inductive Lean.RBColor : Type

• red: Lean.RBColor
• black: Lean.RBColor

inductive Lean.RBNode (α : Type u) (β : α → Type v) : Type (max u v)

• leaf: {α : Type u} → {β : α → Type v} → Lean.RBNode α β
• node: {α : Type u} → {β : α → Type v} → Lean.RBColor → Lean.RBNode α β → (key : α) → β key → Lean.RBNode α β → Lean.RBNode α β

def Lean.RBNode.depth {α : Type u} {β : α → Type v} (f : Nat → Nat → Nat) : Lean.RBNode α β → Nat

Equations

• Lean.RBNode.depth f Lean.RBNode.leaf = 0
• Lean.RBNode.depth f (Lean.RBNode.node color l key val r) = (f (Lean.RBNode.depth f l) (Lean.RBNode.depth f r)).succ

def Lean.RBNode.min {α : Type u} {β : α → Type v} : Lean.RBNode α β → Option ((k : α) × β k)

def Lean.RBNode.max {α : Type u} {β : α → Type v} : Lean.RBNode α β → Option ((k : α) × β k)

Equations

• Lean.RBNode.leaf.max = none
• (Lean.RBNode.node color lchild k v Lean.RBNode.leaf).max = some ⟨k, v⟩
• (Lean.RBNode.node color lchild key val r).max = r.max

@[specialize #[]]

def Lean.RBNode.fold {α : Type u} {β : α → Type v} {σ : Type w} (f : σ → (k : α) → β k → σ) (init : σ) : Lean.RBNode α β → σ

Equations

• Lean.RBNode.fold f x Lean.RBNode.leaf = x
• Lean.RBNode.fold f x (Lean.RBNode.node color l k v r) = Lean.RBNode.fold f (f (Lean.RBNode.fold f x l) k v) r

@[specialize #[]]

def Lean.RBNode.forM {α : Type u} {β : α → Type v} {m : Type → Type u_1} [Monad m] (f : (k : α) → β k → m Unit) : Lean.RBNode α β → m Unit

Equations

• Lean.RBNode.forM f Lean.RBNode.leaf = pure ()
• Lean.RBNode.forM f (Lean.RBNode.node color l key val r) = do Lean.RBNode.forM f l f key val Lean.RBNode.forM f r

@[specialize #[]]

def Lean.RBNode.foldM {α : Type u} {β : α → Type v} {σ : Type w} {m : Type w → Type u_1} [Monad m] (f : σ → (k : α) → β k → m σ) (init : σ) : Lean.RBNode α β → m σ

Equations

• Lean.RBNode.foldM f x Lean.RBNode.leaf = pure x
• Lean.RBNode.foldM f x (Lean.RBNode.node color l k v r) = do let b ← Lean.RBNode.foldM f x l let b ← f b k v Lean.RBNode.foldM f b r

@[inline]

def Lean.RBNode.forIn {α : Type u} {β : α → Type v} {σ : Type w} {m : Type w → Type u_1} [Monad m] (as : Lean.RBNode α β) (init : σ) (f : (k : α) → β k → σ → m (ForInStep σ)) : m σ

Equations

• as.forIn init f = do let __do_lift ← Lean.RBNode.forIn.visit f as init match __do_lift with | ForInStep.done b => pure b | ForInStep.yield b => pure b

@[specialize #[]]

def Lean.RBNode.forIn.visit {α : Type u} {β : α → Type v} {σ : Type w} {m : Type w → Type u_1} [Monad m] (f : (k : α) → β k → σ → m (ForInStep σ)) : Lean.RBNode α β → σ → m (ForInStep σ)

@[specialize #[]]

def Lean.RBNode.revFold {α : Type u} {β : α → Type v} {σ : Type w} (f : σ → (k : α) → β k → σ) (init : σ) : Lean.RBNode α β → σ

Equations

• Lean.RBNode.revFold f x Lean.RBNode.leaf = x
• Lean.RBNode.revFold f x (Lean.RBNode.node color l k v r) = Lean.RBNode.revFold f (f (Lean.RBNode.revFold f x r) k v) l

@[specialize #[]]

def Lean.RBNode.all {α : Type u} {β : α → Type v} (p : (k : α) → β k → Bool) : Lean.RBNode α β → Bool

Equations

• Lean.RBNode.all p Lean.RBNode.leaf = true
• Lean.RBNode.all p (Lean.RBNode.node color l key val r) = (p key val && Lean.RBNode.all p l && Lean.RBNode.all p r)

