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# Homework 4

This is the task corresponding to homework 4.

## Resources

### Definitions File

```theory Defs
imports Main
begin

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

consts set_mtree :: "'a mtree \<Rightarrow> 'a set"

consts mbst :: "'a::linorder mtree \<Rightarrow> bool"

consts max_val :: "'a::{linorder,zero} mtree \<Rightarrow> 'a"

consts mins :: "'a::linorder \<Rightarrow> 'a mtree \<Rightarrow> 'a mtree"

consts misin :: "'a::linorder \<Rightarrow> 'a mtree \<Rightarrow> bool"

consts mtree_in_range :: "'a::linorder mtree \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a list"

end```

### Template File

```theory Submission
imports Defs
begin

fun set_mtree :: "'a mtree \<Rightarrow> 'a set"  where
"set_mtree _ = undefined"

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

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

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

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

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

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

lemma misin_set: "mbst t \<Longrightarrow> misin x t \<longleftrightarrow> x\<in>set_mtree t"
sorry

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

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

end```

### Check File

```theory Check
imports Submission
begin

lemma mbst_max: "mbst (Node l m a r) \<Longrightarrow> max_val (Node l m a r) = m"
by (rule Submission.mbst_max)

lemma mins_mbst: "mbst t \<Longrightarrow> mbst (mins x t)"
by (rule Submission.mins_mbst)

lemma misin_set: "mbst t \<Longrightarrow> misin x t \<longleftrightarrow> x\<in>set_mtree t"
by (rule Submission.misin_set)

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

end```

Terms and Conditions