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# Homework 7

This is the task corresponding to homework 7.

## Resources

### Definitions File

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

definition lefts::"'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "lefts y xs = takeWhile (\<lambda>x. x \<noteq> y) xs"

definition rights::"'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
where "rights y xs = tl (dropWhile (\<lambda>x. x \<noteq> y) xs)"

fun reconstruct::"'a list \<Rightarrow> 'a list \<Rightarrow> 'a tree"
where
"reconstruct [] [] = \<langle>\<rangle>"
| "reconstruct (y#ys) zs =
(let
ls = lefts y zs;
rs = rights y zs
in \<langle>reconstruct (filter (\<lambda>y. y \<in> set ls) ys) ls, y,
reconstruct (filter (\<lambda>y. y \<in> set rs) ys) rs\<rangle>)"

end```

### Template File

```theory Submission
imports Defs
begin

thm preorder.simps
thm inorder.simps

value "reconstruct [3, 1, 6, 5, 8, 7, 9::int] [1, 3, 5, 6, 7, 8, 9]"
value "reconstruct [5, 3, 4, 1, 2] [1, 2, 3, 4, 5::int]"

theorem reconstruct_correct:
"distinct (preorder t) \<Longrightarrow> reconstruct (preorder t) (inorder t) = t"
sorry

end```

### Check File

```theory Check
imports Submission
begin

theorem reconstruct_correct: "distinct (preorder t) \<Longrightarrow> reconstruct (preorder t) (inorder t) = t"
by (rule Submission.reconstruct_correct)

end```

Terms and Conditions