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### Definitions File

### Template File

### Check File

theory Defs imports "HOL-Library.Tree" begin fun join :: "'a list list \<Rightarrow> 'a list list \<Rightarrow> 'a list list" where "join xss [] = xss" | "join [] yss = yss" | "join (xs#xss) (ys#yss) = (xs @ ys) # join xss yss" fun levels :: "'a tree \<Rightarrow> 'a list list" where "levels Leaf = []" | "levels (Node l a r) = [a] # join (levels l) (levels r)" definition set2 :: "'a list list \<Rightarrow> 'a set" where "set2 xss \<equiv> \<Union>(set (map set xss))" end

theory Submission imports Defs begin lemma levels_height: "length(levels t) = height t" sorry lemma levels_set: "set2 (levels t) = set_tree t" sorry end

theory Check imports Submission begin lemma levels_height: "length(levels t) = height t" by (rule Submission.levels_height) lemma levels_set: "set2 (levels t) = set_tree t" by (rule Submission.levels_set) end

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