# Documentation

Std.Data.BinomialHeap.Basic

• nil: {α : Type u} →

An empty forest, which has depth 0.

• node: {α : Type u} →

A forest of rank r + 1 consists of a root a, a forest child of rank r elements greater than a, and another forest sibling of rank r.

A HeapNode is one of the internal nodes of the binomial heap. It is always a perfect binary tree, with the depth of the tree stored in the Heap. However the interpretation of the two pointers is different: we view the child as going to the first child of this node, and sibling goes to the next sibling of this tree. So it actually encodes a forest where each node has children node.child, node.child.sibling, node.child.sibling.sibling, etc.

Each edge in this forest denotes a le a b relation that has been checked, so the root is smaller than everything else under it.

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instance Std.BinomialHeap.Imp.instReprHeapNode :
{α : Type u_1} → [inst : Repr α] →

The "real size" of the node, counting up how many values of type α are stored. This is O(n) and is intended mainly for specification purposes. For a well formed HeapNode the size is always 2^n - 1 where n is the depth.

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A node containing a single element a.

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O(log n). The rank, or the number of trees in the forest. It is also the depth of the forest.

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inductive Std.BinomialHeap.Imp.Heap (α : Type u) :
• nil: {α : Type u} →

An empty heap.

• cons: {α : Type u} →

A cons node contains a tree of root val, children node and rank rank, and then next which is the rest of the forest.

A Heap is the top level structure in a binomial heap. It consists of a forest of HeapNodes with strictly increasing ranks.

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instance Std.BinomialHeap.Imp.instReprHeap :
{α : Type u_1} → [inst : Repr α] →

O(n). The "real size" of the heap, counting up how many values of type α are stored. This is intended mainly for specification purposes. Prefer Heap.size, which is the same for well formed heaps.

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O(log n). The number of elements in the heap.

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@[inline]

O(1). Is the heap empty?

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@[inline]
def Std.BinomialHeap.Imp.Heap.singleton {α : Type u_1} (a : α) :

O(1). The heap containing a single value a.

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O(1). Auxiliary for Heap.merge: Is the minimum rank in Heap strictly larger than n?

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instance Std.BinomialHeap.Imp.instDecidableRankGT :
{α : Type u_1} → {s : } → {n : Nat} →

O(log n). The number of trees in the forest.

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@[inline]
def Std.BinomialHeap.Imp.combine {α : Type u_1} (le : ααBool) (a₁ : α) (a₂ : α) (n₁ : ) (n₂ : ) :

O(1). Auxiliary for Heap.merge: combines two heap nodes of the same rank into one with the next larger rank.

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@[specialize #[]]
def Std.BinomialHeap.Imp.Heap.merge {α : Type u_1} (le : ααBool) :

Merge two forests of binomial trees. The forests are assumed to be ordered by rank and merge maintains this invariant.

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O(log n). Convert a HeapNode to a Heap by reversing the order of the nodes along the sibling spine.

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Computes s.toHeap ++ res tail-recursively, assuming n = s.rank.

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@[specialize #[]]
def Std.BinomialHeap.Imp.Heap.headD {α : Type u_1} (le : ααBool) (a : α) :

O(log n). Get the smallest element in the heap, including the passed in value a.

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@[inline]
def Std.BinomialHeap.Imp.Heap.head? {α : Type u_1} (le : ααBool) :

O(log n). Get the smallest element in the heap, if it has an element.

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structure Std.BinomialHeap.Imp.FindMin (α : Type u_1) :
Type u_1
• before :

The list of elements prior to the minimum element, encoded as a "difference list".

• val : α

The minimum element.

• node :

The children of the minimum element.

• next :

The forest after the minimum element.

The return type of FindMin, which encodes various quantities needed to reconstruct the tree in deleteMin.

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@[specialize #[]]
def Std.BinomialHeap.Imp.Heap.findMin {α : Type u_1} (le : ααBool) :

O(log n). Find the minimum element, and return a data structure FindMin with information needed to reconstruct the rest of the binomial heap.

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def Std.BinomialHeap.Imp.Heap.deleteMin {α : Type u_1} (le : ααBool) :

O(log n). Find and remove the the minimum element from the binomial heap.

