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theory Defs imports "~~/src/HOL/Library/Tree" "HOL-Library.Multiset" 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)" end
theory Submission imports Defs begin fun siftdown::"'a::linorder tree \<Rightarrow> 'a::linorder tree" where "siftdown _ = undefined" fun heapify:: "'a::linorder tree \<Rightarrow> 'a::linorder tree" where "heapify _ = undefined" lemma heapify_is_heap: "heap (heapify t)" sorry lemma heapify_preserve: "mset (inorder (heapify t)) = mset (inorder t)" sorry end
theory Check imports Submission begin lemma heapify_is_heap: "heap (heapify t)" by(rule heapify_is_heap) lemma heapify_preserve: "mset (inorder (heapify t)) = mset (inorder t)" by(rule heapify_preserve) end