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# FDS Week 10 Homework 1

Week 10 homework 1.

## Resources

Download Files

### Definitions File

```theory Defs
imports Main
begin

datatype trie = LeafF | LeafT | Node "trie * trie"

end
```

### Template File

```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
```

### Check File

```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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