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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
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
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