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# Homework 5

This is the task corresponding to homework 5.

## Resources

### Definitions File

```theory Defs
imports "HOL-Library.Tree"
begin

fun sumto :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" where
"sumto f 0 = 0" |
"sumto f (Suc n) = sumto f n + f(Suc n)"

consts num_leafs :: "'a tree \<Rightarrow> nat"

end```

### Template File

```theory Submission
imports Defs
begin

lemma sum_ident:
"sumto (\<lambda>i. i* 2 ^ i) n = n * 2 ^ (n + 1) - (2 ^ (n + 1) - 2)"
sorry

fun num_leafs :: "'a tree \<Rightarrow> nat"  where
"num_leafs _ = 0"

lemma auxl:
assumes IHl: "num_leafs l \<le> 2 ^ height l"
and IHr: "num_leafs r \<le> 2 ^ height r"
and lr: "height l \<le> height r"
shows "num_leafs(Node l x r) \<le> 2 ^ height(Node l x r)"
sorry

lemma auxr:
assumes IHl: "num_leafs l \<le> 2 ^ height l"
and IHr: "num_leafs r \<le> 2 ^ height r"
and rl: "\<not> height l \<le> height r"
shows "num_leafs(Node l x r) \<le> 2 ^ height(Node l x r)"
sorry

theorem num_leafs_est: "num_leafs t \<le> 2^height t"
proof (induction t)
case Leaf show ?case by auto
next
case (Node l x r)
assume IHl: "num_leafs l \<le> 2 ^ height l"
assume IHr: "num_leafs r \<le> 2 ^ height r"
show "num_leafs \<langle>l, x, r\<rangle> \<le> 2 ^ height \<langle>l, x, r\<rangle>"
sorry
qed

end```

### Check File

```theory Check
imports Submission
begin

lemma sum_ident: "sumto (\<lambda>i. i* 2 ^ i) n = n * 2 ^ (n + 1) - (2 ^ (n + 1) - 2)"
by (rule Submission.sum_ident)

lemma auxl: "(num_leafs l \<le> 2 ^ height l) \<Longrightarrow> (num_leafs r \<le> 2 ^ height r) \<Longrightarrow> (height l \<le> height r) \<Longrightarrow> num_leafs(Node l x r) \<le> 2 ^ height(Node l x r)"
by (rule Submission.auxl)

lemma auxr: "(num_leafs l \<le> 2 ^ height l) \<Longrightarrow> (num_leafs r \<le> 2 ^ height r) \<Longrightarrow> (\<not> height l \<le> height r) \<Longrightarrow> num_leafs(Node l x r) \<le> 2 ^ height(Node l x r)"
by (rule Submission.auxr)

theorem num_leafs_est: "num_leafs t \<le> 2^height t"
by (rule Submission.num_leafs_est)

end```

Terms and Conditions