header "Binomial Heaps"
theory BinomialHeap
imports Main "~~/src/HOL/Library/Multiset"
begin
subsection {* Datatype Definition *}
text {* Binomial heaps are lists of binomial trees. *}
datatype ('e, 'a) BinomialTree =
Node 'e "'a::linorder" nat "('e , 'a) BinomialTree list"
type_synonym ('e, 'a) BinomialQueue_inv = "('e, 'a::linorder) BinomialTree list"
locale BinomialHeapStruc_loc
begin
subsubsection "Auxiliary Definitions"
text {* Projections *}
primrec val :: "('e, 'a::linorder) BinomialTree => 'e" where
"val (Node e a r ts) = e"
primrec prio :: "('e, 'a::linorder) BinomialTree => 'a" where
"prio (Node e a r ts) = a"
primrec rank :: "('e, 'a::linorder) BinomialTree => nat" where
"rank (Node e a r ts) = r"
primrec children ::
"('e, 'a::linorder) BinomialTree => ('e, 'a) BinomialTree list" where
"children (Node e a r ts) = ts"
text {* Combine two binomial trees (of rank $r$) to one (of rank $r+1$). *}
fun link :: "('e, 'a::linorder) BinomialTree => ('e, 'a) BinomialTree =>
('e, 'a) BinomialTree" where
"link (Node e1 a1 r1 ts1) (Node e2 a2 r2 ts2) =
(if a1≤a2
then (Node e1 a1 (Suc r1) ((Node e2 a2 r2 ts2)#ts1))
else (Node e2 a2 (Suc r2) ((Node e1 a1 r1 ts1)#ts2)))"
subsubsection "Abstraction to Multiset"
text {* Return a multiset with all (element, priority) pairs from a queue. *}
fun tree_to_multiset
:: "('e, 'a::linorder) BinomialTree => ('e × 'a) multiset"
and queue_to_multiset
:: "('e, 'a::linorder) BinomialQueue_inv => ('e × 'a) multiset" where
"tree_to_multiset (Node e a r ts) = {#(e,a)#} + queue_to_multiset ts" |
"queue_to_multiset [] = {#}" |
"queue_to_multiset (t#q) = tree_to_multiset t + queue_to_multiset q"
lemma qtmset_append_union[simp]: "queue_to_multiset (q @ q') =
queue_to_multiset q + queue_to_multiset q'"
apply(induct q)
apply(simp)
apply(simp add: union_ac)
done
lemma qtmset_rev[simp]: "queue_to_multiset (rev q) = queue_to_multiset q"
apply(induct q)
apply(simp)
apply(simp add: union_ac)
done
subsubsection "Invariant"
text {* We first formulate the invariant for single binomial trees,
and then extend the invariant to binomial heaps (lists of binomial trees).
The invariant for trees claims that a tree labeled rank $0$ has no children,
and a tree labeled rank $r+1$ is the result of a link operation of
two rank $r$ trees.
*}
function tree_invar :: "('e, 'a::linorder) BinomialTree => bool" where
"tree_invar (Node e a 0 ts) = (ts = [])" |
"tree_invar (Node e a (Suc r) ts) =
(∃ e1 a1 ts1 e2 a2 ts2.
tree_invar (Node e1 a1 r ts1) ∧
tree_invar (Node e2 a2 r ts2) ∧
(Node e a (Suc r) ts) = link (Node e1 a1 r ts1) (Node e2 a2 r ts2))"
by pat_completeness auto
termination
apply(relation "measure (λt. rank t)")
apply auto
done
text {* A queue satisfies the invariant, iff all trees inside the queue satisfy
the invariant, and the queue contains only trees of distinct rank and
is ordered by rank *}
text {* First part: All trees of the queue satisfy the tree invariant: *}
definition queue_invar :: "('e, 'a::linorder) BinomialQueue_inv => bool" where
"queue_invar q ≡ (∀t ∈ set q. tree_invar t)"
text {* Second part: Trees have distinct rank, and are ordered by
ascending rank: *}
fun rank_invar :: "('e, 'a::linorder) BinomialQueue_inv => bool" where
"rank_invar [] = True" |
"rank_invar [t] = True" |
"rank_invar (t # t' # bq) = (rank t < rank t' ∧ rank_invar (t' # bq))"
lemma queue_invar_simps[simp]:
"queue_invar []"
"queue_invar (t#q) <-> tree_invar t ∧ queue_invar q"
"queue_invar (q@q') <-> queue_invar q ∧ queue_invar q'"
unfolding queue_invar_def by auto
text {* Invariant for binomial queues: *}
definition "invar q == queue_invar q ∧ rank_invar q"
lemma mset_link[simp]: "(tree_to_multiset (link t1 t2))
= (tree_to_multiset t1) + (tree_to_multiset t2)"
by(cases t1, cases t2, auto simp add: union_ac)
lemma link_tree_invar:
"[|tree_invar t1; tree_invar t2; rank t1 = rank t2|] ==> tree_invar (link t1 t2)"
by (cases t1, cases t2, simp, blast)
lemma invar_children:
assumes "tree_invar ((Node e a r ts)::(('e, 'a::linorder) BinomialTree))"
shows "queue_invar ts" using assms
unfolding queue_invar_def
proof(induct r arbitrary: e a ts, simp)
case goal1
from goal1(2) obtain e1 a1 ts1 e2 a2 ts2 where
O: "tree_invar (Node e1 a1 r ts1)" "tree_invar (Node e2 a2 r ts2)"
"(Node e a (Suc r) ts) = link (Node e1 a1 r ts1) (Node e2 a2 r ts2)"
by (simp only: tree_invar.simps) blast
from goal1(1)[OF O(1)] O(2)
have case1: "queue_invar ((Node e2 a2 r ts2) # ts1)"
unfolding queue_invar_def by simp
from goal1(1)[OF O(2)] O(1)
have case2: "queue_invar ((Node e1 a1 r ts1) # ts2)"
unfolding queue_invar_def by simp
from O(3) have "ts = (if a1≤a2
then (Node e2 a2 r ts2) # ts1
else (Node e1 a1 r ts1) # ts2)" by auto
with case1 case2 show ?case unfolding queue_invar_def by simp
qed
lemma invar_children': "tree_invar t ==> queue_invar (children t)"
by (cases t) (auto simp add: invar_children)
lemma rank_link: "rank t = rank t' ==> rank (link t t') = rank t + 1"
apply (cases t)
apply (cases t')
apply(auto)
done
lemma rank_invar_not_empty_hd: "[|rank_invar (t # bq); bq ≠ []|] ==>
rank t < rank (hd bq)"
apply(induct bq arbitrary: t)
apply(auto)
done
lemma rank_invar_to_set: "rank_invar (t # bq) ==>
∀ t' ∈ set bq. rank t < rank t'"
apply(induct bq arbitrary: t)
