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Homework 11

This is the task corresponding to homework 11.

Resources

Definitions File

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

fun heap::"'a::linorder tree \<Rightarrow> bool" where
"heap Leaf = True"
| "heap (Node l x r) = ((\<forall>y\<in>set_tree l. x \<le> y) \<and> (\<forall>y\<in>set_tree r. x \<le> y) \<and> heap l \<and> heap r)"

consts heapify :: "'a::linorder tree  \<Rightarrow> 'a::linorder tree"

end```

Template File

```theory Submission
imports Defs
begin

fun siftdown::"'a::linorder tree  \<Rightarrow> 'a::linorder tree" fun heapify :: "'a::linorder tree  \<Rightarrow> 'a::linorder tree"  where
"heapify _ = undefined"

theorem heapify_heap: "heap (heapify t)"
sorry

theorem heapify_mset: "mset (inorder (heapify t)) = mset (inorder t)"
sorry

end```

Check File

```theory Check
imports Submission
begin

theorem heapify_heap: "heap (heapify t)"
by (rule Submission.heapify_heap)

theorem heapify_mset: "mset (inorder (heapify t)) = mset (inorder t)"
by (rule Submission.heapify_mset)

end```

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