--- a/src/HOL/Data_Structures/Binomial_Heap.thy Mon Jan 22 15:06:38 2018 +0100
+++ b/src/HOL/Data_Structures/Binomial_Heap.thy Mon Jan 22 15:50:29 2018 +0100
@@ -15,7 +15,7 @@
We formalize the binomial heap presentation from Okasaki's book.
We show the functional correctness and complexity of all operations.
- The presentation is engineered for simplicity, and most
+ The presentation is engineered for simplicity, and most
proofs are straightforward and automatic.
\<close>
@@ -29,39 +29,39 @@
fun mset_tree :: "'a::linorder tree \<Rightarrow> 'a multiset" where
"mset_tree (Node _ a c) = {#a#} + (\<Sum>t\<in>#mset c. mset_tree t)"
-
-definition mset_heap :: "'a::linorder heap \<Rightarrow> 'a multiset" where
+
+definition mset_heap :: "'a::linorder heap \<Rightarrow> 'a multiset" where
"mset_heap c = (\<Sum>t\<in>#mset c. mset_tree t)"
-
-lemma mset_tree_simp_alt[simp]:
+
+lemma mset_tree_simp_alt[simp]:
"mset_tree (Node r a c) = {#a#} + mset_heap c"
unfolding mset_heap_def by auto
-declare mset_tree.simps[simp del]
-
-lemma mset_tree_nonempty[simp]: "mset_tree t \<noteq> {#}"
+declare mset_tree.simps[simp del]
+
+lemma mset_tree_nonempty[simp]: "mset_tree t \<noteq> {#}"
by (cases t) auto
-
-lemma mset_heap_Nil[simp]:
+
+lemma mset_heap_Nil[simp]:
"mset_heap [] = {#}"
by (auto simp: mset_heap_def)
lemma mset_heap_Cons[simp]: "mset_heap (t#ts) = mset_tree t + mset_heap ts"
by (auto simp: mset_heap_def)
-
+
lemma mset_heap_empty_iff[simp]: "mset_heap ts = {#} \<longleftrightarrow> ts=[]"
by (auto simp: mset_heap_def)
-
+
lemma root_in_mset[simp]: "root t \<in># mset_tree t"
-by (cases t) auto
-
-lemma mset_heap_rev_eq[simp]: "mset_heap (rev ts) = mset_heap ts"
+by (cases t) auto
+
+lemma mset_heap_rev_eq[simp]: "mset_heap (rev ts) = mset_heap ts"
by (auto simp: mset_heap_def)
-
-subsubsection \<open>Invariants\<close>
-
-text \<open>Binomial invariant\<close>
+
+subsubsection \<open>Invariants\<close>
+
+text \<open>Binomial invariant\<close>
fun invar_btree :: "'a::linorder tree \<Rightarrow> bool" where
-"invar_btree (Node r x ts) \<longleftrightarrow>
+"invar_btree (Node r x ts) \<longleftrightarrow>
(\<forall>t\<in>set ts. invar_btree t) \<and> map rank ts = rev [0..<r]"
definition invar_bheap :: "'a::linorder heap \<Rightarrow> bool" where
@@ -74,22 +74,22 @@
definition invar_oheap :: "'a::linorder heap \<Rightarrow> bool" where
"invar_oheap ts \<longleftrightarrow> (\<forall>t\<in>set ts. invar_otree t)"
-
+
definition invar :: "'a::linorder heap \<Rightarrow> bool" where
"invar ts \<longleftrightarrow> invar_bheap ts \<and> invar_oheap ts"
-
+
text \<open>The children of a node are a valid heap\<close>
-lemma invar_oheap_children:
- "invar_otree (Node r v ts) \<Longrightarrow> invar_oheap (rev ts)"
+lemma invar_oheap_children:
+ "invar_otree (Node r v ts) \<Longrightarrow> invar_oheap (rev ts)"
by (auto simp: invar_oheap_def)
-lemma invar_bheap_children:
- "invar_btree (Node r v ts) \<Longrightarrow> invar_bheap (rev ts)"
+lemma invar_bheap_children:
+ "invar_btree (Node r v ts) \<Longrightarrow> invar_bheap (rev ts)"
by (auto simp: invar_bheap_def rev_map[symmetric])
-subsection \<open>Operations and Their Functional Correctness\<close>
-
+subsection \<open>Operations and Their Functional Correctness\<close>
+
subsubsection \<open>\<open>link\<close>\<close>
context
@@ -97,7 +97,7 @@
begin
fun link :: "('a::linorder) tree \<Rightarrow> 'a tree \<Rightarrow> 'a tree" where
- "link (Node r x\<^sub>1 ts\<^sub>1 =: t\<^sub>1) (Node r' x\<^sub>2 ts\<^sub>2 =: t\<^sub>2) =
+ "link (Node r x\<^sub>1 ts\<^sub>1 =: t\<^sub>1) (Node r' x\<^sub>2 ts\<^sub>2 =: t\<^sub>2) =
(if x\<^sub>1\<le>x\<^sub>2 then Node (r+1) x\<^sub>1 (t\<^sub>2#ts\<^sub>1) else Node (r+1) x\<^sub>2 (t\<^sub>1#ts\<^sub>2))"
end
@@ -105,65 +105,65 @@
lemma invar_btree_link:
assumes "invar_btree t\<^sub>1"
assumes "invar_btree t\<^sub>2"
- assumes "rank t\<^sub>1 = rank t\<^sub>2"
- shows "invar_btree (link t\<^sub>1 t\<^sub>2)"
-using assms