apply(simp)
apply (metis nat_less_le rank_invar.simps(3) set_ConsD xt1(7))
done
lemma set_to_rank_invar: "[|∀ t' ∈ set bq. rank t < rank t'; rank_invar bq|]
==> rank_invar (t # bq)"
apply(induct bq arbitrary: t)
apply(simp)
by (metis hd.simps hd_in_set list.distinct(1) rank_invar.simps(3))
lemma rank_invar_hd_cons:
"[|rank_invar bq; rank t < rank (hd bq)|] ==> rank_invar (t # bq)"
apply(cases bq)
apply(auto)
done
lemma rank_invar_cons: "rank_invar (t # bq) ==> rank_invar bq"
apply(cases bq)
apply(auto)
done
lemma invar_cons_up:
"[|invar (t # bq); rank t' < rank t; tree_invar t'|] ==> invar (t' # t # bq)"
unfolding invar_def
by (cases bq) simp_all
lemma invar_cons_down: "invar (t # bq) ==> invar bq"
unfolding invar_def
by (cases bq) simp_all
lemma invar_app_single:
"[|invar bq; ∀t ∈ set bq. rank t < rank t'; tree_invar t'|]
==> invar (bq @ [t'])"
proof (induct bq, simp add: invar_def)
case goal1
from `invar (a # bq)` have "invar bq" by (rule invar_cons_down)
with goal1 have "invar (bq @ [t'])" by simp
with goal1 show ?case
apply (cases bq)
apply (simp_all add: invar_def)
done
qed
subsubsection "Heap Ordering"
fun heap_ordered :: "('e, 'a::linorder) BinomialTree => bool" where
"heap_ordered (Node e a r ts) = (∀x ∈ set_of(queue_to_multiset ts). a ≤ snd x)"
text {* The invariant for trees implies heap order. *}
lemma tree_invar_heap_ordered: "tree_invar t ==> heap_ordered t"
proof (cases t)
case goal1 thus ?case
proof (induct nat arbitrary: t e a list, simp)
case goal1
from goal1 obtain t1 e1 a1 ts1 t2 e2 a2 ts2 where
O: "tree_invar t1" "tree_invar t2" "t = link t1 t2"
and t1[simp]: "t1 = (Node e1 a1 nat ts1)"
and t2[simp]: "t2 = (Node e2 a2 nat ts2)"
by (simp only: tree_invar.simps) blast
from O(3) have "t = (if a1≤a2
then (Node e1 a1 (Suc nat) (t2 # ts1))
else (Node e2 a2 (Suc nat) (t1 # ts2)))" by simp
with goal1(1)[OF O(1) t1] goal1(1)[OF O(2) t2]
show ?case by (cases "a1 ≤ a2", auto)
qed
qed
subsubsection "Height and Length"
text {*
Although complexity of HOL-functions cannot be expressed within
HOL, we can express the height and length of a binomial heap.
By showing that both, height and length, are logarithmic in the number
of contained elements, we give strong evidence that our functions have
logarithmic complexity in the number of elements.
*}
text {* Height of a tree and queue *}
fun height_tree :: "('e, ('a::linorder)) BinomialTree => nat" and
height_queue :: "('e, ('a::linorder)) BinomialQueue_inv => nat"
where
"height_tree (Node e a r ts) = height_queue ts" |
"height_queue [] = 0" |
"height_queue (t # ts) = max (Suc (height_tree t)) (height_queue ts)"
lemma link_length: "size (tree_to_multiset (link t1 t2)) =
size (tree_to_multiset t1) + size (tree_to_multiset t2)"
apply(cases t1)
apply(cases t2)
apply simp
done
lemma tree_rank_estimate:
"tree_invar (Node e a r ts) ==>
size (tree_to_multiset (Node e a r ts)) = (2::nat)^r"
apply(induct r arbitrary: e a ts, simp)
proof -
case goal1
from goal1(2) obtain e1 a1 ts1 e2 a2 ts2 where link:
"(Node e a (Suc r) ts) = link (Node e1 a1 r ts1) (Node e2 a2 r ts2)"
and inv1: "tree_invar (Node e1 a1 r ts1) "
and inv2: "tree_invar (Node e2 a2 r ts2)" by simp blast
from link_length[of "(Node e1 a1 r ts1)" "(Node e2 a2 r ts2)"]
goal1(1)[OF inv1] goal1(1)[OF inv2] link
show ?case by simp
qed
lemma tree_rank_height:
"tree_invar (Node e a r ts) ==> height_tree (Node e a r ts) = r"
apply(induct r arbitrary: e a ts, simp)
proof -
case goal1
from goal1(2) obtain e1 a1 ts1 e2 a2 ts2 where link:
"(Node e a (Suc r) ts) = link (Node e1 a1 r ts1) (Node e2 a2 r ts2)"
and inv1: "tree_invar (Node e1 a1 r ts1) "
and inv2: "tree_invar (Node e2 a2 r ts2)" by simp blast
with link goal1(1)[OF inv1] goal1(1)[OF inv2] goal1(2) show ?case
by (cases "a1 ≤ a2") (simp_all)
qed
text {* A binomial tree of height $h$ contains exactly $2^{h}$ elements *}
theorem tree_height_estimate:
"tree_invar t ==> size (tree_to_multiset t) = (2::nat)^(height_tree t)"
apply (cases t, simp only:)
apply (frule tree_rank_estimate)
apply (frule tree_rank_height)
apply (simp only: )
done
lemma size_mset_tree: "tree_invar t ==>
size (tree_to_multiset t) = (2::nat)^(rank t)"
apply (cases t)
by (simp only: tree_rank_estimate rank.simps)
lemma invar_butlast: "invar (bq @ [t]) ==> invar bq"
unfolding invar_def
apply (induct bq) apply simp apply (case_tac bq)
by (simp_all)
lemma invar_last_max: "invar (bq @ [m]) ==> ∀ t ∈ set bq. rank t < rank m"
unfolding invar_def
apply (induct bq) apply simp apply (case_tac bq) apply simp by simp
lemma invar_length: "invar bq ==> length bq ≤ Suc (rank (last bq))"
proof (induct bq rule: rev_induct)
case Nil thus ?case by simp
next
case (snoc x xs)
show ?case proof (cases xs)
case Nil thus ?thesis by simp
next
case (Cons xxs xx)[simp]
from snoc.hyps[OF invar_butlast[OF snoc.prems]] have
IH: "length xs ≤ Suc (rank (last xs))" .
also from invar_last_max[OF snoc.prems] last_in_set[of xs] have
"Suc (rank (last xs)) ≤ rank (last (xs @ [x]))"
by auto
finally show ?thesis by simp
qed
qed
lemma size_queue_listsum:
"size (queue_to_multiset bq) = listsum (map (size o tree_to_multiset) bq)"
by (induct bq) simp_all
text {*
A binomial heap of length $l$ contains at least $2^l - 1$ elements.
