inductive Batteries.BinomialHeap.Imp.HeapNode (α : Type u) : Type u

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.

• nil {α : Type u} : HeapNode α

  An empty forest, which has depth 0.

• node {α : Type u} (a : α) (child sibling : HeapNode α) : HeapNode α

  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.

instance Batteries.BinomialHeap.Imp.instReprHeapNode {α✝ : Type u_1} [Repr α✝] : Repr (HeapNode α✝)

def Batteries.BinomialHeap.Imp.HeapNode.realSize {α : Type u_1} : HeapNode α → Nat

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.

def Batteries.BinomialHeap.Imp.HeapNode.singleton {α : Type u_1} (a : α) : HeapNode α

A node containing a single element a.

def Batteries.BinomialHeap.Imp.HeapNode.rank {α : Type u_1} : HeapNode α → Nat

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

inductive Batteries.BinomialHeap.Imp.Heap (α : Type u) : Type u

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

• nil {α : Type u} : Heap α

  An empty heap.

• cons {α : Type u} (rank : Nat) (val : α) (node : HeapNode α) (next : Heap α) : Heap α

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

instance Batteries.BinomialHeap.Imp.instReprHeap {α✝ : Type u_1} [Repr α✝] : Repr (Heap α✝)

def Batteries.BinomialHeap.Imp.Heap.realSize {α : Type u_1} : Heap α → Nat

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.

def Batteries.BinomialHeap.Imp.Heap.size {α : Type u_1} : Heap α → Nat

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

@[inline]

def Batteries.BinomialHeap.Imp.Heap.isEmpty {α : Type u_1} : Heap α → Bool

O(1). Is the heap empty?

@[inline]

def Batteries.BinomialHeap.Imp.Heap.singleton {α : Type u_1} (a : α) : Heap α

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

def Batteries.BinomialHeap.Imp.Heap.rankGT {α : Type u_1} : Heap α → Nat → Prop

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

instance Batteries.BinomialHeap.Imp.instDecidableRankGT {α✝ : Type u_1} {s : Heap α✝} {n : Nat} : Decidable (s.rankGT n)

def Batteries.BinomialHeap.Imp.Heap.length {α : Type u_1} : Heap α → Nat

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

@[inline]

def Batteries.BinomialHeap.Imp.combine {α : Type u_1} (le : α → α → Bool) (a₁ a₂ : α) (n₁ n₂ : HeapNode α) : α × HeapNode α

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

@[irreducible, specialize #[]]

def Batteries.BinomialHeap.Imp.Heap.merge {α : Type u_1} (le : α → α → Bool) : Heap α → Heap α → Heap α

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

def Batteries.BinomialHeap.Imp.HeapNode.toHeap {α : Type u_1} (s : HeapNode α) : Heap α

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

def Batteries.BinomialHeap.Imp.HeapNode.toHeap.go {α : Type u_1} : HeapNode α → Nat → Heap α → Heap α

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

@[specialize #[]]

def Batteries.BinomialHeap.Imp.Heap.headD {α : Type u_1} (le : α → α → Bool) (a : α) : Heap α → α

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

@[inline]

def Batteries.BinomialHeap.Imp.Heap.head? {α : Type u_1} (le : α → α → Bool) : Heap α → Option α

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

structure Batteries.BinomialHeap.Imp.FindMin (α : Type u_1) : Type u_1

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

• before : Heap α → Heap α

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

• val : α

  The minimum element.

• node : HeapNode α

  The children of the minimum element.

• next : Heap α

  The forest after the minimum element.