+ assumes "rank t\<^sub>1 = rank t\<^sub>2"
+ shows "invar_btree (link t\<^sub>1 t\<^sub>2)"
+using assms
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) simp
-lemma invar_link_otree:
+lemma invar_link_otree:
assumes "invar_otree t\<^sub>1"
assumes "invar_otree t\<^sub>2"
- shows "invar_otree (link t\<^sub>1 t\<^sub>2)"
-using assms
+ shows "invar_otree (link t\<^sub>1 t\<^sub>2)"
+using assms
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) auto
lemma rank_link[simp]: "rank (link t\<^sub>1 t\<^sub>2) = rank t\<^sub>1 + 1"
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) simp
-
+
lemma mset_link[simp]: "mset_tree (link t\<^sub>1 t\<^sub>2) = mset_tree t\<^sub>1 + mset_tree t\<^sub>2"
by (cases "(t\<^sub>1, t\<^sub>2)" rule: link.cases) simp
-
+
subsubsection \<open>\<open>ins_tree\<close>\<close>
fun ins_tree :: "'a::linorder tree \<Rightarrow> 'a heap \<Rightarrow> 'a heap" where
"ins_tree t [] = [t]"
| "ins_tree t\<^sub>1 (t\<^sub>2#ts) =
- (if rank t\<^sub>1 < rank t\<^sub>2 then t\<^sub>1#t\<^sub>2#ts else ins_tree (link t\<^sub>1 t\<^sub>2) ts)"
-
-lemma invar_bheap_Cons[simp]:
- "invar_bheap (t#ts)
+ (if rank t\<^sub>1 < rank t\<^sub>2 then t\<^sub>1#t\<^sub>2#ts else ins_tree (link t\<^sub>1 t\<^sub>2) ts)"
+
+lemma invar_bheap_Cons[simp]:
+ "invar_bheap (t#ts)
\<longleftrightarrow> invar_btree t \<and> invar_bheap ts \<and> (\<forall>t'\<in>set ts. rank t < rank t')"
by (auto simp: sorted_wrt_Cons invar_bheap_def)
-
+
lemma invar_btree_ins_tree:
- assumes "invar_btree t"
+ assumes "invar_btree t"
assumes "invar_bheap ts"
- assumes "\<forall>t'\<in>set ts. rank t \<le> rank t'"
- shows "invar_bheap (ins_tree t ts)"
+ assumes "\<forall>t'\<in>set ts. rank t \<le> rank t'"
+ shows "invar_bheap (ins_tree t ts)"
using assms
by (induction t ts rule: ins_tree.induct) (auto simp: invar_btree_link less_eq_Suc_le[symmetric])
-
-lemma invar_oheap_Cons[simp]:
- "invar_oheap (t#ts) \<longleftrightarrow> invar_otree t \<and> invar_oheap ts"
+
+lemma invar_oheap_Cons[simp]:
+ "invar_oheap (t#ts) \<longleftrightarrow> invar_otree t \<and> invar_oheap ts"
by (auto simp: invar_oheap_def)
lemma invar_oheap_ins_tree:
- assumes "invar_otree t"
+ assumes "invar_otree t"
assumes "invar_oheap ts"
- shows "invar_oheap (ins_tree t ts)"
-using assms
+ shows "invar_oheap (ins_tree t ts)"
+using assms
by (induction t ts rule: ins_tree.induct) (auto simp: invar_link_otree)
-
-lemma mset_heap_ins_tree[simp]:
- "mset_heap (ins_tree t ts) = mset_tree t + mset_heap ts"
-by (induction t ts rule: ins_tree.induct) auto
+
+lemma mset_heap_ins_tree[simp]:
+ "mset_heap (ins_tree t ts) = mset_tree t + mset_heap ts"
+by (induction t ts rule: ins_tree.induct) auto
lemma ins_tree_rank_bound:
- assumes "t' \<in> set (ins_tree t ts)"
+ assumes "t' \<in> set (ins_tree t ts)"
assumes "\<forall>t'\<in>set ts. rank t\<^sub>0 < rank t'"
- assumes "rank t\<^sub>0 < rank t"
+ assumes "rank t\<^sub>0 < rank t"
shows "rank t\<^sub>0 < rank t'"
-using assms
+using assms
by (induction t ts rule: ins_tree.induct) (auto split: if_splits)
subsubsection \<open>\<open>insert\<close>\<close>
@@ -172,10 +172,10 @@
definition insert :: "'a::linorder \<Rightarrow> 'a heap \<Rightarrow> 'a heap" where
"insert x ts = ins_tree (Node 0 x []) ts"
-
+
lemma invar_insert[simp]: "invar t \<Longrightarrow> invar (insert x t)"
-by (auto intro!: invar_btree_ins_tree simp: invar_oheap_ins_tree insert_def invar_def)
-
+by (auto intro!: invar_btree_ins_tree simp: invar_oheap_ins_tree insert_def invar_def)
+
lemma mset_heap_insert[simp]: "mset_heap (insert x t) = {#x#} + mset_heap t"
by(auto simp: insert_def)
@@ -183,7 +183,7 @@
fun merge :: "'a::linorder heap \<Rightarrow> 'a heap \<Rightarrow> 'a heap" where
"merge ts\<^sub>1 [] = ts\<^sub>1"
-| "merge [] ts\<^sub>2 = ts\<^sub>2"
+| "merge [] ts\<^sub>2 = ts\<^sub>2"
| "merge (t\<^sub>1#ts\<^sub>1) (t\<^sub>2#ts\<^sub>2) = (
if rank t\<^sub>1 < rank t\<^sub>2 then t\<^sub>1 # merge ts\<^sub>1 (t\<^sub>2#ts\<^sub>2) else