*}
theorem queue_length_estimate_lower:
"invar bq ==> (size (queue_to_multiset bq)) ≥ 2^(length bq) - 1"
proof (induct bq rule: rev_induct)
case Nil thus ?case by simp
next
case (snoc x xs)
from snoc.hyps[OF invar_butlast[OF snoc.prems]]
have IH: "2 ^ length xs ≤ Suc (size (queue_to_multiset xs))" by simp
have size_q:
"size (queue_to_multiset (xs @ [x])) =
size (queue_to_multiset xs) + size (tree_to_multiset x)"
by (simp add: size_queue_listsum)
also
from snoc.prems have inv_x: "tree_invar x" by (simp add: invar_def)
hence "size (tree_to_multiset x) = 2 ^ rank x" by (simp add: size_mset_tree)
finally have
eq: "size (queue_to_multiset (xs @ [x])) =
size (queue_to_multiset xs) + (2::nat)^(rank x)" .
from invar_length[OF snoc.prems] have "length xs ≤ rank x" by simp
hence snd: "(2::nat) ^ length xs ≤ (2::nat) ^ rank x" by simp
have
"(2::nat) ^ length (xs @ [x]) = (2::nat) ^ (length xs) + (2::nat) ^ (length xs)"
by simp
with IH have
"2 ^ length (xs @ [x]) ≤ Suc (size (queue_to_multiset xs)) + 2 ^ length xs"
by simp
with snd have "2 ^ length (xs @ [x]) ≤
Suc (size (queue_to_multiset xs)) + 2 ^ rank x"
by arith
with eq show ?case by simp
qed
subsection {* Operations *}
subsubsection "Empty"
lemma empty_correct[simp]:
"invar Nil"
"queue_to_multiset Nil = {#}"
by (simp_all add: invar_def)
text {* The empty multiset is represented by exactly the empty queue *}
lemma empty_iff: "t=Nil <-> queue_to_multiset t = {#}"
apply (cases t)
apply auto
apply (case_tac a)
apply auto
done
subsubsection "Insert"
text {* Inserts a binomial tree into a binomial queue, such that the queue
does not contain two trees of same rank. *}
fun ins :: "('e, 'a::linorder) BinomialTree => ('e, 'a) BinomialQueue_inv =>
('e, 'a) BinomialQueue_inv" where
"ins t [] = [t]" |
"ins t' (t # bq) = (if (rank t') < (rank t)
then t' # t # bq
else (if (rank t) < (rank t')
then t # (ins t' bq)
else ins (link t' t) bq))"
text {* Inserts an element with priority into the queue. *}
definition insert :: "'e => 'a::linorder => ('e, 'a) BinomialQueue_inv =>
('e, 'a) BinomialQueue_inv" where
"insert e a bq = ins (Node e a 0 []) bq"
lemma ins_mset:
"[|tree_invar t; queue_invar q|] ==> queue_to_multiset (ins t q)
= tree_to_multiset t + queue_to_multiset q"
proof(induct q arbitrary: t, simp)
case goal1 thus ?case
apply(cases "rank t < rank a")
apply(simp add: union_ac)
apply(cases "rank t = rank a") defer
apply(simp add: union_ac)
proof -
case goal1
from goal1(3) have inv_a: "tree_invar a" by (simp)
from goal1(3) have inv_q: "queue_invar q" by (simp)
note inv_link = link_tree_invar[OF goal1(2) inv_a goal1(5)]
note iv = goal1(1)[OF inv_link inv_q]
with mset_link[of t a] goal1(5) show ?case by (simp add: union_ac)
qed
qed
lemma insert_mset:
assumes "queue_invar q"
shows "queue_to_multiset (insert e a q) =
queue_to_multiset q + {# (e,a) #}" using assms
proof -
have inv: "tree_invar (Node e a 0 [])" by simp
from ins_mset[OF inv assms] show ?thesis by (simp add: union_ac insert_def)
qed
lemma ins_queue_invar: "[|tree_invar t; queue_invar q|] ==> queue_invar (ins t q)"
proof (induct q arbitrary: t)
case (Cons a q)
note iv = Cons.hyps
show ?case
proof (cases "rank t = rank a")
case True[simp]
from Cons.prems have
inv_a: "tree_invar a" and inv_q: "queue_invar q"
by (simp_all)
note inv_link = link_tree_invar[OF `tree_invar t` inv_a True]
from iv[OF inv_link inv_q] show ?thesis by simp
next
case False
with Cons show ?thesis by auto
qed
qed simp
lemma insert_queue_invar:
assumes "queue_invar q"
shows "queue_invar (insert e a q)"
proof -
have inv: "tree_invar (Node e a 0 [])" by simp
from ins_queue_invar[OF inv assms] show ?thesis by (simp add: insert_def)
qed
lemma rank_ins: "(rank_invar (t # bq) ==>
(rank (hd (ins t' (t # bq))) ≥ rank t) ∨
(rank (hd (ins t' (t # bq))) ≥ rank t'))"
apply(auto)
apply(induct bq arbitrary: t t')
apply(simp add: rank_link)
proof -
case goal1
thus ?case
apply(cases "rank (link t' t) = rank a")
apply(auto simp add: rank_link)
proof -
case goal1
from goal1 and `!! t' t. [|rank_invar (t # bq); rank t' = rank t|]
==> rank t ≤ rank (hd (ins (link t' t) bq))`[of a "(link t' t)"]
show ?case
apply(cases "rank (hd (ins (link (link t' t) a) bq)) = rank a")
proof -
case goal1
thus ?case by simp
next
case goal2
hence "rank a ≤ rank (hd (ins (link (link t' t) a) bq))"
by (simp add: rank_link)
with goal2 show ?case by simp
qed
qed
qed
lemma rank_ins2: "rank_invar bq ==>
rank t ≤ rank (hd (ins t bq)) ∨
(rank (hd (ins t bq)) = rank (hd bq) ∧ bq ≠ [])"
apply(induct bq arbitrary: t)
apply(auto)
proof -
case goal1
hence r: "rank (link t a) = rank a + 1" by (simp add: rank_link)
from goal1 r and goal1(1)[of "(link t a)"] show ?case
apply(cases bq)
apply(auto)
done
qed
lemma rank_invar_ins: "rank_invar bq ==> rank_invar (ins t bq)"
apply(induct bq arbitrary: t)
apply(simp)
apply(auto)
proof -
case goal1
hence inv: "rank_invar (ins t bq)" by (cases bq, simp_all)
