobias Nipkow *)

theory Tree_Real
imports
Complex_Main
Tree
begin

text ‹
This theory is separate from \<^theory>‹HOL-Library.Tree› because the former is discrete and
builds on \<^theory>‹Main› whereas this theory builds on \<^theory>‹Complex_Main›.
›

lemma size1_height_log: "log 2 (size1 t) ≤ height t"

lemma min_height_size1_log: "min_height t ≤ log 2 (size1 t)"

lemma size1_log_if_complete: "complete t ⟹ height t = log 2 (size1 t)"

lemma min_height_size1_log_if_incomplete:
"¬ complete t ⟹ min_height t < log 2 (size1 t)"

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"
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]
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]
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)
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)