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### Definitions File

### Template File

### Check File

theory Defs imports Main begin 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 xs) = xs" by (induct xs) (auto simp add: reverse_snoc) consts lmax :: "nat list \<Rightarrow> nat" consts contains :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" end

theory Submission imports Defs begin fun lmax :: "nat list \<Rightarrow> nat" where "lmax _ = undefined" lemma max_greater: "x \<in> set xs \<Longrightarrow> x\<le>lmax xs" sorry lemma lmax_reverse: "lmax (reverse xs) = lmax xs" sorry fun contains :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" where "contains _ = undefined" lemma contains_reverse: "contains a (reverse xs) = contains a xs" sorry end

theory Check imports Submission begin lemma max_greater: "x \<in> set xs \<Longrightarrow> x\<le>lmax xs" by (rule Submission.max_greater) lemma lmax_reverse: "lmax (reverse xs) = lmax xs" by (rule Submission.lmax_reverse) lemma contains_reverse: "contains a (reverse xs) = contains a xs" by (rule Submission.contains_reverse) end

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