from goal1 have hd: "bq ≠ [] ==> rank a < rank (hd bq)"
by (cases bq, auto)
from goal1 have "rank t ≤ rank (hd (ins t bq)) ∨
(rank (hd (ins t bq)) = rank (hd bq) ∧ bq ≠ [])"
by (simp add: rank_ins2 rank_invar_cons)
with goal1 have "rank a < rank (hd (ins t bq)) ∨
(rank (hd (ins t bq)) = rank (hd bq) ∧ bq ≠ [])" by auto
with goal1 and inv and hd show ?case
apply(auto simp add: rank_invar_hd_cons)
done
next
case goal2
hence inv: "rank_invar bq" by (cases bq, simp_all)
with goal2 and goal2(1)[of "(link t a)"] show ?case by simp
qed
lemma rank_invar_insert: "rank_invar bq ==> rank_invar (insert e a bq)"
by (simp add: rank_invar_ins insert_def)
lemma insert_correct:
assumes I: "invar q"
shows
"invar (insert e a q)"
"queue_to_multiset (insert e a q) = queue_to_multiset q + {# (e,a) #}"
using insert_queue_invar[of q] rank_invar_insert[of q] insert_mset[of q] I
unfolding invar_def by auto
subsubsection "Meld"
text {* Melds two queues. *}
fun meld :: "('e, 'a::linorder) BinomialQueue_inv => ('e, 'a) BinomialQueue_inv
=> ('e, 'a) BinomialQueue_inv"
where
"meld [] bq = bq" |
"meld bq [] = bq" |
"meld (t1#bq1) (t2#bq2) =
(if (rank t1) < (rank t2)
then t1 # (meld bq1 (t2 # bq2))
else (
if (rank t2 < rank t1)
then t2 # (meld (t1 # bq1) bq2)
else ins (link t1 t2) (meld bq1 bq2)
)
)"
lemma meld_queue_invar:
"[|queue_invar q; queue_invar q'|] ==> queue_invar (meld q q')"
proof (induct q q' rule: meld.induct, simp, simp)
case goal1
from goal1 show ?case
proof (cases "rank t1 < rank t2")
case goal1
from goal1(4) have inv_bq1: "queue_invar bq1" by simp
from goal1(4) have inv_t1: "tree_invar t1" by simp
from goal1(1)[OF goal1(6) inv_bq1 goal1(5)] inv_t1 goal1(6)
show ?case by simp
next
case goal2 thus ?case
proof(cases "rank t2 < rank t1")
case goal1
from goal1(5) have inv_bq2: "queue_invar bq2" by simp
from goal1(5) have inv_t2: "tree_invar t2" by simp
from goal1(2)[OF goal1(6) goal1(7) goal1(4) inv_bq2] inv_t2 goal1(6,7)
show ?case by simp
next
case goal2
from goal2(6,7) have eq: "rank t1 = rank t2" by simp
from goal2(4) have inv_bq1: "queue_invar bq1" by simp
from goal2(4) have inv_t1: "tree_invar t1" by simp
from goal2(5) have inv_bq2: "queue_invar bq2" by simp
from goal2(5) have inv_t2: "tree_invar t2" by simp
note inv_link = link_tree_invar[OF inv_t1 inv_t2 eq]
note inv_meld = goal2(3)[OF goal2(6,7) inv_bq1 inv_bq2]
from ins_queue_invar[OF inv_link inv_meld] goal2(6,7)
show ?case by simp
qed
qed
qed
lemma rank_ins_min: "rank_invar bq ==>
rank (hd (ins t bq)) ≥ min (rank t) (rank (hd bq))"
apply(induct bq arbitrary: t)
apply(auto)
proof -
case goal1
hence inv: "rank_invar bq" by (cases bq, simp_all)
from goal1 have r: "rank (link t a) = rank a + 1" by (simp add: rank_link)
with goal1 and inv and goal1(1)[of "(link t a)"] show ?case
apply(cases bq)
apply(auto)
done
qed
lemma rank_invar_meld_strong:
"[|rank_invar bq1; rank_invar bq2|] ==> rank_invar (meld bq1 bq2) ∧
rank (hd (meld bq1 bq2)) ≥ min (rank (hd bq1)) (rank (hd bq2))"
apply(induct bq1 bq2 rule: meld.induct)
apply(simp, simp)
proof -
case goal1
from goal1 have inv1: "rank_invar bq1" by (cases bq1, simp_all)
from goal1 have inv2: "rank_invar bq2" by (cases bq2, simp_all)
from inv1 and inv2 and goal1 show ?case
apply(auto)
proof -
let ?t = "t2"
let ?bq = "bq2"
let ?meld = "rank t2 < rank (hd (meld (t1 # bq1) bq2))"
case goal1
hence "?bq ≠ [] ==> rank ?t < rank (hd ?bq)"
by (simp add: rank_invar_not_empty_hd)
with goal1 have ne: "?bq ≠ [] ==> ?meld" by simp
from goal1 have "?bq = [] ==> ?meld" by simp
with ne have "?meld" by (cases "?bq = []")
with goal1 show ?case by (simp add: rank_invar_hd_cons)
next -- analog
let ?t = "t1"
let ?bq = "bq1"
let ?meld = "rank t1 < rank (hd (meld bq1 (t2 # bq2)))"
case goal2
hence "?bq ≠ [] ==> rank ?t < rank (hd ?bq)"
by (simp add: rank_invar_not_empty_hd)
with goal2 have ne: "?bq ≠ [] ==> ?meld" by simp
from goal2 have "?bq = [] ==> ?meld" by simp
with ne have "?meld" by (cases "?bq = []")
with goal2 show ?case by (simp add: rank_invar_hd_cons)
next
case goal3
thus ?case by (simp add: rank_invar_ins)
next
case goal4
from goal4 have r: "rank (link t1 t2) = rank t2 + 1"
by (simp add: rank_link)
have m: "meld bq1 [] = bq1" by (cases bq1, auto)
from inv1 and inv2 and goal4
have mm: "min (rank (hd bq1)) (rank (hd bq2)) ≤ rank (hd (meld bq1 bq2))"
by simp
from `rank_invar (t1 # bq1)` have "bq1 ≠ [] ==> rank t1 < rank (hd bq1)"
by (simp add: rank_invar_not_empty_hd)
with goal4 have r1: "bq1 ≠ [] ==> rank t2 < rank (hd bq1)" by simp
from `rank_invar (t2 # bq2)`
have r2: "bq2 ≠ [] ==> rank t2 < rank (hd bq2)"
by (simp add: rank_invar_not_empty_hd)
from inv1 r r1 rank_ins_min[of bq1 "(link t1 t2)"]
have abc1: "bq1 ≠ [] ==> rank t2 ≤ rank (hd (ins (link t1 t2) bq1))"
by simp
from inv2 r r2 rank_ins_min[of bq2 "(link t1 t2)"]
have abc2: "bq2 ≠ [] ==> rank t2 ≤ rank (hd (ins (link t1 t2) bq2))"
by simp
from r1 r2 mm have
