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# Homework 5

This is the task corresponding to homework 5.

## Resources

### Definitions File

```theory Defs
imports Main
begin

fun a :: "nat \<Rightarrow> int" where
"a 0 = 0" |
"a (Suc n) = a n ^ 2 + 1"

end```

### Template File

```theory Submission
imports Defs
begin

theorem split_list: "\<exists>ys zs. length ys = length xs div n \<and> xs=ys@zs"
sorry

thm power_mono[where n=2]

theorem a_bound: "a n \<le> 2 ^ (2 ^ n) - 1"
proof(induction n)
case 0 thus ?case by simp
next
case (Suc n)
assume IH: "a n \<le> 2 ^ 2 ^ n - 1"
show "a (Suc n) \<le> 2 ^ 2 ^ Suc n - 1"
sorry
qed

end```

### Check File

```theory Check
imports Submission
begin

theorem split_list: "\<exists>ys zs. length ys = length xs div n \<and> xs=ys@zs"
by (rule Submission.split_list)

theorem a_bound: "a n \<le> 2 ^ (2 ^ n) - 1"
by (rule Submission.a_bound)

end```

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