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@[inline]
def Std.BinomialHeap.Imp.Heap.tail? {α : Type u_1} (le : ααBool) (h : ) :

O(log n). Get the tail of the binomial heap after removing the minimum element.

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@[inline]
def Std.BinomialHeap.Imp.Heap.tail {α : Type u_1} (le : ααBool) (h : ) :

O(log n). Remove the minimum element of the heap.

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theorem Std.BinomialHeap.Imp.Heap.realSize_merge {α : Type u_1} (le : ααBool) (s₁ : ) (s₂ : ) :
theorem Std.BinomialHeap.Imp.Heap.realSize_deleteMin {α : Type u_1} {le : ααBool} {a : α} {s' : } {s : } (eq : = some (a, s')) :
theorem Std.BinomialHeap.Imp.Heap.realSize_tail? {α : Type u_1} {le : ααBool} {s' : } {s : } :
theorem Std.BinomialHeap.Imp.Heap.realSize_tail {α : Type u_1} (le : ααBool) (s : ) :
@[specialize #[]]
def Std.BinomialHeap.Imp.Heap.foldM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [] (le : ααBool) (s : ) (init : β) (f : βαm β) :
m β

O(n log n). Monadic fold over the elements of a heap in increasing order, by repeatedly pulling the minimum element out of the heap.

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• One or more equations did not get rendered due to their size.
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@[inline]
def Std.BinomialHeap.Imp.Heap.fold {α : Type u_1} {β : Type u_2} (le : ααBool) (s : ) (init : β) (f : βαβ) :
β

O(n log n). Fold over the elements of a heap in increasing order, by repeatedly pulling the minimum element out of the heap.

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@[inline]
def Std.BinomialHeap.Imp.Heap.toArray {α : Type u_1} (le : ααBool) (s : ) :

O(n log n). Convert the heap to an array in increasing order.

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@[inline]
def Std.BinomialHeap.Imp.Heap.toList {α : Type u_1} (le : ααBool) (s : ) :
List α

O(n log n). Convert the heap to a list in increasing order.

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@[specialize #[]]
def Std.BinomialHeap.Imp.HeapNode.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [] (nil : β) (join : αββm β) :

O(n). Fold a monadic function over the tree structure to accumulate a value.

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@[specialize #[]]
def Std.BinomialHeap.Imp.Heap.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [] (nil : β) (join : αββm β) :

O(n). Fold a monadic function over the tree structure to accumulate a value.

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@[inline]
def Std.BinomialHeap.Imp.Heap.foldTree {β : Type u_1} {α : Type u_2} (nil : β) (join : αβββ) (s : ) :
β

O(n). Fold a function over the tree structure to accumulate a value.

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O(n). Convert the heap to a list in arbitrary order.

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O(n). Convert the heap to an array in arbitrary order.

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def Std.BinomialHeap.Imp.HeapNode.WF {α : Type u_1} (le : ααBool) (a : α) :

The well formedness predicate for a heap node. It asserts that:

• If a is added at the top to make the forest into a tree, the resulting tree is a le-min-heap (if le is well-behaved)
• When interpreting child and sibling as left and right children of a binary tree, it is a perfect binary tree with depth r
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def Std.BinomialHeap.Imp.Heap.WF {α : Type u_1} (le : ααBool) (n : Nat) :

The well formedness predicate for a binomial heap. It asserts that:

