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### Definitions File

### Template File

### Check File

theory Defs imports Main begin datatype trie = LeafF | LeafT | Node "trie * trie" end

theory Template imports Defs begin fun isin :: "trie \<Rightarrow> bool list \<Rightarrow> bool" where "isin _ _ = undefined" fun ins :: "bool list \<Rightarrow> trie \<Rightarrow> trie" where "ins _ _ = undefined" fun delete2 :: "bool list \<Rightarrow> trie \<Rightarrow> trie" where "delete2 _ _ = undefined" fun is_trie :: "nat \<Rightarrow> trie \<Rightarrow> bool" where "is_trie _ _ = undefined" lemma isin_correct: "\<lbrakk> is_trie n t; length as = n \<rbrakk> \<Longrightarrow> isin (ins as t) bs = (as = bs \<or> isin t bs) \<and> is_trie n (ins as t)" sorry lemma delete2_correct: "is_trie n t \<Longrightarrow> isin (delete2 as t) bs = (as\<noteq>bs \<and> isin t bs) \<and> (is_trie n (delete2 as t))" sorry end

theory Check imports Template begin lemma isin_correct: "\<lbrakk> is_trie n t; length as = n \<rbrakk> \<Longrightarrow> isin (ins as t) bs = (as = bs \<or> isin t bs) \<and> is_trie n (ins as t)" by(rule isin_correct) lemma delete2_correct: "is_trie n t \<Longrightarrow> isin (delete2 as t) bs = (as\<noteq>bs \<and> isin t bs) \<and> (is_trie n (delete2 as t))" by(rule delete2_correct) end

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