if rank t\<^sub>2 < rank t\<^sub>1 then t\<^sub>2 # merge (t\<^sub>1#ts\<^sub>1) ts\<^sub>2
@@ -192,7 +192,7 @@
lemma merge_simp2[simp]: "merge [] ts\<^sub>2 = ts\<^sub>2"
by (cases ts\<^sub>2) auto
-
+
lemma merge_rank_bound:
assumes "t' \<in> set (merge ts\<^sub>1 ts\<^sub>2)"
assumes "\<forall>t'\<in>set ts\<^sub>1. rank t < rank t'"
@@ -205,16 +205,16 @@
lemma invar_bheap_merge:
assumes "invar_bheap ts\<^sub>1"
assumes "invar_bheap ts\<^sub>2"
- shows "invar_bheap (merge ts\<^sub>1 ts\<^sub>2)"
+ shows "invar_bheap (merge ts\<^sub>1 ts\<^sub>2)"
using assms
proof (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct)
case (3 t\<^sub>1 ts\<^sub>1 t\<^sub>2 ts\<^sub>2)
-
- from "3.prems" have [simp]: "invar_btree t\<^sub>1" "invar_btree t\<^sub>2"
+
+ from "3.prems" have [simp]: "invar_btree t\<^sub>1" "invar_btree t\<^sub>2"
by auto
-
- consider (LT) "rank t\<^sub>1 < rank t\<^sub>2"
- | (GT) "rank t\<^sub>1 > rank t\<^sub>2"
+
+ consider (LT) "rank t\<^sub>1 < rank t\<^sub>2"
+ | (GT) "rank t\<^sub>1 > rank t\<^sub>2"
| (EQ) "rank t\<^sub>1 = rank t\<^sub>2"
using antisym_conv3 by blast
then show ?case proof cases
@@ -230,7 +230,7 @@
from "3.IH"(3) "3.prems" have [simp]: "invar_bheap (merge ts\<^sub>1 ts\<^sub>2)"
by auto
-
+
have "rank t\<^sub>2 < rank t'" if "t' \<in> set (merge ts\<^sub>1 ts\<^sub>2)" for t'
using that
apply (rule merge_rank_bound)
@@ -239,66 +239,66 @@
by (auto simp: Suc_le_eq invar_btree_ins_tree invar_btree_link)
qed
qed simp_all
-
+
lemma invar_oheap_merge:
assumes "invar_oheap ts\<^sub>1"
assumes "invar_oheap ts\<^sub>2"
- shows "invar_oheap (merge ts\<^sub>1 ts\<^sub>2)"
+ shows "invar_oheap (merge ts\<^sub>1 ts\<^sub>2)"
using assms
by (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct)
- (auto simp: invar_oheap_ins_tree invar_link_otree)
-
+ (auto simp: invar_oheap_ins_tree invar_link_otree)
+
lemma invar_merge[simp]: "\<lbrakk> invar ts\<^sub>1; invar ts\<^sub>2 \<rbrakk> \<Longrightarrow> invar (merge ts\<^sub>1 ts\<^sub>2)"
by (auto simp: invar_def invar_bheap_merge invar_oheap_merge)
-
-lemma mset_heap_merge[simp]:
+
+lemma mset_heap_merge[simp]:
"mset_heap (merge ts\<^sub>1 ts\<^sub>2) = mset_heap ts\<^sub>1 + mset_heap ts\<^sub>2"
-by (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct) auto
-
+by (induction ts\<^sub>1 ts\<^sub>2 rule: merge.induct) auto
+
subsubsection \<open>\<open>get_min\<close>\<close>
fun get_min :: "'a::linorder heap \<Rightarrow> 'a" where
"get_min [t] = root t"
| "get_min (t#ts) = min (root t) (get_min ts)"
-
+
lemma invar_otree_root_min:
assumes "invar_otree t"
- assumes "x \<in># mset_tree t"
- shows "root t \<le> x"
+ assumes "x \<in># mset_tree t"
+ shows "root t \<le> x"
using assms
by (induction t arbitrary: x rule: mset_tree.induct) (fastforce simp: mset_heap_def)
-
-lemma get_min_mset_aux:
- assumes "ts\<noteq>[]"
+
+lemma get_min_mset_aux:
+ assumes "ts\<noteq>[]"
assumes "invar_oheap ts"
- assumes "x \<in># mset_heap ts"
+ assumes "x \<in># mset_heap ts"
shows "get_min ts \<le> x"
- using assms
-apply (induction ts arbitrary: x rule: get_min.induct)
-apply (auto
+ using assms
+apply (induction ts arbitrary: x rule: get_min.induct)
+apply (auto
simp: invar_otree_root_min min_def intro: order_trans;
meson linear order_trans invar_otree_root_min
)+
-done
+done
-lemma get_min_mset:
- assumes "ts\<noteq>[]"
+lemma get_min_mset:
+ assumes "ts\<noteq>[]"
assumes "invar ts"
- assumes "x \<in># mset_heap ts"
+ assumes "x \<in># mset_heap ts"
shows "get_min ts \<le> x"
using assms by (auto simp: invar_def get_min_mset_aux)
-lemma get_min_member:
- "ts\<noteq>[] \<Longrightarrow> get_min ts \<in># mset_heap ts"
+lemma get_min_member:
+ "ts\<noteq>[] \<Longrightarrow> get_min ts \<in># mset_heap ts"
by (induction ts rule: get_min.induct) (auto simp: min_def)
-lemma get_min:
+lemma get_min:
assumes "mset_heap ts \<noteq> {#}"
assumes "invar ts"
shows "get_min ts = Min_mset (mset_heap ts)"
-using assms get_min_member get_min_mset