• It consists of a list of well formed trees with the specified ranks
• The ranks are in strictly increasing order, and all are at least n
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theorem Std.BinomialHeap.Imp.Heap.WF.nil :
∀ {α : Type u_1} {le : ααBool} {n : Nat}, Std.BinomialHeap.Imp.Heap.WF le n Std.BinomialHeap.Imp.Heap.nil
theorem Std.BinomialHeap.Imp.Heap.WF.singleton :
∀ {α : Type u_1} {a : α} {le : ααBool},
theorem Std.BinomialHeap.Imp.Heap.WF.of_rankGT :
∀ {α : Type u_1} {le : ααBool} {n' : Nat} {s : } {n : Nat}, Std.BinomialHeap.Imp.Heap.WF le (n + 1) s
theorem Std.BinomialHeap.Imp.Heap.WF.of_le {n : Nat} {n' : Nat} :
∀ {α : Type u_1} {le : ααBool} {s : }, n n'
theorem Std.BinomialHeap.Imp.Heap.rankGT.of_le :
∀ {α : Type u_1} {s : } {n n' : Nat}, n' n
theorem Std.BinomialHeap.Imp.Heap.WF.rankGT :
∀ {α : Type u_1} {lt : ααBool} {n : Nat} {s : }, Std.BinomialHeap.Imp.Heap.WF lt (n + 1) s
theorem Std.BinomialHeap.Imp.Heap.WF.merge' :
∀ {α : Type u_1} {le : ααBool} {s₁ s₂ : } {n : Nat}, ()
theorem Std.BinomialHeap.Imp.Heap.WF.merge :
∀ {α : Type u_1} {le : ααBool} {s₁ s₂ : } {n : Nat},
theorem Std.BinomialHeap.Imp.HeapNode.WF.rank_eq {α : Type u_1} {le : ααBool} {a : α} {n : Nat} :
theorem Std.BinomialHeap.Imp.HeapNode.WF.toHeap {α : Type u_1} {le : ααBool} {a : α} {n : Nat} (h : ) :
theorem Std.BinomialHeap.Imp.HeapNode.WF.toHeap.go {α : Type u_1} {le : ααBool} {a : α} {res : } {n : Nat} :
structure Std.BinomialHeap.Imp.FindMin.WF {α : Type u_1} (le : ααBool) (res : ) :
• rank : Nat

The rank of the minimum element

• before : ∀ {s_1 : }, Std.BinomialHeap.Imp.Heap.WF le s.rank s_1

before is a difference list which can be appended to a binomial heap with ranks at least rank to produce another well formed heap.

• node : Std.BinomialHeap.Imp.HeapNode.WF le res.val res.node s.rank

node is a well formed forest of rank rank with val at the root.

• next : Std.BinomialHeap.Imp.Heap.WF le (s.rank + 1) res.next

next is a binomial heap with ranks above rank + 1.

The well formedness predicate for a FindMin value. This is not actually a predicate, as it contains an additional data value rank corresponding to the rank of the returned node, which is omitted from findMin.

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def Std.BinomialHeap.Imp.Heap.WF.findMin {α : Type u_1} {le : ααBool} {n : Nat} {res : } {s : } (h : ) (hr : ) (hk : ∀ {s : }, Std.BinomialHeap.Imp.Heap.WF le 0 (k s)) :

The conditions under which findMin is well-formed.

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• One or more equations did not get rendered due to their size.
• = hr
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theorem Std.BinomialHeap.Imp.Heap.WF.deleteMin {α : Type u_1} {le : ααBool} {n : Nat} {a : α} {s' : } {s : } (h : ) (eq : = some (a, s')) :
theorem Std.BinomialHeap.Imp.Heap.WF.tail? {α : Type u_1} {s : } {le : ααBool} {n : Nat} {tl : } (hwf : ) :
theorem Std.BinomialHeap.Imp.Heap.WF.tail {α : Type u_1} {s : } {le : ααBool} {n : Nat} (hwf : ) :
def Std.BinomialHeap (α : Type u) (le : ααBool) :

A binomial heap is a data structure which supports the following primary operations:

• insert : α → BinomialHeap α → BinomialHeap α: add an element to the heap
• deleteMin : BinomialHeap α → Option (α × BinomialHeap α): remove the minimum element from the heap
• merge : BinomialHeap α → BinomialHeap α → BinomialHeap α: combine two heaps

The first two operations are known as a "priority queue", so this could be called a "mergeable priority queue". The standard choice for a priority queue is a binary heap, which supports insert and deleteMin in O(log n), but merge is O(n). With a BinomialHeap, all three operations are O(log n).

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@[inline]
def Std.mkBinomialHeap (α : Type u) (le : ααBool) :

O(1). Make a new empty binomial heap.

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@[inline]
def Std.BinomialHeap.empty {α : Type u} {le : ααBool} :

O(1). Make a new empty binomial heap.