"[|bq1 ≠ []; bq2 ≠ []|] ==> rank t2 < rank (hd (meld bq1 bq2))" by simp
with `rank_invar (meld bq1 bq2)`
r rank_ins_min[of "meld bq1 bq2" "link t1 t2"]
have "[|bq1 ≠ []; bq2 ≠ []|] ==>
rank t2 < rank (hd (ins (link t1 t2) (meld bq1 bq2)))" by simp
thm rank_ins_min[of "meld bq1 bq2" "link t1 t2"]
with inv1 and inv2 and r m r1 show ?case
apply(cases "bq2 = []")
apply(cases "bq1 = []")
apply(simp)
apply(auto simp add: abc1)
apply(cases "bq1 = []")
apply(simp)
apply(auto simp add: abc2)
done
qed
qed
lemma rank_invar_meld:
"[|rank_invar bq1; rank_invar bq2|] ==> rank_invar (meld bq1 bq2)"
by (simp only: rank_invar_meld_strong)
lemma meld_mset: "[|queue_invar q; queue_invar q'|] ==>
queue_to_multiset (meld q q') =
queue_to_multiset q + queue_to_multiset q'"
proof(induct q q' rule: meld.induct)
case (3 t1 bq1 t2 bq2)
note iv="3.hyps"
note prems="3.prems"
show ?case
proof (cases rule: nat_less_cases[of "rank t1" "rank t2",
case_names less eq greater])
case less
from prems have inv_bq1: "queue_invar bq1" by simp
from iv(1)[OF less inv_bq1 prems(2)] less
show ?thesis by (simp add: union_ac)
next
case greater with prems iv show ?thesis
by (auto simp add: union_ac)
next
case eq[simp]
from prems have
inv_bq1: "queue_invar bq1" and
inv_t1: "tree_invar t1" and
inv_bq2: "queue_invar bq2" and
inv_t2: "tree_invar t2" by simp_all
note inv_link = link_tree_invar[OF inv_t1 inv_t2 eq]
note inv_meld = meld_queue_invar[OF inv_bq1 inv_bq2]
note mset_meld = iv(3)[OF _ _ inv_bq1 inv_bq2, simplified]
note mset_link = mset_link[of t1 t2]
from ins_mset[OF inv_link inv_meld] mset_meld mset_link
show ?thesis by (simp add: union_ac)
qed
qed simp_all
lemma meld_correct:
assumes "invar q" "invar q'"
shows
"invar (meld q q')"
"queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'"
using assms
unfolding invar_def
by (simp_all add: meld_queue_invar rank_invar_meld meld_mset)
subsubsection "Find Minimal Element"
text {* Finds the tree containing the minimal element. *}
fun getMinTree :: "('e, 'a::linorder) BinomialQueue_inv =>
('e, 'a) BinomialTree" where
"getMinTree [t] = t" |
"getMinTree (t#bq) = (if prio t ≤ prio (getMinTree bq)
then t else (getMinTree bq))"
lemma mintree_exists: "(bq ≠ []) = (getMinTree bq ∈ set bq)"
proof (induct bq, simp)
case goal1 thus ?case by (cases bq) simp_all
qed
lemma treehead_in_multiset:
"t ∈ set bq ==> (val t, prio t) ∈# queue_to_multiset bq"
by (induct bq, simp, cases t, auto)
lemma heap_ordered_single:
"heap_ordered t = (∀x ∈ set_of (tree_to_multiset t). prio t ≤ snd x)"
by (cases t) auto
lemma getMinTree_cons:
"prio (getMinTree (y # x # xs)) ≤ prio (getMinTree (x # xs))"
by (induct xs rule: getMinTree.induct) simp_all
lemma getMinTree_min_tree:
"t ∈ set bq ==> prio (getMinTree bq) ≤ prio t"
apply(induct bq arbitrary: t rule: getMinTree.induct)
apply simp
defer
apply simp
proof -
case goal1 thus ?case
apply (cases "ta = t")
apply auto[1]
apply (metis getMinTree_cons goal1(1) goal1(3) set_ConsD xt1(6))
done
qed
lemma getMinTree_min_prio:
"[|queue_invar bq; y ∈ set_of (queue_to_multiset bq)|]
==> prio (getMinTree bq) ≤ snd y"
proof -
case goal1
hence "bq ≠ []" by (cases bq) simp_all
with goal1 have "∃ t ∈ set bq. (y ∈ set_of ((tree_to_multiset t)))"
apply(induct bq)
apply simp
proof -
case goal1 thus ?case
apply(cases "y ∈ set_of (tree_to_multiset a)")
apply simp
apply(cases bq)
apply simp_all
done
qed
from this obtain t where O:
"t ∈ set bq"
"y ∈ set_of (tree_to_multiset t)" by blast
obtain e a r ts where [simp]: "t = (Node e a r ts)" by (cases t) blast
from O goal1(1) have inv: "tree_invar t" by (simp add: queue_invar_def)
from tree_invar_heap_ordered[OF inv] heap_ordered.simps[of e a r ts] O
have "prio t ≤ snd y" by auto
with getMinTree_min_tree[OF O(1)] show ?case by simp
qed
text {* Finds the minimal Element in the queue. *}
definition findMin :: "('e, 'a::linorder) BinomialQueue_inv => ('e × 'a)" where
"findMin bq = (let min = getMinTree bq in (val min, prio min))"
lemma findMin_correct:
assumes I: "invar q"
assumes NE: "q ≠ Nil"
shows
"findMin q ∈# queue_to_multiset q"
"∀y∈set_of (queue_to_multiset q). snd (findMin q) ≤ snd y"
proof -
from NE have "getMinTree q ∈ set q" by (simp only: mintree_exists)
thus "findMin q ∈# queue_to_multiset q"
by (simp add: treehead_in_multiset Let_def findMin_def)
show "∀y∈set_of (queue_to_multiset q). snd (findMin q) ≤ snd y"
using I[unfolded invar_def]
by (auto simp add: getMinTree_min_prio Let_def findMin_def)
qed
subsubsection "Delete Minimal Element"
text {* Removes the first tree, which has the priority $a$ within his root. *}
fun remove1Prio :: "'a => ('e, 'a::linorder) BinomialQueue_inv =>
('e, 'a) BinomialQueue_inv" where
"remove1Prio a [] = []" |
"remove1Prio a (t#bq) =
(if (prio t) = a then bq else t # (remove1Prio a bq))"
text {* Returns the queue without the minimal element. *}
definition deleteMin :: "('e, 'a::linorder) BinomialQueue_inv =>
('e, 'a) BinomialQueue_inv" where