@[specialize #[]]

def Batteries.BinomialHeap.Imp.Heap.findMin {α : Type u_1} (le : α → α → Bool) (k : Heap α → Heap α) : Heap α → FindMin α → FindMin α

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

def Batteries.BinomialHeap.Imp.Heap.deleteMin {α : Type u_1} (le : α → α → Bool) : Heap α → Option (α × Heap α)

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

@[inline]

def Batteries.BinomialHeap.Imp.Heap.tail? {α : Type u_1} (le : α → α → Bool) (h : Heap α) : Option (Heap α)

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

@[inline]

def Batteries.BinomialHeap.Imp.Heap.tail {α : Type u_1} (le : α → α → Bool) (h : Heap α) : Heap α

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

@[irreducible]

theorem Batteries.BinomialHeap.Imp.Heap.realSize_merge {α : Type u_1} (le : α → α → Bool) (s₁ s₂ : Heap α) : (merge le s₁ s₂).realSize = s₁.realSize + s₂.realSize

theorem Batteries.BinomialHeap.Imp.HeapNode.realSize_toHeap {α : Type u_1} (s : HeapNode α) : s.toHeap.realSize = s.realSize

theorem Batteries.BinomialHeap.Imp.HeapNode.realSize_toHeap.go {α : Type u_1} {n : Nat} {res : Heap α} (s : HeapNode α) : (toHeap.go s n res).realSize = s.realSize + res.realSize

theorem Batteries.BinomialHeap.Imp.Heap.realSize_deleteMin {α : Type u_1} {le : α → α → Bool} {a : α} {s' s : Heap α} (eq : deleteMin le s = some (a, s')) : s.realSize = s'.realSize + 1

theorem Batteries.BinomialHeap.Imp.Heap.realSize_tail? {α : Type u_1} {le : α → α → Bool} {s' s : Heap α} : tail? le s = some s' → s.realSize = s'.realSize + 1

theorem Batteries.BinomialHeap.Imp.Heap.realSize_tail {α : Type u_1} (le : α → α → Bool) (s : Heap α) : (tail le s).realSize = s.realSize - 1

@[irreducible, specialize #[]]

def Batteries.BinomialHeap.Imp.Heap.foldM {m : Type u_1 → Type u_2} {α : Type u_3} {β : Type u_1} [Monad m] (le : α → α → Bool) (s : Heap α) (init : β) (f : β → α → m 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.

@[inline]

def Batteries.BinomialHeap.Imp.Heap.fold {α : Type u_1} {β : Type u_2} (le : α → α → Bool) (s : Heap α) (init : β) (f : β → α → β) : 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.

@[inline]

def Batteries.BinomialHeap.Imp.Heap.toArray {α : Type u_1} (le : α → α → Bool) (s : Heap α) : Array α

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

@[inline]

def Batteries.BinomialHeap.Imp.Heap.toList {α : Type u_1} (le : α → α → Bool) (s : Heap α) : List α

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

@[specialize #[]]

def Batteries.BinomialHeap.Imp.HeapNode.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] (nil : β) (join : α → β → β → m β) : HeapNode α → m β

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

@[specialize #[]]

def Batteries.BinomialHeap.Imp.Heap.foldTreeM {m : Type u_1 → Type u_2} {β : Type u_1} {α : Type u_3} [Monad m] (nil : β) (join : α → β → β → m β) : Heap α → m β

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

@[inline]

def Batteries.BinomialHeap.Imp.Heap.foldTree {β : Type u_1} {α : Type u_2} (nil : β) (join : α → β → β → β) (s : Heap α) : β

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

def Batteries.BinomialHeap.Imp.Heap.toListUnordered {α : Type u_1} (s : Heap α) : List α

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

def Batteries.BinomialHeap.Imp.Heap.toArrayUnordered {α : Type u_1} (s : Heap α) : Array α

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

def Batteries.BinomialHeap.Imp.HeapNode.WF {α : Type u_1} (le : α → α → Bool) (a : α) : HeapNode α → Nat → Prop

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

def Batteries.BinomialHeap.Imp.Heap.WF {α : Type u_1} (le : α → α → Bool) (n : Nat) : Heap α → Prop