+using assms get_min_member get_min_mset
by (auto simp: eq_Min_iff)
-
+
subsubsection \<open>\<open>get_min_rest\<close>\<close>
fun get_min_rest :: "'a::linorder heap \<Rightarrow> 'a tree \<times> 'a heap" where
@@ -306,51 +306,51 @@
| "get_min_rest (t#ts) = (let (t',ts') = get_min_rest ts
in if root t \<le> root t' then (t,ts) else (t',t#ts'))"
-lemma get_min_rest_get_min_same_root:
+lemma get_min_rest_get_min_same_root:
assumes "ts\<noteq>[]"
- assumes "get_min_rest ts = (t',ts')"
- shows "root t' = get_min ts"
-using assms
+ assumes "get_min_rest ts = (t',ts')"
+ shows "root t' = get_min ts"
+using assms
by (induction ts arbitrary: t' ts' rule: get_min.induct) (auto simp: min_def split: prod.splits)
-lemma mset_get_min_rest:
- assumes "get_min_rest ts = (t',ts')"
+lemma mset_get_min_rest:
+ assumes "get_min_rest ts = (t',ts')"
assumes "ts\<noteq>[]"
- shows "mset ts = {#t'#} + mset ts'"
-using assms
+ shows "mset ts = {#t'#} + mset ts'"
+using assms
by (induction ts arbitrary: t' ts' rule: get_min.induct) (auto split: prod.splits if_splits)
-
+
lemma set_get_min_rest:
- assumes "get_min_rest ts = (t', ts')"
+ assumes "get_min_rest ts = (t', ts')"
assumes "ts\<noteq>[]"
shows "set ts = Set.insert t' (set ts')"
using mset_get_min_rest[OF assms, THEN arg_cong[where f=set_mset]]
-by auto
+by auto
-lemma invar_bheap_get_min_rest:
- assumes "get_min_rest ts = (t',ts')"
+lemma invar_bheap_get_min_rest:
+ assumes "get_min_rest ts = (t',ts')"
assumes "ts\<noteq>[]"
- assumes "invar_bheap ts"
+ assumes "invar_bheap ts"
shows "invar_btree t'" and "invar_bheap ts'"
proof -
have "invar_btree t' \<and> invar_bheap ts'"
- using assms
+ using assms
proof (induction ts arbitrary: t' ts' rule: get_min.induct)
case (2 t v va)
then show ?case
apply (clarsimp split: prod.splits if_splits)
apply (drule set_get_min_rest; fastforce)
- done
+ done
qed auto
thus "invar_btree t'" and "invar_bheap ts'" by auto
qed
-lemma invar_oheap_get_min_rest:
- assumes "get_min_rest ts = (t',ts')"
+lemma invar_oheap_get_min_rest:
+ assumes "get_min_rest ts = (t',ts')"
assumes "ts\<noteq>[]"
- assumes "invar_oheap ts"
+ assumes "invar_oheap ts"
shows "invar_otree t'" and "invar_oheap ts'"
-using assms
+using assms
by (induction ts arbitrary: t' ts' rule: get_min.induct) (auto split: prod.splits if_splits)
subsubsection \<open>\<open>del_min\<close>\<close>
@@ -358,30 +358,30 @@
definition del_min :: "'a::linorder heap \<Rightarrow> 'a::linorder heap" where
"del_min ts = (case get_min_rest ts of
(Node r x ts\<^sub>1, ts\<^sub>2) \<Rightarrow> merge (rev ts\<^sub>1) ts\<^sub>2)"
-
+
lemma invar_del_min[simp]:
assumes "ts \<noteq> []"
assumes "invar ts"
shows "invar (del_min ts)"
-using assms
-unfolding invar_def del_min_def
-by (auto
- split: prod.split tree.split
+using assms
+unfolding invar_def del_min_def
+by (auto
+ split: prod.split tree.split
intro!: invar_bheap_merge invar_oheap_merge
dest: invar_bheap_get_min_rest invar_oheap_get_min_rest
intro!: invar_oheap_children invar_bheap_children
)
-
-lemma mset_heap_del_min:
+
+lemma mset_heap_del_min:
assumes "ts \<noteq> []"
shows "mset_heap ts = mset_heap (del_min ts) + {# get_min ts #}"
using assms
unfolding del_min_def
apply (clarsimp split: tree.split prod.split)
-apply (frule (1) get_min_rest_get_min_same_root)
-apply (frule (1) mset_get_min_rest)
+apply (frule (1) get_min_rest_get_min_same_root)
+apply (frule (1) mset_get_min_rest)
apply (auto simp: mset_heap_def)
-done
+done
subsubsection \<open>Instantiating the Priority Queue Locale\<close>
@@ -401,11 +401,11 @@
case 3 thus ?case by auto
next
case (4 q)
- thus ?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) thus ?case using get_min[of q] by auto
-next
+next
case 6 thus ?case by (auto simp add: invar_def invar_bheap_def invar_oheap_def)
next
case 7 thus ?case by simp
@@ -419,57 +419,57 @@
subsection \<open>Complexity\<close>
-
-text \<open>The size of a binomial tree is determined by its rank\<close>
+
