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theory Defs imports Main begin text ‹Definitions and lemmas from the tutorial› fun snoc :: "'a list ⇒ 'a ⇒ 'a list" where "snoc  x = [x]" | "snoc (y # ys) x = y # (snoc ys x)" fun reverse :: "'a list ⇒ '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) end
theory Submission imports Defs begin fun list_sum :: "nat list ⇒ nat" where "list_sum _ = undefined" theorem list_sum_reverse: "list_sum (reverse xs) = list_sum xs" sorry fun upto :: "nat ⇒ nat list" where "upto _ = undefined" theorem gauss: "list_sum (upto n) = n * (n + 1) div 2" sorry end
theory Check imports Submission begin theorem list_sum_reverse: "list_sum (reverse xs) = list_sum xs" by (rule Submission.list_sum_reverse) theorem gauss: "list_sum (upto n) = n * (n + 1) div 2" by (rule Submission.gauss) end