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### Definitions File

### Template File

### Check File

theory Defs imports Main begin fun a :: "nat \<Rightarrow> int" where "a 0 = 0" | "a (Suc n) = a n ^ 2 + 1" end

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

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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