"deleteMin bq ≡ (let min = getMinTree bq in
meld (rev (children min))
(remove1Prio (prio min) bq))"
lemma queue_invar_rev: "queue_invar q ==> queue_invar (rev q)"
by (simp add: queue_invar_def)
lemma queue_invar_remove1: "queue_invar q ==> queue_invar (remove1 t q)"
by (auto simp add: queue_invar_def)
lemma qtm_in_set_subset: "t ∈ set q ==>
tree_to_multiset t ≤ queue_to_multiset q"
proof(induct q, simp)
case goal1 thus ?case
proof(cases "t = a", simp)
case goal1
hence t_in_q: "t ∈ set q" by simp
have "queue_to_multiset q ≤ queue_to_multiset (a # q)"
by simp
from order_trans[OF goal1(1)[OF t_in_q] this] show ?case .
qed
qed
lemma remove1_mset: "t ∈ set q ==>
queue_to_multiset (remove1 t q) =
queue_to_multiset q - tree_to_multiset t"
proof (induct q)
case (Cons a q) thus ?case
proof (cases "t = a", simp add: union_ac)
case goal1
from goal1(2,3) have t_in_q: "t ∈ set q" by simp
note iv = goal1(1)[OF t_in_q]
note t_subset_q = qtm_in_set_subset[OF t_in_q]
note assoc =
multiset_diff_union_assoc[OF t_subset_q, of "tree_to_multiset a"]
from iv goal1(3) assoc show ?case by (simp add: union_ac)
qed
qed simp
lemma remove1Prio_remove1[simp]:
"remove1Prio (prio (getMinTree bq)) bq = remove1 (getMinTree bq) bq"
proof (induct bq)
case Nil thus ?case by simp
next
case (Cons t bq)
note iv = Cons
thus ?case
proof (cases "t = getMinTree (t # bq)")
case True
with iv show ?thesis by simp
next
case False
hence ne: "bq ≠ []" by auto
with False have down: "getMinTree (t # bq) = getMinTree bq"
by (induct bq rule: getMinTree.induct) auto
from ne False have "prio t ≠ prio (getMinTree bq)"
by (induct bq rule: getMinTree.induct) auto
with down iv False ne show ?thesis by simp
qed
qed
lemma deleteMin_queue_invar:
assumes INV: "queue_invar q"
assumes NE: "q ≠ Nil"
shows "queue_invar (deleteMin q)"
using assms
proof (cases q, simp)
case goal1
from goal1(3) have q_ne: "q ≠ []" by simp
with mintree_exists[of q] goal1(1)
have inv_min: "tree_invar (getMinTree q)" by (simp add: queue_invar_def)
note inv_children = invar_children'[OF inv_min]
note inv_rev = queue_invar_rev[OF inv_children]
note inv_rem = queue_invar_remove1[OF goal1(1), of "getMinTree q"]
from meld_queue_invar[OF inv_rev inv_rem] show ?case
by (simp add: deleteMin_def Let_def)
qed
lemma children_rank_less:
"tree_invar t ==> ∀t' ∈ set (children t). rank t' < rank t"
proof (cases t)
case goal1 thus ?case
proof (induct nat arbitrary: t e a list, simp)
case goal1
from goal1 obtain e1 a1 ts1 e2 a2 ts2 where
O: "tree_invar (Node e1 a1 nat ts1) ∧ tree_invar (Node e2 a2 nat ts2)
∧ t = link (Node e1 a1 nat ts1) (Node e2 a2 nat ts2)"
by (simp only: tree_invar.simps) blast
hence ch_id: "children t =
(if a1 ≤ a2 then (Node e2 a2 nat ts2)#ts1
else (Node e1 a1 nat ts1)#ts2)" by simp
from O goal1(1)[of "Node e1 a1 nat ts1" "e1" "a1" "ts1"]
have p1: "∀t'∈set ((Node e2 a2 nat ts2) # ts1). rank t' < Suc nat" by auto
from O goal1(1)[of "Node e2 a2 nat ts2" "e2" "a2" "ts2"]
have p2: "∀t'∈set ((Node e1 a1 nat ts1) # ts2). rank t' < Suc nat" by auto
from goal1(3) p1 p2 ch_id show ?case by simp
qed
qed
lemma strong_rev_children: "tree_invar t ==> invar (rev (children t))"
unfolding invar_def
proof (cases t)
case goal1 thus ?case
proof (induct "nat" arbitrary: t e a list, simp)
case goal1
from goal1 obtain e1 a1 ts1 e2 a2 ts2 where
O: "tree_invar (Node e1 a1 nat ts1) ∧ tree_invar (Node e2 a2 nat ts2)
∧ t = link (Node e1 a1 nat ts1) (Node e2 a2 nat ts2)"
by (simp only: tree_invar.simps) blast
hence ch_id: "children t =
(if a1 ≤ a2 then (Node e2 a2 nat ts2)#ts1
else (Node e1 a1 nat ts1)#ts2)" by simp
from O goal1(1)[of "Node e1 a1 nat ts1" "e1" "a1" "ts1"] have
rev_ts1: "invar (rev ts1)" by (simp add: invar_def)
from O children_rank_less[of "Node e1 a1 nat ts1"] have
"∀t∈set (rev ts1). rank t < rank (Node e2 a2 nat ts2)" by simp
with O rev_ts1 invar_app_single[of "rev ts1" "Node e2 a2 nat ts2"]
have p1: "invar (rev ((Node e2 a2 nat ts2) # ts1))" by simp
from O goal1(1)[of "Node e2 a2 nat ts2" "e2" "a2" "ts2"] have
rev_ts2: "invar (rev ts2)" by (simp add: invar_def)
from O children_rank_less[of "Node e2 a2 nat ts2"] have
"∀t∈set (rev ts2). rank t < rank (Node e1 a1 nat ts1)" by simp
with O rev_ts2 invar_app_single[of "rev ts2" "Node e1 a1 nat ts1"]
have p2: "invar (rev ((Node e1 a1 nat ts1) # ts2))" by simp
from p1 p2 ch_id show ?case by (simp add: invar_def)
qed
qed
lemma first_less: "rank_invar (t # bq) ==> ∀t' ∈ set bq. rank t < rank t'"
apply(induct bq arbitrary: t)
apply (simp)
apply (metis order_le_less rank_invar.simps(3) set_ConsD xt1(7))
done
lemma strong_remove1: "invar bq ==> invar (remove1 t bq)"
proof (induct bq arbitrary: t, simp)
case goal1
thus ?case
apply(cases "t=a")
proof -
case goal1
from goal1(2) have "invar bq" by (rule invar_cons_down)
with goal1(3) show ?case by simp
next
case goal2
from goal2(2) have "invar bq" by (rule invar_cons_down)
with goal2(1)[of "t"] have si1: "invar (remove1 t bq)" .
