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### Definitions File

### Template File

### Check File

theory Defs imports Main begin fun contains :: "'a \<Rightarrow> 'a list \<Rightarrow> bool" where "contains x [] = False" | "contains x (y#ys) \<longleftrightarrow> x=y \<or> contains x ys" lemma contains_append[simp]: "contains a (xs@[b]) = (contains a xs \<or> a = b)" by (induction xs) auto lemma contains_rev: "contains a (rev xs) = contains a xs" by (induction xs) auto consts ldistinct :: "'a list \<Rightarrow> bool" consts length_fold :: "'a list \<Rightarrow> nat" consts length_foldr :: "'a list \<Rightarrow> nat" consts slice :: "'a list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a list" end

theory Submission imports Defs begin fun ldistinct :: "'a list \<Rightarrow> bool" where "ldistinct _ = undefined" lemma ldistinct_rev: "ldistinct (rev xs) = ldistinct xs" sorry thm fold.simps thm foldr.simps definition length_fold :: "'a list \<Rightarrow> nat" where "length_fold _ = undefined" definition length_foldr :: "'a list \<Rightarrow> nat" where "length_foldr _ = undefined" lemma length_fold: "length_fold xs = length xs" sorry lemma length_foldr: "length_foldr xs = length xs" sorry fun slice :: "'a list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a list" where "slice _ = undefined" value "slice [0,1,2,3,4,5,6::int] 2 3 = [2,3,4]" text \<open>(In range)\<close> value "slice [0,1,2,3,4,5,6::int] 2 10 = [2,3,4,5,6]" text \<open>(Length out of range)\<close> value "slice [0,1,2,3,4,5,6::int] 10 10 = []" text \<open>(Start index out of range)\<close> lemma slice_append: "slice xs s l1 @ slice xs (s+l1) l2 = slice xs s (l1+l2)" sorry lemma ldistinct_slice: "ldistinct xs \<Longrightarrow> ldistinct (slice xs s l)" sorry end

theory Check imports Submission begin lemma ldistinct_rev: "ldistinct (rev xs) = ldistinct xs" by (rule Submission.ldistinct_rev) lemma length_fold: "length_fold xs = length xs" by (rule Submission.length_fold) lemma length_foldr: "length_foldr xs = length xs" by (rule Submission.length_foldr) lemma slice_append: "slice xs s l1 @ slice xs (s+l1) l2 = slice xs s (l1+l2)" by (rule Submission.slice_append) lemma ldistinct_slice: "ldistinct xs \<Longrightarrow> ldistinct (slice xs s l)" by (rule Submission.ldistinct_slice) end

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