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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 fold_right :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" end

theory Submission imports Defs begin fun fold_right :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b" where "fold_right _ = undefined" value "fold_right (+) [1,2,3] (4 :: nat) = 10" value "fold_right (#) [a,b,c] [] = [a,b,c]" lemma fold_map: "fold_right f (map g xs) a = fold_right (f o g) xs a" sorry lemma fold_append: "fold_right f (xs @ ys) a = fold_right f xs (fold_right f ys a)" sorry lemma fold_append_plus: "fold_right (+) (xs @ ys) (0 :: nat) = fold_right (+) xs 0 + fold_right (+) ys 0" sorry theorem fold_add_reverse: "fold_right (+) (reverse xs) (x :: nat) = fold_right (+) xs x" sorry end

theory Check imports Submission begin lemma fold_map: "fold_right f (map g xs) a = fold_right (f o g) xs a" by (rule Submission.fold_map) lemma fold_append: "fold_right f (xs @ ys) a = fold_right f xs (fold_right f ys a)" by (rule Submission.fold_append) lemma fold_append_plus: "fold_right (+) (xs @ ys) (0 :: nat) = fold_right (+) xs 0 + fold_right (+) ys 0" by (rule Submission.fold_append_plus) theorem fold_add_reverse: "fold_right (+) (reverse xs) (x :: nat) = fold_right (+) xs x" by (rule Submission.fold_add_reverse) end

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