@[specialize #[]]

def Lean.RBNode.any {α : Type u} {β : α → Type v} (p : (k : α) → β k → Bool) : Lean.RBNode α β → Bool

def Lean.RBNode.singleton {α : Type u} {β : α → Type v} (k : α) (v : β k) : Lean.RBNode α β

def Lean.RBNode.isSingleton {α : Type u} {β : α → Type v} : Lean.RBNode α β → Bool

Equations

• (Lean.RBNode.node color Lean.RBNode.leaf key val Lean.RBNode.leaf).isSingleton = true
• x.isSingleton = false

@[inline]

def Lean.RBNode.balance1 {α : Type u} {β : α → Type v} : Lean.RBNode α β → (a : α) → β a → Lean.RBNode α β → Lean.RBNode α β

@[inline]

def Lean.RBNode.balance2 {α : Type u} {β : α → Type v} : Lean.RBNode α β → (a : α) → β a → Lean.RBNode α β → Lean.RBNode α β

Equations

• One or more equations did not get rendered due to their size.
• x✝².balance2 x✝¹ x✝ x = Lean.RBNode.node Lean.RBColor.black x✝² x✝¹ x✝ x

def Lean.RBNode.isRed {α : Type u} {β : α → Type v} : Lean.RBNode α β → Bool

def Lean.RBNode.isBlack {α : Type u} {β : α → Type v} : Lean.RBNode α β → Bool

@[specialize #[]]

def Lean.RBNode.ins {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) : Lean.RBNode α β → (k : α) → β k → Lean.RBNode α β

Equations

• One or more equations did not get rendered due to their size.
• Lean.RBNode.ins cmp Lean.RBNode.leaf x✝ x = Lean.RBNode.node Lean.RBColor.red Lean.RBNode.leaf x✝ x Lean.RBNode.leaf

def Lean.RBNode.setBlack {α : Type u} {β : α → Type v} : Lean.RBNode α β → Lean.RBNode α β

@[specialize #[]]

def Lean.RBNode.insert {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) (t : Lean.RBNode α β) (k : α) (v : β k) : Lean.RBNode α β

Equations

• Lean.RBNode.insert cmp t k v = if t.isRed = true then (Lean.RBNode.ins cmp t k v).setBlack else Lean.RBNode.ins cmp t k v

def Lean.RBNode.setRed {α : Type u} {β : α → Type v} : Lean.RBNode α β → Lean.RBNode α β

Equations

• (Lean.RBNode.node color l k v r).setRed = Lean.RBNode.node Lean.RBColor.red l k v r
• x.setRed = x

def Lean.RBNode.balLeft {α : Type u} {β : α → Type v} : Lean.RBNode α β → (k : α) → β k → Lean.RBNode α β → Lean.RBNode α β

Equations

• One or more equations did not get rendered due to their size.
• (Lean.RBNode.node Lean.RBColor.red a kx vx b).balLeft x✝¹ x✝ x = Lean.RBNode.node Lean.RBColor.red (Lean.RBNode.node Lean.RBColor.black a kx vx b) x✝¹ x✝ x
• x✝¹.balLeft x✝ x (Lean.RBNode.node Lean.RBColor.black a ky vy b) = x✝¹.balance2 x✝ x (Lean.RBNode.node Lean.RBColor.red a ky vy b)
• x✝².balLeft x✝¹ x✝ x = Lean.RBNode.node Lean.RBColor.red x✝² x✝¹ x✝ x

def Lean.RBNode.balRight {α : Type u} {β : α → Type v} (l : Lean.RBNode α β) (k : α) (v : β k) (r : Lean.RBNode α β) : Lean.RBNode α β

def Lean.RBNode.size {α : Type u} {β : α → Type v} : Lean.RBNode α β → Nat

The number of nodes in the tree.

Equations

• Lean.RBNode.leaf.size = 0
• (Lean.RBNode.node color l key val r).size = l.size + r.size + 1

@[irreducible]

def Lean.RBNode.appendTrees {α : Type u} {β : α → Type v} : Lean.RBNode α β → Lean.RBNode α β → Lean.RBNode α β

@[specialize #[]]

def Lean.RBNode.del {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) (x : α) : Lean.RBNode α β → Lean.RBNode α β

@[specialize #[]]

def Lean.RBNode.erase {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) (x : α) (t : Lean.RBNode α β) : Lean.RBNode α β

Equations

• Lean.RBNode.erase cmp x t = (Lean.RBNode.del cmp x t).setBlack

@[specialize #[]]

def Lean.RBNode.findCore {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) : Lean.RBNode α β → (k : α) → Option ((k : α) × β k)

