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(* Author: Tobias Nipkow *)
section \<open>Heaps\<close>
theory Heaps
imports
"HOL-Library.Tree_Multiset"
Priority_Queue_Specs
begin
text \<open>Heap = priority queue on trees:\<close>
locale Heap =
fixes insert :: "('a::linorder) \ 'a tree \ 'a tree"
and del_min :: "'a tree \ 'a tree"
assumes mset_insert: "heap q \ mset_tree (insert x q) = {#x#} + mset_tree q"
and mset_del_min: "\ heap q; q \ Leaf \ \ mset_tree (del_min q) = mset_tree q - {#value q#}"
and heap_insert: "heap q \ heap(insert x q)"
and heap_del_min: "heap q \ heap(del_min q)"
begin
definition empty :: "'a tree" where
"empty = Leaf"
fun is_empty :: "'a tree \ bool" where
"is_empty t = (t = Leaf)"
sublocale Priority_Queue where empty = empty and is_empty = is_empty and insert = insert
and get_min = "value" and del_min = del_min and invar = heap and mset = mset_tree
proof (standard, goal_cases)
case 1 thus ?case by (simp add: empty_def)
next
case 2 thus ?case by(auto)
next
case 3 thus ?case by(simp add: mset_insert)
next
case 4 thus ?case by(simp add: mset_del_min)
next
case 5 thus ?case by(auto simp: neq_Leaf_iff Min_insert2 simp del: Un_iff)
next
case 6 thus ?case by(simp add: empty_def)
next
case 7 thus ?case by(simp add: heap_insert)
next
case 8 thus ?case by(simp add: heap_del_min)
qed
end
text \<open>Once you have \<open>merge\<close>, \<open>insert\<close> and \<open>del_min\<close> are easy:\<close>
locale Heap_Merge =
fixes merge :: "'a::linorder tree \ 'a tree \ 'a tree"
assumes mset_merge: "\ heap q1; heap q2 \ \ mset_tree (merge q1 q2) = mset_tree q1 + mset_tree q2"
and invar_merge: "\ heap q1; heap q2 \ \ heap (merge q1 q2)"
begin
fun insert :: "'a \ 'a tree \ 'a tree" where
"insert x t = merge (Node Leaf x Leaf) t"
fun del_min :: "'a tree \ 'a tree" where
"del_min Leaf = Leaf" |
"del_min (Node l x r) = merge l r"
interpretation Heap insert del_min
proof(standard, goal_cases)
case 1 thus ?case by(simp add:mset_merge)
next
case (2 q) thus ?case by(cases q)(auto simp: mset_merge)
next
case 3 thus ?case by (simp add: invar_merge)
next
case (4 q) thus ?case by (cases q)(auto simp: invar_merge)
qed
sublocale PQM: Priority_Queue_Merge where empty = empty and is_empty = is_empty and insert = insert
and get_min = "value" and del_min = del_min and invar = heap and mset = mset_tree and merge = merge
proof(standard, goal_cases)
case 1 thus ?case by (simp add: mset_merge)
next
case 2 thus ?case by (simp add: invar_merge)
qed
end
end
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