from goal2(3) have
"invar (remove1 t (a # bq)) = invar (a # (remove1 t bq))"
by simp
with si1 goal2(2) show ?case
proof (cases "remove1 t bq", simp add: invar_def)
fix aa list
assume ass: "invar (remove1 t bq)" "invar (a # bq)"
"invar (remove1 t (a # bq)) = invar (a # remove1 t bq)"
"remove1 t bq = aa # list"
from ass(2) have "tree_invar a" by (simp add: invar_def)
from ass(2) first_less[of "a" "bq"] have
"∀t ∈ set (remove1 t bq). rank a < rank t"
by (metis notin_set_remove1 invar_def)
with ass(4) have "rank a < rank aa" by simp
with ass invar_cons_up[of "aa" "list" "a"] show ?case
by (simp add: invar_def)
qed
qed
qed
theorem deleteMin_invar:
"[|invar bq; bq ≠ []|] ==> invar (deleteMin bq)"
proof -
case goal1
have eq: "invar (deleteMin bq) =
invar (meld (rev (children (getMinTree bq))) (remove1 (getMinTree bq) bq))"
by (simp add: deleteMin_def Let_def)
from goal1 mintree_exists[of "bq"] have ti: "tree_invar (getMinTree bq)"
by (simp add: invar_def Let_def queue_invar_def)
with strong_rev_children[of "getMinTree bq"] have
m1: "invar (rev (children (getMinTree bq)))" .
from strong_remove1[of "bq" "getMinTree bq"] goal1(1) have
m2: "invar (remove1 (getMinTree bq) bq)" .
from meld_correct(1)[of "rev (children (getMinTree bq))"
"remove1 (getMinTree bq) bq"] m1 m2
have
"invar (meld (rev (children (getMinTree bq))) (remove1 (getMinTree bq) bq))" .
with eq show ?case ..
qed
lemma children_mset: "queue_to_multiset (children t) =
tree_to_multiset t - {# (val t, prio t) #}"
proof (cases t)
case goal1
thus ?case by (induct list, simp add: diff_cancel, simp)
qed
lemma deleteMin_mset: "[|queue_invar q; q ≠ Nil|] ==>
queue_to_multiset (deleteMin q) =
queue_to_multiset q - {# (findMin q) #}"
proof -
case goal1
with mintree_exists[of "q"] have min_in_q: "getMinTree q ∈ set q" by auto
with goal1(1) have inv_min: "tree_invar (getMinTree q)"
by (simp add: queue_invar_def)
from goal1(2) have q_ne: "q ≠ []" .
note inv_children = invar_children'[OF inv_min]
note inv_rev = queue_invar_rev[OF inv_children]
note inv_rem = queue_invar_remove1[OF goal1(1), of "getMinTree q"]
note m_meld = meld_mset[OF inv_rev inv_rem]
note m_rem = remove1_mset[OF min_in_q]
note m_rev = qtmset_rev[of "children (getMinTree q)"]
note m_children = children_mset[of "getMinTree q"]
note min_subset_q = qtm_in_set_subset[OF min_in_q]
let ?Q = "queue_to_multiset q"
let ?MT = "tree_to_multiset (getMinTree q)"
from q_ne have head_subset_min:
"{# (val (getMinTree q), prio (getMinTree q)) #} ≤ ?MT"
by(cases "getMinTree q") simp
let ?Q = "queue_to_multiset q"
let ?MT = "tree_to_multiset (getMinTree q)"
from m_meld m_rem m_rev m_children
multiset_diff_union_assoc[OF head_subset_min, of "?Q - ?MT"]
mset_le_multiset_union_diff_commute[OF min_subset_q, of "?MT"]
show ?case by (simp add: deleteMin_def union_ac Let_def findMin_def)
qed
lemma deleteMin_correct:
assumes INV: "invar q"
assumes NE: "q ≠ Nil"
shows
"invar (deleteMin q)"
"queue_to_multiset (deleteMin q) = queue_to_multiset q - {# (findMin q) #}"
using deleteMin_invar deleteMin_mset INV NE
unfolding invar_def
by auto
end
interpretation BinomialHeapStruc: BinomialHeapStruc_loc .