Equations

• One or more equations did not get rendered due to their size.
• Lean.RBNode.findCore cmp Lean.RBNode.leaf x = none

@[specialize #[]]

def Lean.RBNode.find {α : Type u} (cmp : α → α → Ordering) {β : Type v} : (Lean.RBNode α fun (x : α) => β) → α → Option β

Equations

• Lean.RBNode.find cmp Lean.RBNode.leaf x = none
• Lean.RBNode.find cmp (Lean.RBNode.node color a ky vy b) x = match cmp x ky with | Ordering.lt => Lean.RBNode.find cmp a x | Ordering.gt => Lean.RBNode.find cmp b x | Ordering.eq => some vy

@[specialize #[]]

def Lean.RBNode.lowerBound {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) : Lean.RBNode α β → α → Option (Sigma β) → Option (Sigma β)

Equations

• One or more equations did not get rendered due to their size.
• Lean.RBNode.lowerBound cmp Lean.RBNode.leaf x✝ x = x

inductive Lean.RBNode.WellFormed {α : Type u} {β : α → Type v} (cmp : α → α → Ordering) : Lean.RBNode α β → Prop

• leafWff: ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering}, Lean.RBNode.WellFormed cmp Lean.RBNode.leaf
• insertWff: ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {n n' : Lean.RBNode α β} {k : α} {v : β k}, Lean.RBNode.WellFormed cmp n → n' = Lean.RBNode.insert cmp n k v → Lean.RBNode.WellFormed cmp n'
• eraseWff: ∀ {α : Type u} {β : α → Type v} {cmp : α → α → Ordering} {n n' : Lean.RBNode α β} {k : α}, Lean.RBNode.WellFormed cmp n → n' = Lean.RBNode.erase cmp k n → Lean.RBNode.WellFormed cmp n'

@[specialize #[]]

def Lean.RBNode.mapM {α : Type v} {β γ : α → Type v} {M : Type v → Type v} [Applicative M] (f : (a : α) → β a → M (γ a)) : Lean.RBNode α β → M (Lean.RBNode α γ)

Equations

• One or more equations did not get rendered due to their size.
• Lean.RBNode.mapM f Lean.RBNode.leaf = pure Lean.RBNode.leaf

@[specialize #[]]

def Lean.RBNode.map {α : Type u} {β γ : α → Type v} (f : (a : α) → β a → γ a) : Lean.RBNode α β → Lean.RBNode α γ

def Lean.RBNode.toArray {α : Type u} {β : α → Type v} (n : Lean.RBNode α β) : Array (Sigma β)

instance Lean.RBNode.instEmptyCollection {α : Type u} {β : α → Type v} : EmptyCollection (Lean.RBNode α β)

def Lean.RBMap (α : Type u) (β : Type v) (cmp : α → α → Ordering) : Type (max u v)

Equations

• Lean.RBMap α β cmp = { t : Lean.RBNode α fun (x : α) => β // Lean.RBNode.WellFormed cmp t }

@[inline]

def Lean.mkRBMap (α : Type u) (β : Type v) (cmp : α → α → Ordering) : Lean.RBMap α β cmp

@[inline]

def Lean.RBMap.empty {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp

Equations

• Lean.RBMap.empty = Lean.mkRBMap α β cmp

instance Lean.instEmptyCollectionRBMap (α : Type u) (β : Type v) (cmp : α → α → Ordering) : EmptyCollection (Lean.RBMap α β cmp)

Equations

• Lean.instEmptyCollectionRBMap α β cmp = { emptyCollection := Lean.RBMap.empty }

instance Lean.instInhabitedRBMap (α : Type u) (β : Type v) (cmp : α → α → Ordering) : Inhabited (Lean.RBMap α β cmp)

def Lean.RBMap.depth {α : Type u} {β : Type v} {cmp : α → α → Ordering} (f : Nat → Nat → Nat) (t : Lean.RBMap α β cmp) : Nat

def Lean.RBMap.isSingleton {α : Type u} {β : Type v} {cmp : α → α → Ordering} (t : Lean.RBMap α β cmp) : Bool

@[inline]

def Lean.RBMap.fold {α : Type u} {β : Type v} {σ : Type w} {cmp : α → α → Ordering} (f : σ → α → β → σ) (init : σ) : Lean.RBMap α β cmp → σ

Equations

• Lean.RBMap.fold f x ⟨t, property⟩ = Lean.RBNode.fold f x t

@[inline]

def Lean.RBMap.revFold {α : Type u} {β : Type v} {σ : Type w} {cmp : α → α → Ordering} (f : σ → α → β → σ) (init : σ) : Lean.RBMap α β cmp → σ

