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# Homework 8_1

This is the task corresponding to homework 8_1.

## Resources

### Definitions File

```theory Defs
imports Complex_Main "HOL-Data_Structures.Tree23_Set"
begin

lemma height_bound_upper: "complete t \<Longrightarrow> height t \<le> log 2 (size t + 1)"
using ht_sz_if_complete le_log2_of_power by blast

lemma height_bound_lower_aux:
assumes "complete t"
shows "size t + 1 \<le> 3^(height t)"
using assms
by (induction t) auto

lemma height_bound_lower: assumes "complete t"
shows "log 3 (size t + 1) \<le> height t"
proof -
from log_le_cancel_iff[of 3 "size t + 1" "3^height t"]
and height_bound_lower_aux[OF assms]
have "log 3 (size t + 1) \<le> log 3 (3 ^ height t)"
using of_nat_mono
by fastforce
also have "\<dots> = height t"
finally show ?thesis .
qed

consts num_leaves :: "'a' tree23 \<Rightarrow> nat"

consts is_2_tree :: "'a tree23 \<Rightarrow> bool"

end```

### Template File

```theory Submission
imports Defs
begin

fun num_leaves :: "'a tree23 \<Rightarrow> nat"  where
"num_leaves _ = undefined"

fun is_2_tree :: "'a tree23 \<Rightarrow> bool"  where
"is_2_tree _ = undefined"

theorem complete_2_tree_height: "complete t \<Longrightarrow> is_2_tree t \<longleftrightarrow> num_leaves t = 2^height t"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem complete_2_tree_height: "complete t \<Longrightarrow> is_2_tree t \<longleftrightarrow> num_leaves t = 2^height t"
by (rule Submission.complete_2_tree_height)

end```

Terms and Conditions