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

theorem Batteries.BinomialHeap.Imp.Heap.WF.nil {α✝ : Type u_1} {le : α✝ → α✝ → Bool} {n : Nat} : WF le n Heap.nil

theorem Batteries.BinomialHeap.Imp.Heap.WF.singleton {α✝ : Type u_1} {a : α✝} {le : α✝ → α✝ → Bool} : WF le 0 (Heap.singleton a)

theorem Batteries.BinomialHeap.Imp.Heap.WF.of_rankGT {α✝ : Type u_1} {le : α✝ → α✝ → Bool} {n' : Nat} {s : Heap α✝} {n : Nat} (hlt : s.rankGT n) (h : WF le n' s) : WF le (n + 1) s

theorem Batteries.BinomialHeap.Imp.Heap.WF.of_le {n n' : Nat} {α✝ : Type u_1} {le : α✝ → α✝ → Bool} {s : Heap α✝} (hle : n ≤ n') (h : WF le n' s) : WF le n s

theorem Batteries.BinomialHeap.Imp.Heap.rankGT.of_le {α✝ : Type u_1} {s : Heap α✝} {n n' : Nat} (h : s.rankGT n) (h' : n' ≤ n) : s.rankGT n'

theorem Batteries.BinomialHeap.Imp.Heap.WF.rankGT {α✝ : Type u_1} {lt : α✝ → α✝ → Bool} {n : Nat} {s : Heap α✝} (h : WF lt (n + 1) s) : s.rankGT n

@[irreducible]

theorem Batteries.BinomialHeap.Imp.Heap.WF.merge' {α✝ : Type u_1} {le : α✝ → α✝ → Bool} {s₁ s₂ : Heap α✝} {n : Nat} (h₁ : WF le n s₁) (h₂ : WF le n s₂) : WF le n (Heap.merge le s₁ s₂) ∧ ((s₁.rankGT n ↔ s₂.rankGT n) → (Heap.merge le s₁ s₂).rankGT n)

theorem Batteries.BinomialHeap.Imp.Heap.WF.merge {α✝ : Type u_1} {le : α✝ → α✝ → Bool} {s₁ s₂ : Heap α✝} {n : Nat} (h₁ : WF le n s₁) (h₂ : WF le n s₂) : WF le n (Heap.merge le s₁ s₂)

theorem Batteries.BinomialHeap.Imp.HeapNode.WF.rank_eq {α : Type u_1} {le : α → α → Bool} {a : α} {n : Nat} {s : HeapNode α} : WF le a s n → s.rank = n

theorem Batteries.BinomialHeap.Imp.HeapNode.WF.toHeap {α : Type u_1} {le : α → α → Bool} {a : α} {n : Nat} {s : HeapNode α} (h : WF le a s n) : Heap.WF le 0 s.toHeap

theorem Batteries.BinomialHeap.Imp.HeapNode.WF.toHeap.go {α : Type u_1} {le : α → α → Bool} {a : α} {res : Heap α} {n : Nat} {s : HeapNode α} : WF le a s n → Heap.WF le s.rank res → Heap.WF le 0 (HeapNode.toHeap.go s s.rank res)

structure Batteries.BinomialHeap.Imp.FindMin.WF {α : Type u_1} (le : α → α → Bool) (res : FindMin α) : Type

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.

• rank : Nat

  The rank of the minimum element

• before {s : Heap α} : Heap.WF le self.rank s → Heap.WF le 0 (res.before s)

  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 : HeapNode.WF le res.val res.node self.rank

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

• next : Heap.WF le (self.rank + 1) res.next

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

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

The conditions under which findMin is well-formed.

theorem Batteries.BinomialHeap.Imp.Heap.WF.deleteMin {α : Type u_1} {le : α → α → Bool} {n : Nat} {a : α} {s' s : Heap α} (h : WF le n s) (eq : Heap.deleteMin le s = some (a, s')) : WF le 0 s'

theorem Batteries.BinomialHeap.Imp.Heap.WF.tail? {α : Type u_1} {s : Heap α} {le : α → α → Bool} {n : Nat} {tl : Heap α} (hwf : WF le n s) : Heap.tail? le s = some tl → WF le 0 tl

theorem Batteries.BinomialHeap.Imp.Heap.WF.tail {α : Type u_1} {s : Heap α} {le : α → α → Bool} {n : Nat} (hwf : WF le n s) : WF le 0 (Heap.tail le s)

def Batteries.BinomialHeap (α : Type u) (le : α → α → Bool) : Type u

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).