@[inline]

def Lean.RBMap.foldM {α : Type u} {β : Type v} {σ : Type w} {cmp : α → α → Ordering} {m : Type w → Type u_1} [Monad m] (f : σ → α → β → m σ) (init : σ) : Lean.RBMap α β cmp → m σ

@[inline]

def Lean.RBMap.forM {α : Type u} {β : Type v} {cmp : α → α → Ordering} {m : Type u_1 → Type u_2} [Monad m] (f : α → β → m PUnit) (t : Lean.RBMap α β cmp) : m PUnit

Equations

• Lean.RBMap.forM f t = Lean.RBMap.foldM (fun (x : PUnit) (k : α) (v : β) => f k v) PUnit.unit t

@[inline]

def Lean.RBMap.forIn {α : Type u} {β : Type v} {σ : Type w} {cmp : α → α → Ordering} {m : Type w → Type u_1} [Monad m] (t : Lean.RBMap α β cmp) (init : σ) (f : α × β → σ → m (ForInStep σ)) : m σ

Equations

• t.forIn init f = t.val.forIn init fun (a : α) (b : β) (acc : σ) => f (a, b) acc

instance Lean.RBMap.instForInProd {α : Type u} {β : Type v} {cmp : α → α → Ordering} {m : Type u_1 → Type u_2} : ForIn m (Lean.RBMap α β cmp) (α × β)

@[inline]

def Lean.RBMap.isEmpty {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → Bool

@[specialize #[]]

def Lean.RBMap.toList {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → List (α × β)

Returns a List of the key/value pairs in order.

Equations

• Lean.RBMap.toList ⟨t, property⟩ = Lean.RBNode.revFold (fun (ps : List (α × β)) (k : α) (v : β) => (k, v) :: ps) [] t

@[specialize #[]]

def Lean.RBMap.toArray {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → Array (α × β)

Returns an Array of the key/value pairs in order.

@[inline]

def Lean.RBMap.min {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → Option (α × β)

Returns the kv pair (a,b) such that a ≤ k for all keys in the RBMap.

@[inline]

def Lean.RBMap.max {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → Option (α × β)

Returns the kv pair (a,b) such that a ≥ k for all keys in the RBMap.

instance Lean.RBMap.instRepr {α : Type u} {β : Type v} {cmp : α → α → Ordering} [Repr α] [Repr β] : Repr (Lean.RBMap α β cmp)

@[inline]

def Lean.RBMap.insert {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → α → β → Lean.RBMap α β cmp

Equations

• Lean.RBMap.insert ⟨t, w⟩ x✝ x = ⟨Lean.RBNode.insert cmp t x✝ x, ⋯⟩

@[inline]

def Lean.RBMap.erase {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → α → Lean.RBMap α β cmp

Equations

• Lean.RBMap.erase ⟨t, w⟩ x = ⟨Lean.RBNode.erase cmp x t, ⋯⟩

@[specialize #[]]

def Lean.RBMap.ofList {α : Type u} {β : Type v} {cmp : α → α → Ordering} : List (α × β) → Lean.RBMap α β cmp

Equations

• Lean.RBMap.ofList [] = Lean.mkRBMap α β cmp
• Lean.RBMap.ofList ((k, v) :: xs) = (Lean.RBMap.ofList xs).insert k v

@[inline]

def Lean.RBMap.findCore? {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → α → Option ((_ : α) × β)

@[inline]

def Lean.RBMap.find? {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → α → Option β

@[inline]

def Lean.RBMap.findD {α : Type u} {β : Type v} {cmp : α → α → Ordering} (t : Lean.RBMap α β cmp) (k : α) (v₀ : β) : β

@[inline]

def Lean.RBMap.lowerBound {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → α → Option ((_ : α) × β)

(lowerBound k) retrieves the kv pair of the largest key smaller than or equal to k, if it exists.

@[inline]

def Lean.RBMap.contains {α : Type u} {β : Type v} {cmp : α → α → Ordering} (t : Lean.RBMap α β cmp) (a : α) : Bool

Returns true if the given key a is in the RBMap.