+text \<open>The size of a binomial tree is determined by its rank\<close>
lemma size_mset_btree:
assumes "invar_btree t"
- shows "size (mset_tree t) = 2^rank t"
+ shows "size (mset_tree t) = 2^rank t"
using assms
proof (induction t)
case (Node r v ts)
hence IH: "size (mset_tree t) = 2^rank t" if "t \<in> set ts" for t
using that by auto
-
+
from Node have COMPL: "map rank ts = rev [0..<r]" by auto
-
+
have "size (mset_heap ts) = (\<Sum>t\<leftarrow>ts. size (mset_tree t))"
by (induction ts) auto
also have "\<dots> = (\<Sum>t\<leftarrow>ts. 2^rank t)" using IH
- by (auto cong: sum_list_cong)
- also have "\<dots> = (\<Sum>r\<leftarrow>map rank ts. 2^r)"
+ by (auto cong: map_cong)
+ also have "\<dots> = (\<Sum>r\<leftarrow>map rank ts. 2^r)"
by (induction ts) auto
- also have "\<dots> = (\<Sum>i\<in>{0..<r}. 2^i)"
- unfolding COMPL
+ also have "\<dots> = (\<Sum>i\<in>{0..<r}. 2^i)"
+ unfolding COMPL
by (auto simp: rev_map[symmetric] interv_sum_list_conv_sum_set_nat)
- also have "\<dots> = 2^r - 1"
+ also have "\<dots> = 2^r - 1"
by (induction r) auto
- finally show ?case
+ finally show ?case
by (simp)
qed
-
-text \<open>The length of a binomial heap is bounded by the number of its elements\<close>
-lemma size_mset_bheap:
+
+text \<open>The length of a binomial heap is bounded by the number of its elements\<close>
+lemma size_mset_bheap:
assumes "invar_bheap ts"
shows "2^length ts \<le> size (mset_heap ts) + 1"
proof -
- from \<open>invar_bheap ts\<close> have
+ from \<open>invar_bheap ts\<close> have
ASC: "sorted_wrt (<) (map rank ts)" and
TINV: "\<forall>t\<in>set ts. invar_btree t"
unfolding invar_bheap_def by auto
-
- have "(2::nat)^length ts = (\<Sum>i\<in>{0..<length ts}. 2^i) + 1"
+
+ have "(2::nat)^length ts = (\<Sum>i\<in>{0..<length ts}. 2^i) + 1"
by (simp add: sum_power2)
also have "\<dots> \<le> (\<Sum>t\<leftarrow>ts. 2^rank t) + 1"
using sorted_wrt_less_sum_mono_lowerbound[OF _ ASC, of "(^) (2::nat)"]
- using power_increasing[where a="2::nat"]
+ using power_increasing[where a="2::nat"]
by (auto simp: o_def)
- also have "\<dots> = (\<Sum>t\<leftarrow>ts. size (mset_tree t)) + 1" using TINV
- by (auto cong: sum_list_cong simp: size_mset_btree)
+ also have "\<dots> = (\<Sum>t\<leftarrow>ts. size (mset_tree t)) + 1" using TINV
+ by (auto cong: map_cong simp: size_mset_btree)
also have "\<dots> = size (mset_heap ts) + 1"
unfolding mset_heap_def by (induction ts) auto
finally show ?thesis .
-qed
-
+qed
+
subsubsection \<open>Timing Functions\<close>
text \<open>
@@ -477,14 +477,14 @@
estimations of their complexity.
\<close>
definition t_link :: "'a::linorder tree \<Rightarrow> 'a tree \<Rightarrow> nat" where
-[simp]: "t_link _ _ = 1"
+[simp]: "t_link _ _ = 1"
fun t_ins_tree :: "'a::linorder tree \<Rightarrow> 'a heap \<Rightarrow> nat" where
"t_ins_tree t [] = 1"
| "t_ins_tree t\<^sub>1 (t\<^sub>2 # rest) = (
- (if rank t\<^sub>1 < rank t\<^sub>2 then 1
+ (if rank t\<^sub>1 < rank t\<^sub>2 then 1
else t_link t\<^sub>1 t\<^sub>2 + t_ins_tree (link t\<^sub>1 t\<^sub>2) rest)
- )"
+ )"
definition t_insert :: "'a::linorder \<Rightarrow> 'a heap \<Rightarrow> nat" where
"t_insert x ts = t_ins_tree (Node 0 x []) ts"
@@ -494,74 +494,74 @@
subsubsection \<open>\<open>t_insert\<close>\<close>
-lemma t_insert_bound:
+lemma t_insert_bound:
assumes "invar ts"
shows "t_insert x ts \<le> log 2 (size (mset_heap ts) + 1) + 1"
proof -
- have 1: "t_insert x ts \<le> length ts + 1"
+ have 1: "t_insert x ts \<le> length ts + 1"
unfolding t_insert_def by (rule t_ins_tree_simple_bound)
- also have "\<dots> \<le> log 2 (2 * (size (mset_heap ts) + 1))"
+ also have "\<dots> \<le> log 2 (2 * (size (mset_heap ts) + 1))"
proof -
- from size_mset_bheap[of ts] assms
+ from size_mset_bheap[of ts] assms
have "2 ^ length ts \<le> size (mset_heap ts) + 1"
unfolding invar_def by auto
hence "2 ^ (length ts + 1) \<le> 2 * (size (mset_heap ts) + 1)" by auto
thus ?thesis using le_log2_of_power by blast
qed
- finally show ?thesis
+ finally show ?thesis
