--- a/src/HOL/Data_Structures/Binomial_Heap.thy Thu Aug 31 08:41:41 2017 +0200
+++ b/src/HOL/Data_Structures/Binomial_Heap.thy Thu Aug 31 09:50:11 2017 +0200
@@ -386,35 +386,35 @@
subsubsection \<open>Instantiating the Priority Queue Locale\<close>
-interpretation binheap: Priority_Queue
+text \<open>Last step of functional correctness proof: combine all the above lemmas
+to show that binomial heaps satisfy the specification of priority queues with merge.\<close>
+
+interpretation binheap: Priority_Queue_Merge
where empty = "[]" and is_empty = "op = []" and insert = insert
- and get_min = get_min and del_min = del_min
+ and get_min = get_min and del_min = del_min and merge = merge
and invar = invar and mset = mset_heap
proof (unfold_locales, goal_cases)
- case 1
- then show ?case by simp
+ case 1 thus ?case by simp
next
- case (2 q)
- then show ?case by auto
+ case 2 thus ?case by auto
next
- case (3 q x)
- then show ?case by auto
+ case 3 thus ?case by auto
next
case (4 q)
- then show ?case using mset_heap_del_min[of q] get_min[OF _ \<open>invar q\<close>]
+ thus ?case using mset_heap_del_min[of q] get_min[OF _ \<open>invar q\<close>]
by (auto simp: union_single_eq_diff)
next
- case (5 q)
- then show ?case using get_min[of q] by auto
+ case (5 q) thus ?case using get_min[of q] by auto
next
- case 6
- then show ?case by (auto simp add: invar_def invar_bheap_def invar_oheap_def)
+ case 6 thus ?case by (auto simp add: invar_def invar_bheap_def invar_oheap_def)
+next
+ case 7 thus ?case by simp
next
- case (7 q x)
- then show ?case by simp
+ case 8 thus ?case by simp
next
- case (8 q)
- then show ?case by simp
+ case 9 thus ?case by simp
+next
+ case 10 thus ?case by simp
qed
--- a/src/HOL/Data_Structures/Leftist_Heap.thy Thu Aug 31 08:41:41 2017 +0200
+++ b/src/HOL/Data_Structures/Leftist_Heap.thy Thu Aug 31 09:50:11 2017 +0200
@@ -136,12 +136,14 @@
lemma heap_del_min: "heap t \<Longrightarrow> heap(del_min t)"
by(cases t)(auto simp add: heap_merge simp del: merge.simps)
+text \<open>Last step of functional correctness proof: combine all the above lemmas
+to show that leftist heaps satisfy the specification of priority queues with merge.\<close>
-interpretation lheap: Priority_Queue
+interpretation lheap: Priority_Queue_Merge
where empty = Leaf and is_empty = "\<lambda>h. h = Leaf"
and insert = insert and del_min = del_min
-and get_min = get_min and invar = "\<lambda>h. heap h \<and> ltree h"
-and mset = mset_tree
+and get_min = get_min and merge = merge
+and invar = "\<lambda>h. heap h \<and> ltree h" and mset = mset_tree
proof(standard, goal_cases)
case 1 show ?case by simp
next
@@ -158,6 +160,10 @@
case 7 thus ?case by(simp add: heap_insert ltree_insert)
next
case 8 thus ?case by(simp add: heap_del_min ltree_del_min)
+next
+ case 9 thus ?case by (simp add: mset_merge)
+next
+ case 10 thus ?case by (simp add: heap_merge ltree_merge)
qed
--- a/src/HOL/Data_Structures/Priority_Queue.thy Thu Aug 31 08:41:41 2017 +0200
+++ b/src/HOL/Data_Structures/Priority_Queue.thy Thu Aug 31 09:50:11 2017 +0200
@@ -25,17 +25,12 @@
and invar_empty: "invar empty"
and invar_insert: "invar q \<Longrightarrow> invar (insert x q)"
and invar_del_min: "invar q \<Longrightarrow> mset q \<noteq> {#} \<Longrightarrow> invar (del_min q)"
-begin
-(* FIXME why? *)
+text \<open>Extend locale with \<open>merge\<close>. Need to enforce that \<open>'q\<close> is the same in both locales.\<close>
-lemma get_min_alt: "invar q \<Longrightarrow> mset q \<noteq> {#} \<Longrightarrow>
- get_min q \<in># mset q \<and> (\<forall>x\<in>#mset q. get_min q \<le> x)"
- by (simp add: mset_get_min)
-
-lemmas invar_simps[simp] = invar_empty invar_insert invar_del_min
-lemmas mset_simps[simp] = mset_empty is_empty mset_insert mset_del_min mset_get_min
+locale Priority_Queue_Merge = Priority_Queue where empty = empty for empty :: 'q +
+fixes merge :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"
+assumes mset_merge: "\<lbrakk> invar q1; invar q2 \<rbrakk> \<Longrightarrow> mset (merge q1 q2) = mset q1 + mset q2"
+and invar_merge: "\<lbrakk> invar q1; invar q2 \<rbrakk> \<Longrightarrow> invar (merge q1 q2)"
end
-
-end