Equations

• t.contains a = (t.find? a).isSome

@[inline]

def Lean.RBMap.fromList {α : Type u} {β : Type v} (l : List (α × β)) (cmp : α → α → Ordering) : Lean.RBMap α β cmp

Equations

• Lean.RBMap.fromList l cmp = List.foldl (fun (r : Lean.RBMap α β cmp) (p : α × β) => r.insert p.fst p.snd) (Lean.mkRBMap α β cmp) l

@[inline]

def Lean.RBMap.fromArray {α : Type u} {β : Type v} (l : Array (α × β)) (cmp : α → α → Ordering) : Lean.RBMap α β cmp

Equations

• Lean.RBMap.fromArray l cmp = Array.foldl (fun (r : Lean.RBMap α β cmp) (p : α × β) => r.insert p.fst p.snd) (Lean.mkRBMap α β cmp) l

@[inline]

def Lean.RBMap.all {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → (α → β → Bool) → Bool

Returns true if the given predicate is true for all items in the RBMap.

Equations

• Lean.RBMap.all ⟨t, property⟩ x = Lean.RBNode.all x t

@[inline]

def Lean.RBMap.any {α : Type u} {β : Type v} {cmp : α → α → Ordering} : Lean.RBMap α β cmp → (α → β → Bool) → Bool

Returns true if the given predicate is true for any item in the RBMap.

Equations

• Lean.RBMap.any ⟨t, property⟩ x = Lean.RBNode.any x t

def Lean.RBMap.size {α : Type u} {β : Type v} {cmp : α → α → Ordering} (m : Lean.RBMap α β cmp) : Nat

The number of items in the RBMap.

def Lean.RBMap.maxDepth {α : Type u} {β : Type v} {cmp : α → α → Ordering} (t : Lean.RBMap α β cmp) : Nat

Equations

• t.maxDepth = Lean.RBNode.depth Nat.max t.val

@[inline]

def Lean.RBMap.min! {α : Type u} {β : Type v} {cmp : α → α → Ordering} [Inhabited α] [Inhabited β] (t : Lean.RBMap α β cmp) : α × β

@[inline]

def Lean.RBMap.max! {α : Type u} {β : Type v} {cmp : α → α → Ordering} [Inhabited α] [Inhabited β] (t : Lean.RBMap α β cmp) : α × β

Equations

• t.max! = match t.max with | some p => p | none => panicWithPosWithDecl "Lean.Data.RBMap" "Lean.RBMap.max!" 383 14 "map is empty"

@[inline]

def Lean.RBMap.find! {α : Type u} {β : Type v} {cmp : α → α → Ordering} [Inhabited β] (t : Lean.RBMap α β cmp) (k : α) : β

Attempts to find the value with key k : α in t and panics if there is no such key.

def Lean.RBMap.mergeBy {α : Type u} {β : Type v} {cmp : α → α → Ordering} (mergeFn : α → β → β → β) (t₁ t₂ : Lean.RBMap α β cmp) : Lean.RBMap α β cmp

Merges the maps t₁ and t₂, if a key a : α exists in both, then use mergeFn a b₁ b₂ to produce the new merged value.

def Lean.RBMap.intersectBy {α : Type u} {β : Type v} {cmp : α → α → Ordering} {γ : Type v₁} {δ : Type v₂} (mergeFn : α → β → γ → δ) (t₁ : Lean.RBMap α β cmp) (t₂ : Lean.RBMap α γ cmp) : Lean.RBMap α δ cmp

Intersects the maps t₁ and t₂ using mergeFn a b₁ b₂ to produce the new value.

def Lean.RBMap.filter {α : Type u} {β : Type v} {cmp : α → α → Ordering} (f : α → β → Bool) (m : Lean.RBMap α β cmp) : Lean.RBMap α β cmp

filter f m returns the RBMap consisting of all "key/val"-pairs in m where f key val returns true.

Equations

• Lean.RBMap.filter f m = Lean.RBMap.fold (fun (r : Lean.RBMap α β cmp) (k : α) (v : β) => if f k v = true then r.insert k v else r) ∅ m

def Lean.RBMap.filterMap {α : Type u} {β : Type v} {cmp : α → α → Ordering} {γ : Type u_1} (f : α → β → Option γ) (m : Lean.RBMap α β cmp) : Lean.RBMap α γ cmp

filterMap f m filters an RBMap and simultaneously modifies the filtered values.

It takes a function f : α → β → Option γ and applies f k v to the value with key k. The resulting entries with non-none value are collected to form the output RBMap.

Equations

• Lean.RBMap.filterMap f m = Lean.RBMap.fold (fun (r : Lean.RBMap α γ cmp) (k : α) (v : β) => match f k v with | none => r | some b => r.insert k b) ∅ m

def Lean.rbmapOf {α : Type u} {β : Type v} (l : List (α × β)) (cmp : α → α → Ordering) : Lean.RBMap α β cmp

Equations

• Lean.rbmapOf l cmp = Lean.RBMap.fromList l cmp