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### Definitions File

### Template File

### Check File

theory Defs imports Main begin text \<open>Definitions and lemmas from the tutorial\<close> fun snoc :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where "snoc [] x = [x]" | "snoc (y # ys) x = y # (snoc ys x)" fun reverse :: "'a list \<Rightarrow> 'a list" where "reverse [] = []" | "reverse (x # xs) = snoc (reverse xs) x" lemma reverse_snoc: "reverse (snoc xs y) = y # reverse xs" by (induct xs) auto theorem reverse_reverse: "reverse (reverse xs) = xs" by (induct xs) (auto simp add: reverse_snoc) consts repeat :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" end

theory Submission imports Defs begin fun repeat :: "nat \<Rightarrow> 'a \<Rightarrow> 'a list" where "repeat _ = undefined" value "repeat 5 (0::nat) = [0, 0, 0, 0, 0]" value "repeat 3 (1::nat) = [1, 1, 1]" theorem rep_len: "length (repeat n a) = n" sorry theorem rep_rev: "reverse (repeat n a) = repeat n a" sorry end

theory Check imports Submission begin theorem rep_len: "length (repeat n a) = n" by (rule Submission.rep_len) theorem rep_rev: "reverse (repeat n a) = repeat n a" by (rule Submission.rep_rev) end

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