by (simp only: log_mult of_nat_mult) auto
-qed
+qed
subsubsection \<open>\<open>t_merge\<close>\<close>
fun t_merge :: "'a::linorder heap \<Rightarrow> 'a heap \<Rightarrow> nat" where
"t_merge ts\<^sub>1 [] = 1"
-| "t_merge [] ts\<^sub>2 = 1"
+| "t_merge [] ts\<^sub>2 = 1"
| "t_merge (t\<^sub>1#ts\<^sub>1) (t\<^sub>2#ts\<^sub>2) = 1 + (
if rank t\<^sub>1 < rank t\<^sub>2 then t_merge ts\<^sub>1 (t\<^sub>2#ts\<^sub>2)
else if rank t\<^sub>2 < rank t\<^sub>1 then t_merge (t\<^sub>1#ts\<^sub>1) ts\<^sub>2
else t_ins_tree (link t\<^sub>1 t\<^sub>2) (merge ts\<^sub>1 ts\<^sub>2) + t_merge ts\<^sub>1 ts\<^sub>2
- )"
-
-text \<open>A crucial idea is to estimate the time in correlation with the
- result length, as each carry reduces the length of the result.\<close>
+ )"
+
+text \<open>A crucial idea is to estimate the time in correlation with the
+ result length, as each carry reduces the length of the result.\<close>
lemma t_ins_tree_length:
"t_ins_tree t ts + length (ins_tree t ts) = 2 + length ts"
by (induction t ts rule: ins_tree.induct) auto
-lemma t_merge_length:
+lemma t_merge_length:
"length (merge ts\<^sub>1 ts\<^sub>2) + t_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1"
-by (induction ts\<^sub>1 ts\<^sub>2 rule: t_merge.induct)
+by (induction ts\<^sub>1 ts\<^sub>2 rule: t_merge.induct)
(auto simp: t_ins_tree_length algebra_simps)
text \<open>Finally, we get the desired logarithmic bound\<close>
lemma t_merge_bound_aux:
fixes ts\<^sub>1 ts\<^sub>2
- defines "n\<^sub>1 \<equiv> size (mset_heap ts\<^sub>1)"
+ defines "n\<^sub>1 \<equiv> size (mset_heap ts\<^sub>1)"
defines "n\<^sub>2 \<equiv> size (mset_heap ts\<^sub>2)"
assumes BINVARS: "invar_bheap ts\<^sub>1" "invar_bheap ts\<^sub>2"
shows "t_merge ts\<^sub>1 ts\<^sub>2 \<le> 4*log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 2"
proof -
- define n where "n = n\<^sub>1 + n\<^sub>2"
-
- from t_merge_length[of ts\<^sub>1 ts\<^sub>2]
+ define n where "n = n\<^sub>1 + n\<^sub>2"
+
+ from t_merge_length[of ts\<^sub>1 ts\<^sub>2]
have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1" by auto
- hence "(2::nat)^t_merge ts\<^sub>1 ts\<^sub>2 \<le> 2^(2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1)"
+ hence "(2::nat)^t_merge ts\<^sub>1 ts\<^sub>2 \<le> 2^(2 * (length ts\<^sub>1 + length ts\<^sub>2) + 1)"
by (rule power_increasing) auto
- also have "\<dots> = 2*(2^length ts\<^sub>1)\<^sup>2*(2^length ts\<^sub>2)\<^sup>2"
+ also have "\<dots> = 2*(2^length ts\<^sub>1)\<^sup>2*(2^length ts\<^sub>2)\<^sup>2"
by (auto simp: algebra_simps power_add power_mult)
also note BINVARS(1)[THEN size_mset_bheap]
also note BINVARS(2)[THEN size_mset_bheap]
- finally have "2 ^ t_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (n\<^sub>1 + 1)\<^sup>2 * (n\<^sub>2 + 1)\<^sup>2"
+ finally have "2 ^ t_merge ts\<^sub>1 ts\<^sub>2 \<le> 2 * (n\<^sub>1 + 1)\<^sup>2 * (n\<^sub>2 + 1)\<^sup>2"
by (auto simp: power2_nat_le_eq_le n\<^sub>1_def n\<^sub>2_def)
- from le_log2_of_power[OF this] have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 \<dots>"
+ from le_log2_of_power[OF this] have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 \<dots>"
by simp
also have "\<dots> = log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n\<^sub>2 + 1)"
by (simp add: log_mult log_nat_power)
also have "n\<^sub>2 \<le> n" by (auto simp: n_def)
- finally have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n + 1)"
+ finally have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 2 + 2*log 2 (n\<^sub>1 + 1) + 2*log 2 (n + 1)"
by auto
also have "n\<^sub>1 \<le> n" by (auto simp: n_def)
finally have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> log 2 2 + 4*log 2 (n + 1)"
@@ -569,15 +569,15 @@
also have "log 2 2 \<le> 2" by auto
finally have "t_merge ts\<^sub>1 ts\<^sub>2 \<le> 4*log 2 (n + 1) + 2" by auto
thus ?thesis unfolding n_def by (auto simp: algebra_simps)
-qed
-
+qed
+
lemma t_merge_bound:
fixes ts\<^sub>1 ts\<^sub>2
- defines "n\<^sub>1 \<equiv> size (mset_heap ts\<^sub>1)"
