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# FDS Week 4 Homework

Week 4 homework.

## Resources

### Definitions File

```theory Defs
imports Main
begin

datatype 'a mtree = Leaf | Node (left: "'a mtree") (minimum: 'a) (element: 'a) (right: "'a mtree")

end

```

### Template File

```theory Template
imports Defs
begin

fun set_mtree2 where
"set_mtree2 _ = undefined"

fun mbst :: "'a::{linorder,zero} mtree \<Rightarrow> bool" where
"mbst _ = undefined"

fun min_val :: "'a::{linorder,zero} mtree \<Rightarrow> 'a" where
"min_val _ = undefined"

lemma mbst_works: "mbst (Node l m a r) \<Longrightarrow> min_val (Node l m a r) = m"
sorry

fun mins :: "'a::{linorder,zero} \<Rightarrow> 'a mtree \<Rightarrow> 'a mtree" where
"mins _ _ = undefined"

lemma mins_set: "mbst t \<Longrightarrow> set_mtree2 (mins x t) = insert x (set_mtree2 t)"
sorry

lemma mins_works: "mbst t \<Longrightarrow> mbst (mins x t)"
sorry

fun misin :: "'a::linorder \<Rightarrow> 'a mtree \<Rightarrow> bool" where
"misin _ _ = undefined"

lemma misin_works: "mbst t \<Longrightarrow> misin x t \<longleftrightarrow> x\<in>set_mtree2 t"
sorry

fun mtree_in_range :: "'a::linorder mtree \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a list"
where
"mtree_in_range _ _ _ = undefined"

lemma mtree_in_range_works: "mbst t \<Longrightarrow> set (mtree_in_range t u v) = {x\<in>set_mtree2 t. u\<le>x \<and> x\<le>v}"
sorry

end

```

### Check File

```theory Check
imports Template
begin

lemma mbst_works: "mbst (Node l m a r) \<Longrightarrow> min_val (Node l m a r) = m"
by (rule mbst_works)

lemma mins_works: "mbst t \<Longrightarrow> mbst (mins x t)"
by (rule mins_works)

lemma misin_works: "mbst t \<Longrightarrow> misin x t \<longleftrightarrow> x\<in>set_mtree2 t"
by (rule misin_works)

lemma mtree_in_range_works: "mbst t \<Longrightarrow> set (mtree_in_range t u v) = {x\<in>set_mtree2 t. u\<le>x \<and> x\<le>v}"
by (rule mtree_in_range_works)

end```

Terms and Conditions