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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