+ defines "n\<^sub>1 \<equiv> size (mset_heap ts\<^sub>1)"
defines "n\<^sub>2 \<equiv> size (mset_heap ts\<^sub>2)"
assumes "invar ts\<^sub>1" "invar ts\<^sub>2"
shows "t_merge ts\<^sub>1 ts\<^sub>2 \<le> 4*log 2 (n\<^sub>1 + n\<^sub>2 + 1) + 2"
-using assms t_merge_bound_aux unfolding invar_def by blast
+using assms t_merge_bound_aux unfolding invar_def by blast
subsubsection \<open>\<open>t_get_min\<close>\<close>
@@ -585,10 +585,10 @@
"t_get_min [t] = 1"
| "t_get_min (t#ts) = 1 + t_get_min ts"
-lemma t_get_min_estimate: "ts\<noteq>[] \<Longrightarrow> t_get_min ts = length ts"
+lemma t_get_min_estimate: "ts\<noteq>[] \<Longrightarrow> t_get_min ts = length ts"
by (induction ts rule: t_get_min.induct) auto
-
-lemma t_get_min_bound:
+
+lemma t_get_min_bound:
assumes "invar ts"
assumes "ts\<noteq>[]"
shows "t_get_min ts \<le> log 2 (size (mset_heap ts) + 1)"
@@ -600,8 +600,8 @@
unfolding invar_def by auto
thus ?thesis using le_log2_of_power by blast
qed
- finally show ?thesis by auto
-qed
+ finally show ?thesis by auto
+qed
subsubsection \<open>\<open>t_del_min\<close>\<close>
@@ -609,10 +609,10 @@
"t_get_min_rest [t] = 1"
| "t_get_min_rest (t#ts) = 1 + t_get_min_rest ts"
-lemma t_get_min_rest_estimate: "ts\<noteq>[] \<Longrightarrow> t_get_min_rest ts = length ts"
+lemma t_get_min_rest_estimate: "ts\<noteq>[] \<Longrightarrow> t_get_min_rest ts = length ts"
by (induction ts rule: t_get_min_rest.induct) auto
-
-lemma t_get_min_rest_bound_aux:
+
+lemma t_get_min_rest_bound_aux:
assumes "invar_bheap ts"
assumes "ts\<noteq>[]"
shows "t_get_min_rest ts \<le> log 2 (size (mset_heap ts) + 1)"
@@ -624,14 +624,14 @@
by auto
thus ?thesis using le_log2_of_power by blast
qed
- finally show ?thesis by auto
-qed
+ finally show ?thesis by auto
+qed
-lemma t_get_min_rest_bound:
+lemma t_get_min_rest_bound:
assumes "invar ts"
assumes "ts\<noteq>[]"
shows "t_get_min_rest ts \<le> log 2 (size (mset_heap ts) + 1)"
-using assms t_get_min_rest_bound_aux unfolding invar_def by blast
+using assms t_get_min_rest_bound_aux unfolding invar_def by blast
text\<open>Note that although the definition of function @{const rev} has quadratic complexity,
it can and is implemented (via suitable code lemmas) as a linear time function.
@@ -643,8 +643,8 @@
"t_del_min ts = t_get_min_rest ts + (case get_min_rest ts of (Node _ x ts\<^sub>1, ts\<^sub>2)
\<Rightarrow> t_rev ts\<^sub>1 + t_merge (rev ts\<^sub>1) ts\<^sub>2
)"
-
-lemma t_rev_ts1_bound_aux:
+
+lemma t_rev_ts1_bound_aux:
fixes ts
defines "n \<equiv> size (mset_heap ts)"
assumes BINVAR: "invar_bheap (rev ts)"
@@ -657,16 +657,16 @@
from le_log2_of_power[OF this] have "t_rev ts \<le> log 2 (2 * (n + 1))"
by auto
also have "\<dots> = 1 + log 2 (n+1)"
- by (simp only: of_nat_mult log_mult) auto
+ by (simp only: of_nat_mult log_mult) auto
finally show ?thesis by (auto simp: algebra_simps)
-qed
+qed
lemma t_del_min_bound_aux:
fixes ts
defines "n \<equiv> size (mset_heap ts)"
assumes BINVAR: "invar_bheap ts"
assumes "ts\<noteq>[]"
- shows "t_del_min ts \<le> 6 * log 2 (n+1) + 3"
+ shows "t_del_min ts \<le> 6 * log 2 (n+1) + 3"
proof -
obtain r x ts\<^sub>1 ts\<^sub>2 where GM: "get_min_rest ts = (Node r x ts\<^sub>1, ts\<^sub>2)"
by (metis surj_pair tree.exhaust_sel)
@@ -676,17 +676,17 @@
define n\<^sub>1 where "n\<^sub>1 = size (mset_heap ts\<^sub>1)"
define n\<^sub>2 where "n\<^sub>2 = size (mset_heap ts\<^sub>2)"
-
+
have t_rev_ts1_bound: "t_rev ts\<^sub>1 \<le> 1 + log 2 (n+1)"
proof -
note t_rev_ts1_bound_aux[OF BINVAR1, simplified, folded n\<^sub>1_def]
- also have "n\<^sub>1 \<le> n"
+ also have "n\<^sub>1 \<le> n"
unfolding n\<^sub>1_def n_def
using mset_get_min_rest[OF GM \<open>ts\<noteq>[]\<close>]
by (auto simp: mset_heap_def)
finally show ?thesis by (auto simp: algebra_simps)
- qed
-
+ qed
+
have "t_del_min ts = t_get_min_rest ts + t_rev ts\<^sub>1 + t_merge (rev ts\<^sub>1) ts\<^sub>2"
unfolding t_del_min_def by (simp add: GM)