@[inline]

def Batteries.mkBinomialHeap (α : Type u) (le : α → α → Bool) : BinomialHeap α le

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

@[inline]

def Batteries.BinomialHeap.empty {α : Type u} {le : α → α → Bool} : BinomialHeap α le

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

instance Batteries.BinomialHeap.instEmptyCollection {α : Type u} {le : α → α → Bool} : EmptyCollection (BinomialHeap α le)

instance Batteries.BinomialHeap.instInhabited {α : Type u} {le : α → α → Bool} : Inhabited (BinomialHeap α le)

@[inline]

def Batteries.BinomialHeap.isEmpty {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Bool

O(1). Is the heap empty?

@[inline]

def Batteries.BinomialHeap.size {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Nat

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

@[inline]

def Batteries.BinomialHeap.singleton {α : Type u} {le : α → α → Bool} (a : α) : BinomialHeap α le

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

@[inline]

def Batteries.BinomialHeap.merge {α : Type u} {le : α → α → Bool} : BinomialHeap α le → BinomialHeap α le → BinomialHeap α le

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

@[inline]

def Batteries.BinomialHeap.insert {α : Type u} {le : α → α → Bool} (a : α) (h : BinomialHeap α le) : BinomialHeap α le

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

def Batteries.BinomialHeap.ofList {α : Type u} (le : α → α → Bool) (as : List α) : BinomialHeap α le

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

def Batteries.BinomialHeap.ofArray {α : Type u} (le : α → α → Bool) (as : Array α) : BinomialHeap α le

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

@[inline]

def Batteries.BinomialHeap.deleteMin {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Option (α × BinomialHeap α le)

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

instance Batteries.BinomialHeap.instStream {α : Type u} {le : α → α → Bool} : Stream (BinomialHeap α le) α

def Batteries.BinomialHeap.forIn {α : Type u} {le : α → α → Bool} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] (b : BinomialHeap α le) (x : β) (f : α → β → m (ForInStep β)) : 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.

instance Batteries.BinomialHeap.instForIn {α : Type u} {le : α → α → Bool} {m : Type u_1 → Type u_2} : ForIn m (BinomialHeap α le) α

@[inline]

def Batteries.BinomialHeap.head? {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Option α

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

@[inline]

def Batteries.BinomialHeap.head! {α : Type u} {le : α → α → Bool} [Inhabited α] (b : BinomialHeap α le) : α

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

@[inline]

def Batteries.BinomialHeap.headI {α : Type u} {le : α → α → Bool} [Inhabited α] (b : BinomialHeap α le) : α

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

@[inline]

def Batteries.BinomialHeap.tail? {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Option (BinomialHeap α le)

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

@[inline]

def Batteries.BinomialHeap.tail {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : BinomialHeap α le

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

@[inline]

def Batteries.BinomialHeap.foldM {α : Type u} {le : α → α → Bool} {m : Type u_1 → Type u_2} {β : Type u_1} [Monad m] (b : BinomialHeap α le) (init : β) (f : β → α → m 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.

@[inline]

def Batteries.BinomialHeap.fold {α : Type u} {le : α → α → Bool} {β : Type u_1} (b : BinomialHeap α le) (init : β) (f : β → α → β) : 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.

@[inline]

def Batteries.BinomialHeap.toList {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : List α

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

@[inline]

def Batteries.BinomialHeap.toArray {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Array α

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

@[inline]

def Batteries.BinomialHeap.toListUnordered {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : List α

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

@[inline]

def Batteries.BinomialHeap.toArrayUnordered {α : Type u} {le : α → α → Bool} (b : BinomialHeap α le) : Array α

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