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### Definitions File

### Template File

### Check File

theory Defs imports Main begin end

theory Submission imports Defs begin fun find_pfx :: "nat list => nat => nat list" where "find_pfx _ _ = undefined" lemma find_pfx_append: "(find_pfx (xs1 @ [x] @ xs2) x) = (find_pfx (xs1 @ [x]) x)" sorry lemma last_find_pfx_val: "last (find_pfx (xs @ [x]) x) = x" sorry lemma find_pfx_append_2: "\<exists>xs2. xs = find_pfx xs x @ xs2" sorry end

theory Check imports Submission begin text ‹Test cases› lemma "find_pfx [1::nat,2,3] 2 = [1,2]" "find_pfx [] (1::nat) = []" by simp+ lemma find_pfx_append: "(find_pfx (xs1 @ [x] @ xs2) x) = (find_pfx (xs1 @ [x]) x)" by (rule Submission.find_pfx_append) lemma last_find_pfx_val: "last (find_pfx (xs @ [x]) x) = x" by (rule Submission.last_find_pfx_val) lemma find_pfx_append_2: "\<exists>xs2. xs = find_pfx xs x @ xs2" by (rule Submission.find_pfx_append_2) end

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