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instance Std.BinomialHeap.instInhabitedBinomialHeap {α : Type u} {le : ααBool} :
@[inline]
def Std.BinomialHeap.isEmpty {α : Type u} {le : ααBool} (b : ) :

O(1). Is the heap empty?

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@[inline]
def Std.BinomialHeap.size {α : Type u} {le : ααBool} (b : ) :

O(log n). The number of elements in the heap.

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@[inline]
def Std.BinomialHeap.singleton {α : Type u} {le : ααBool} (a : α) :

O(1). Make a new heap containing a.

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@[inline]
def Std.BinomialHeap.merge {α : Type u} {le : ααBool} :

O(log n). Merge the contents of two heaps.

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@[inline]
def Std.BinomialHeap.insert {α : Type u} {le : ααBool} (a : α) (h : ) :

O(log n). Add element a to the given heap h.

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def Std.BinomialHeap.ofList {α : Type u} (le : ααBool) (as : List α) :

O(n log n). Construct a heap from a list by inserting all the elements.

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def Std.BinomialHeap.ofArray {α : Type u} (le : ααBool) (as : ) :

O(n log n). Construct a heap from a list by inserting all the elements.

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@[inline]
def Std.BinomialHeap.deleteMin {α : Type u} {le : ααBool} (b : ) :
Option (α × )

O(log n). Remove and return the minimum element from the heap.

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instance Std.BinomialHeap.instStreamBinomialHeap {α : Type u} {le : ααBool} :
Stream () α
def Std.BinomialHeap.forIn {α : Type u} {le : ααBool} {m : Type u_1 → Type u_2} {β : Type u_1} [] (b : ) (x : β) (f : αβm ()) :
m β

O(n log n). Implementation of for x in (b : BinomialHeap α le) ... notation, which iterates over the elements in the heap in increasing order.

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instance Std.BinomialHeap.instForInBinomialHeap {α : Type u} {le : ααBool} {m : Type u_1 → Type u_2} :
ForIn m () α
@[inline]
def Std.BinomialHeap.head? {α : Type u} {le : ααBool} (b : ) :

O(log n). Returns the smallest element in the heap, or none if the heap is empty.

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@[inline]
def Std.BinomialHeap.head! {α : Type u} {le : ααBool} [] (b : ) :
α

O(log n). Returns the smallest element in the heap, or panics if the heap is empty.

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@[inline]
def Std.BinomialHeap.headI {α : Type u} {le : ααBool} [] (b : ) :
α

O(log n). Returns the smallest element in the heap, or default if the heap is empty.

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@[inline]
def Std.BinomialHeap.tail? {α : Type u} {le : ααBool} (b : ) :

O(log n). Removes the smallest element from the heap, or none if the heap is empty.

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@[inline]
def Std.BinomialHeap.tail {α : Type u} {le : ααBool} (b : ) :

O(log n). Removes the smallest element from the heap, if possible.

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@[inline]
def Std.BinomialHeap.foldM {α : Type u} {le : ααBool} {m : Type u_1 → Type u_2} {β : Type u_1} [] (b : ) (init : β) (f : βαm β) :
m β

O(n log n). Monadic fold over the elements of a heap in increasing order, by repeatedly pulling the minimum element out of the heap.

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@[inline]
def Std.BinomialHeap.fold {α : Type u} {le : ααBool} {β : Type u_1} (b : ) (init : β) (f : βαβ) :
β

O(n log n). Fold over the elements of a heap in increasing order, by repeatedly pulling the minimum element out of the heap.

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@[inline]
def Std.BinomialHeap.toList {α : Type u} {le : ααBool} (b : ) :
List α

O(n log n). Convert the heap to a list in increasing order.

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@[inline]
def Std.BinomialHeap.toArray {α : Type u} {le : ααBool} (b : ) :

O(n log n). Convert the heap to an array in increasing order.

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@[inline]
def Std.BinomialHeap.toListUnordered {α : Type u} {le : ααBool} (b : ) :
List α

O(n). Convert the heap to a list in arbitrary order.

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@[inline]
def Std.BinomialHeap.toArrayUnordered {α : Type u} {le : ααBool} (b : ) :

O(n). Convert the heap to an array in arbitrary order.

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