subsection "Hiding the Invariant"
subsubsection "Datatype"
typedef ('e, 'a) BinomialHeap =
"{q :: ('e,'a::linorder) BinomialQueue_inv. BinomialHeapStruc.invar q }"
apply (rule_tac x="Nil" in exI)
apply auto
done
lemma Rep_BinomialHeap_invar[simp]:
"BinomialHeapStruc.invar (Rep_BinomialHeap x)"
using Rep_BinomialHeap
by (auto)
lemma [simp]:
"BinomialHeapStruc.invar q ==> Rep_BinomialHeap (Abs_BinomialHeap q) = q"
using Abs_BinomialHeap_inverse by auto
lemma [simp, code abstype]: "Abs_BinomialHeap (Rep_BinomialHeap q) = q"
by (rule Rep_BinomialHeap_inverse)
locale BinomialHeap_loc
begin
subsubsection "Operations"
definition [code]:
"to_mset t == BinomialHeapStruc.queue_to_multiset (Rep_BinomialHeap t)"
definition empty where "empty == Abs_BinomialHeap Nil"
lemma [code abstract, simp]: "Rep_BinomialHeap empty = []"
by (unfold empty_def) simp
definition [code]: "isEmpty q == Rep_BinomialHeap q = Nil"
lemma empty_rep: "q=empty <-> Rep_BinomialHeap q = Nil"
apply (auto simp add: empty_def)
apply (metis Rep_BinomialHeap_inverse)
done
lemma isEmpty_correct: "isEmpty q <-> q=empty"
by (simp add: empty_rep isEmpty_def)
definition
insert
:: "'e => ('a::linorder) => ('e,'a) BinomialHeap => ('e,'a) BinomialHeap"
where "insert e a q ==
Abs_BinomialHeap (BinomialHeapStruc.insert e a (Rep_BinomialHeap q))"
lemma [code abstract]:
"Rep_BinomialHeap (insert e a q)
= BinomialHeapStruc.insert e a (Rep_BinomialHeap q)"
by (simp add: insert_def BinomialHeapStruc.insert_correct)
definition [code]: "findMin q == BinomialHeapStruc.findMin (Rep_BinomialHeap q)"
definition "deleteMin q ==
if q=empty then empty
else Abs_BinomialHeap (BinomialHeapStruc.deleteMin (Rep_BinomialHeap q))"
text {*
In this lemma, we do not use equality, but case-distinction for checking
non-emptyness. That prevents the code generator from introducing
an equality-class parameter for the entry type @{text 'a}.
*}
lemma [code abstract]: "Rep_BinomialHeap (deleteMin q) =
(case (Rep_BinomialHeap q) of [] => [] |
_ => BinomialHeapStruc.deleteMin (Rep_BinomialHeap q))"
proof (cases "Rep_BinomialHeap q")
case Nil
show ?thesis
apply (simp add: Nil)
apply (auto simp add: deleteMin_def BinomialHeapStruc.deleteMin_correct
BinomialHeapStruc.empty_iff empty_rep Nil)
done
next
case (Cons a b)
hence NE: "Rep_BinomialHeap q ≠ []" by auto
show ?thesis
apply (simp add: Cons)
apply (fold Cons)
using NE
by (auto simp add: deleteMin_def BinomialHeapStruc.deleteMin_correct
BinomialHeapStruc.empty_iff empty_rep)
qed
definition "meld q1 q2 ==
Abs_BinomialHeap (BinomialHeapStruc.meld (Rep_BinomialHeap q1)
(Rep_BinomialHeap q2))"
lemma [code abstract]:
"Rep_BinomialHeap (meld q1 q2)
= BinomialHeapStruc.meld (Rep_BinomialHeap q1) (Rep_BinomialHeap q2)"
by (simp add: meld_def BinomialHeapStruc.meld_correct)
subsubsection "Correctness"
lemma empty_correct: "to_mset q = {#} <-> q=empty"
by (simp add: to_mset_def BinomialHeapStruc.empty_iff empty_rep)
lemma to_mset_of_empty[simp]: "to_mset empty = {#}"
by (simp add: empty_correct)
lemma insert_correct: "to_mset (insert e a q) = to_mset q + {#(e,a)#}"
apply (unfold insert_def to_mset_def)
apply (simp add: BinomialHeapStruc.insert_correct)
done
lemma findMin_correct:
assumes "q≠empty"
shows
"findMin q ∈# to_mset q"
"∀y∈set_of (to_mset q). snd (findMin q) ≤ snd y"
using assms
apply (unfold findMin_def to_mset_def)
apply (simp_all add: empty_rep BinomialHeapStruc.findMin_correct)
done
lemma deleteMin_correct:
assumes "q≠empty"
shows "to_mset (deleteMin q) = to_mset q - {# findMin q #}"
using assms
apply (unfold findMin_def deleteMin_def to_mset_def)
apply (simp_all add: empty_rep BinomialHeapStruc.deleteMin_correct)
done
lemma meld_correct:
shows "to_mset (meld q q') = to_mset q + to_mset q'"
apply (unfold to_mset_def meld_def)
apply (simp_all add: BinomialHeapStruc.meld_correct)
done
text {* Correctness lemmas to be used with simplifier *}
lemmas correct = empty_correct deleteMin_correct meld_correct
end
interpretation BinomialHeap: BinomialHeap_loc .
subsection "Documentation"
text {*
\underline{@{term_type "BinomialHeap.to_mset"}}\\
Abstraction to multiset.\\
\underline{@{term_type "BinomialHeap.empty"}}\\
The empty heap. ($O(1)$)\\
{\bf Spec} @{text "BinomialHeap.empty_correct"}:
@{thm [display] BinomialHeap.empty_correct[no_vars]}
\underline{@{term_type "BinomialHeap.isEmpty"}}\\
Checks whether heap is empty. Mainly used to work around
code-generation issues. ($O(1)$)\\
{\bf Spec} @{text "BinomialHeap.isEmpty_correct"}:
@{thm [display] BinomialHeap.isEmpty_correct[no_vars]}
\underline{@{term_type "BinomialHeap.insert"}}\\
Inserts element ($O(\log(n))$)\\
{\bf Spec} @{text "BinomialHeap.insert_correct"}:
@{thm [display] BinomialHeap.insert_correct[no_vars]}
\underline{@{term_type "BinomialHeap.findMin"}}\\
Returns a minimal element ($O(\log(n))$)\\
{\bf Spec} @{text "BinomialHeap.findMin_correct"}:
@{thm [display] BinomialHeap.findMin_correct[no_vars]}
\underline{@{term_type "BinomialHeap.deleteMin"}}\\
Deletes {\em the} element that is returned by {\em find\_min}\\
{\bf Spec} @{text "BinomialHeap.deleteMin_correct"}:
@{thm [display] BinomialHeap.deleteMin_correct[no_vars]}
\underline{@{term "BinomialHeap.meld"}}
@{term_type [display] "BinomialHeap.meld"}
Melds two heaps ($O(\log(n+m))$)\\
{\bf Spec} @{text "BinomialHeap.meld_correct"}:
@{thm [display] BinomialHeap.meld_correct[no_vars]}
*}
end