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FDS Week 9 Homework

Week 9 homework.

Resources

Definitions File

```theory Defs
imports Main "HOL-Data_Structures.Tree23" "HOL-Data_Structures.RBT_Set"
begin

fun bhs :: "'a rbt \<Rightarrow> nat set" where
"bhs Tree.Leaf = {0}" |
"bhs (Tree.Node l (_,c) r) = (let H = bhs l \<union> bhs r in if c=Black then Suc ` H else H)"

end
```

Template File

```theory Template
imports Defs
begin

fun mk_rbt :: "'a tree23 \<Rightarrow> 'a rbt" where
"mk_rbt _ = undefined"

lemma mk_rbt_inorder_btree: "Tree2.inorder (mk_rbt t) = Tree23.inorder t"
sorry

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

lemma mk_rbt_invh_btree: "Tree23.complete t \<Longrightarrow> invh (mk_rbt t)"
sorry

lemma mk_rbt_invc_btree: "invc (mk_rbt t)"
sorry

corollary invh_iff_bhs: "invh t \<longleftrightarrow> bhs t = {bheight t}"
sorry
end
```

Check File

```theory Check
imports Template
begin

lemma mk_rbt_inorder_btree: "Tree2.inorder (mk_rbt t) = Tree23.inorder t"
by(rule mk_rbt_inorder_btree)

lemma mk_rbt_color_btree: "color (mk_rbt t) = Black"
by(rule mk_rbt_color_btree)

lemma mk_rbt_invh_btree: "Tree23.complete t \<Longrightarrow> invh (mk_rbt t)"
by(rule mk_rbt_invh_btree)

lemma mk_rbt_invc_btree: "invc (mk_rbt t)"
by(rule mk_rbt_invc_btree)

corollary invh_iff_bhs: "invh t \<longleftrightarrow> bhs t = {bheight t}"
by(rule invh_iff_bhs)

end```

Terms and Conditions