also have "\<dots> \<le> log 2 (n+1) + t_rev ts\<^sub>1 + t_merge (rev ts\<^sub>1) ts\<^sub>2"
@@ -700,17 +700,17 @@
unfolding n\<^sub>1_def n\<^sub>2_def n_def
using mset_get_min_rest[OF GM \<open>ts\<noteq>[]\<close>]
by (auto simp: mset_heap_def)
- finally have "t_del_min ts \<le> 6 * log 2 (n+1) + 3"
+ finally have "t_del_min ts \<le> 6 * log 2 (n+1) + 3"
by auto
thus ?thesis by (simp add: algebra_simps)
-qed
-
+qed
+
lemma t_del_min_bound:
fixes ts
defines "n \<equiv> size (mset_heap ts)"
assumes "invar ts"
assumes "ts\<noteq>[]"
- shows "t_del_min ts \<le> 6 * log 2 (n+1) + 3"
+ shows "t_del_min ts \<le> 6 * log 2 (n+1) + 3"
using assms t_del_min_bound_aux unfolding invar_def by blast
-end
+end
\ No newline at end of file
--- a/src/HOL/Probability/Probability_Mass_Function.thy Mon Jan 22 15:06:38 2018 +0100
+++ b/src/HOL/Probability/Probability_Mass_Function.thy Mon Jan 22 15:50:29 2018 +0100
@@ -530,11 +530,11 @@
by (simp add: integral_distr map_pmf_rep_eq)
lemma pmf_abs_summable [intro]: "pmf p abs_summable_on A"
- by (rule abs_summable_on_subset[OF _ subset_UNIV])
+ by (rule abs_summable_on_subset[OF _ subset_UNIV])
(auto simp: abs_summable_on_def integrable_iff_bounded nn_integral_pmf)
lemma measure_pmf_conv_infsetsum: "measure (measure_pmf p) A = infsetsum (pmf p) A"
- unfolding infsetsum_def by (simp add: integral_eq_nn_integral nn_integral_pmf measure_def)
+ unfolding infsetsum_def by (simp add: integral_eq_nn_integral nn_integral_pmf measure_def)
lemma infsetsum_pmf_eq_1:
assumes "set_pmf p \<subseteq> A"
@@ -774,7 +774,7 @@
apply (subst lebesgue_integral_count_space_finite_support)
apply (auto intro!: finite_subset[OF _ \<open>finite A\<close>] sum.mono_neutral_left simp: pmf_eq_0_set_pmf)
done
-
+
lemma expectation_return_pmf [simp]:
fixes f :: "'a \<Rightarrow> 'b::{banach, second_countable_topology}"
shows "measure_pmf.expectation (return_pmf x) f = f x"
@@ -793,7 +793,7 @@
proof (intro sum.cong refl, goal_cases)
case (1 x)
thus ?case
- by (subst pmf_bind, subst integral_measure_pmf[of A])
+ by (subst pmf_bind, subst integral_measure_pmf[of A])
(insert assms, auto simp: scaleR_sum_left)
qed
also have "\<dots> = (\<Sum>j\<in>A. pmf p j *\<^sub>R (\<Sum>i\<in>(\<Union>x\<in>A. set_pmf (f x)). pmf (f j) i *\<^sub>R h i))"
@@ -802,7 +802,7 @@
proof (intro sum.cong refl, goal_cases)
case (1 x)
thus ?case
- by (subst integral_measure_pmf[of "(\<Union>x\<in>A. set_pmf (f x))"])
+ by (subst integral_measure_pmf[of "(\<Union>x\<in>A. set_pmf (f x))"])
(insert assms, auto simp: scaleR_sum_left)
qed
finally show ?thesis .
@@ -1648,7 +1648,7 @@
shows "transp (rel_pmf R)"
using assms by (fact pmf.rel_transp)
-
+
subsection \<open> Distributions \<close>
context
@@ -1778,7 +1778,7 @@
by (simp add: measure_nonneg measure_pmf.emeasure_eq_measure)
end
-
+
lemma pmf_expectation_bind_pmf_of_set:
fixes A :: "'a set" and f :: "'a \<Rightarrow> 'b pmf"
and h :: "'b \<Rightarrow> 'c::{banach, second_countable_topology}"
@@ -2010,7 +2010,7 @@
bind_return_pmf' binomial_pmf_0 intro!: bind_pmf_cong)
-subsection \<open>PMFs from assiciation lists\<close>
+subsection \<open>PMFs from association lists\<close>
definition pmf_of_list ::" ('a \<times> real) list \<Rightarrow> 'a pmf" where
"pmf_of_list xs = embed_pmf (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
@@ -2032,7 +2032,10 @@
proof -
have "(\<integral>\<^sup>+ x. ennreal (sum_list (map snd (filter (\<lambda>z. fst z = x) xs))) \<partial>count_space UNIV) =
(\<integral>\<^sup>+ x. ennreal (sum_list (map (\<lambda>(x',p). indicator {x'} x * p) xs)) \<partial>count_space UNIV)"
- by (intro nn_integral_cong ennreal_cong, subst sum_list_map_filter') (auto intro: sum_list_cong)
+ apply (intro nn_integral_cong ennreal_cong, subst sum_list_map_filter')
+ apply (rule arg_cong[where f = sum_list])
+ apply (auto cong: map_cong)
+ done
also have "\<dots> = (\<Sum>(x',p)\<leftarrow>xs. (\<integral>\<^sup>+ x. ennreal (indicator {x'} x * p) \<partial>count_space UNIV))"
using assms(1)
proof (induction xs)