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# Homework 8

This is the task corresponding to homework 8.

## Resources

### Definitions File

```theory Defs
imports "HOL-Library.Tree"
begin

fun avl :: "nat tree \<Rightarrow> bool" where
"avl \<langle>\<rangle> = True"
| "avl \<langle>l, n, r\<rangle> = (abs(int(height l) - int(height r)) \<le> 1 \<and>
n = max (height l) (height r) + 1 \<and> avl l \<and> avl r)"

fun ht :: "nat tree \<Rightarrow> nat" where
"ht \<langle>\<rangle> = 0"
| "ht \<langle>l, n, r\<rangle> = n"

lemma ht_height[simp]: "avl t \<Longrightarrow> ht t = height t"
by (induction t) auto

declare [[names_short]]

fun fib_tree :: "nat \<Rightarrow> nat tree" where
"fib_tree 0 = \<langle>\<rangle>" |
"fib_tree (Suc 0) = \<langle>\<langle>\<rangle>, 1, \<langle>\<rangle>\<rangle>" |
"fib_tree (Suc(Suc n)) = \<langle>fib_tree (Suc n), Suc (Suc n), fib_tree n\<rangle>"

datatype 'a itree = iLeaf | iNode "'a itree" "'a \<times> 'a" "'a itree"

fun set_itree2:: "'a::ord itree \<Rightarrow> 'a set" where
"set_itree2 iLeaf = {}"
| "set_itree2 (iNode l (low, high) r) = {low .. high} \<union> ((set_itree2 l) \<union> (set_itree2 r))"

fun set_itree3:: "'a itree \<Rightarrow> ('a \<times> 'a) set" where
"set_itree3 iLeaf = {}"
| "set_itree3 (iNode l (low, high) r) = {(low, high)} \<union> ((set_itree3 l) \<union> (set_itree3 r))"

consts ibst :: "'a::linorder itree \<Rightarrow> bool"

consts delete :: "int \<Rightarrow> int itree \<Rightarrow> int itree"

end```

### Template File

```theory Submission
imports Defs
begin

lemma fib_tree_minimal: "avl t \<Longrightarrow> size1 (fib_tree (ht t)) \<le> size1 t"
sorry

fun ibst :: "'a::linorder itree \<Rightarrow> bool"  where
"ibst _ = undefined"

fun delete :: "int \<Rightarrow> int itree \<Rightarrow> int itree"  where
"delete _ = undefined"

theorem delete_set_minus: "ibst t \<Longrightarrow> set_itree2 (delete x t) = (set_itree2 t) - {x}"
sorry

theorem delete_ibst: "ibst t \<Longrightarrow> ibst (delete x t)"
sorry

end```

### Check File

```theory Check
imports Submission
begin

lemma fib_tree_minimal: "avl t \<Longrightarrow> size1 (fib_tree (ht t)) \<le> size1 t"
by (rule Submission.fib_tree_minimal)

theorem delete_set_minus: "ibst t \<Longrightarrow> set_itree2 (delete x t) = (set_itree2 t) - {x}"
by (rule Submission.delete_set_minus)

theorem delete_ibst: "ibst t \<Longrightarrow> ibst (delete x t)"
by (rule Submission.delete_ibst)

end```

Terms and Conditions