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# Homework 9

This is the task corresponding to homework 9.

## Resources

### Definitions File

```theory Defs
imports "HOL-Data_Structures.RBT_Set"
begin

fun min_height :: "'a tree \<Rightarrow> nat" where
"min_height Leaf = 0" |
"min_height (Node l _ r) = min (min_height l) (min_height r) + 1"

definition "balanced t \<equiv> height t - min_height t \<le> 1"

fun mk_rbt :: "'a tree \<Rightarrow> 'a rbt" where
"mk_rbt Leaf = Leaf"
| "mk_rbt (Node l a r) = (let
l'=mk_rbt l;
r'=mk_rbt r
in
if min_height l > min_height r then
B (paint Red l') a r'
else if min_height l < min_height r then
B l' a (paint Red r')
else
B l' a r'
)"

consts mk_rbt' :: "'a tree \<Rightarrow> 'a rbt \<times> nat"

end```

### Template File

```theory Submission
imports Defs
begin

lemma balanced_subt: "balanced (Node l a r) \<Longrightarrow> balanced l \<and> balanced r"
sorry

lemma balanced_alt:
"balanced t \<longleftrightarrow> height t = min_height t \<or> height t = min_height t + 1"
sorry

lemma mk_rbt_inorder: "inorder (mk_rbt t) = Tree.inorder t"
sorry

lemma mk_rbt_color: "color (mk_rbt t) = Black"
sorry

lemma mk_rbt_bheight: "balanced t \<Longrightarrow> bheight (mk_rbt t) = min_height t"
sorry

lemma mk_rbt_invh: "balanced t \<Longrightarrow> invh (mk_rbt t)"
sorry

lemma mk_rbt_invc: "balanced t \<Longrightarrow> invc (mk_rbt t)"
sorry

fun mk_rbt' :: "'a tree \<Rightarrow> 'a rbt \<times> nat"
where
"mk_rbt' _ = undefined"

lemma mk_rbt'_refine: "fst (mk_rbt' t) = mk_rbt t"
sorry

end```

### Check File

```theory Check
imports Submission
begin

lemma balanced_subt: "balanced (Node l a r) \<Longrightarrow> balanced l \<and> balanced r"
by (rule Submission.balanced_subt)

lemma balanced_alt: "balanced t \<longleftrightarrow> height t = min_height t \<or> height t = min_height t + 1"
by (rule Submission.balanced_alt)

lemma mk_rbt_inorder: "inorder (mk_rbt t) = Tree.inorder t"
by (rule Submission.mk_rbt_inorder)

lemma mk_rbt_color: "color (mk_rbt t) = Black"
by (rule Submission.mk_rbt_color)

lemma mk_rbt_bheight: "balanced t \<Longrightarrow> bheight (mk_rbt t) = min_height t"
by (rule Submission.mk_rbt_bheight)

lemma mk_rbt_invh: "balanced t \<Longrightarrow> invh (mk_rbt t)"
by (rule Submission.mk_rbt_invh)

lemma mk_rbt_invc: "balanced t \<Longrightarrow> invc (mk_rbt t)"
by (rule Submission.mk_rbt_invc)

lemma mk_rbt'_refine: "fst (mk_rbt' t) = mk_rbt t"
by (rule Submission.mk_rbt'_refine)

end```

Terms and Conditions