General operations on functions

theorem Function.comp_def {α : Sort u_1} {β : Sort u_2} {δ : Sort u_3} (f : β → δ) (g : α → β) : f ∘ g = fun x => f (g x)

@[inline, reducible]

def Function.dcomp {α : Sort u₁} {β : α → Sort u₂} {φ : {x : α} → β x → Sort u₃} (f : {x : α} → (y : β x) → φ x y) (g : (x : α) → β x) (x : α) : φ x (g x)

Composition of dependent functions: (f ∘' g) x = f (g x), where type of g x depends on x and type of f (g x) depends on x and g x.

def Function.«term_∘'_» : Lean.TrailingParserDescr

@[reducible]

def Function.compRight {α : Sort u₁} {β : Sort u₂} (f : β → β → β) (g : α → β) : β → α → β

@[reducible]

def Function.compLeft {α : Sort u₁} {β : Sort u₂} (f : β → β → β) (g : α → β) : α → β → β

@[reducible]

def Function.onFun {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} (f : β → β → φ) (g : α → β) : α → α → φ

Given functions f : β → β → φ and g : α → β, produce a function α → α → φ that evaluates g on each argument, then applies f to the results. Can be used, e.g., to transfer a relation from β to α.

@[reducible]

def Function.combine {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {δ : Sort u₄} {ζ : Sort u₅} (f : α → β → φ) (op : φ → δ → ζ) (g : α → β → δ) : α → β → ζ

Given functions f : α → β → φ, g : α → β → δ and a binary operator op : φ → δ → ζ, produce a function α → β → ζ that applies f and g on each argument and then applies op to the results.

@[reducible]

def Function.swap {α : Sort u₁} {β : Sort u₂} {φ : α → β → Sort u₃} (f : (x : α) → (y : β) → φ x y) (y : β) (x : α) : φ x y

@[reducible]

def Function.app {α : Sort u₁} {β : α → Sort u₂} (f : (x : α) → β x) (x : α) : β x

def Function.term_On_ : Lean.TrailingParserDescr

Given functions f : β → β → φ and g : α → β, produce a function α → α → φ that evaluates g on each argument, then applies f to the results. Can be used, e.g., to transfer a relation from β to α.

theorem Function.left_id {α : Sort u₁} {β : Sort u₂} (f : α → β) : id ∘ f = f

theorem Function.right_id {α : Sort u₁} {β : Sort u₂} (f : α → β) : f ∘ id = f

theorem Function.comp.assoc {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {δ : Sort u₄} (f : φ → δ) (g : β → φ) (h : α → β) : (f ∘ g) ∘ h = f ∘ g ∘ h

@[simp]

theorem Function.comp.left_id {α : Sort u₁} {β : Sort u₂} (f : α → β) : id ∘ f = f

@[simp]

theorem Function.comp.right_id {α : Sort u₁} {β : Sort u₂} (f : α → β) : f ∘ id = f

theorem Function.comp_const_right {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} (f : β → φ) (b : β) : f ∘ Function.const α b = Function.const α (f b)

def Function.Injective {α : Sort u₁} {β : Sort u₂} (f : α → β) : Prop

A function f : α → β is called injective if f x = f y implies x = y.

theorem Function.Injective.comp {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {g : β → φ} {f : α → β} (hg : Function.Injective g) (hf : Function.Injective f) : Function.Injective (g ∘ f)

def Function.Surjective {α : Sort u₁} {β : Sort u₂} (f : α → β) : Prop

A function f : α → β is called surjective if every b : β is equal to f a for some a : α.

theorem Function.Surjective.comp {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {g : β → φ} {f : α → β} (hg : Function.Surjective g) (hf : Function.Surjective f) : Function.Surjective (g ∘ f)

def Function.Bijective {α : Sort u₁} {β : Sort u₂} (f : α → β) : Prop

A function is called bijective if it is both injective and surjective.

theorem Function.Bijective.comp {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {g : β → φ} {f : α → β} : Function.Bijective g → Function.Bijective f → Function.Bijective (g ∘ f)

def Function.LeftInverse {α : Sort u₁} {β : Sort u₂} (g : β → α) (f : α → β) : Prop

LeftInverse g f means that g is a left inverse to f. That is, g ∘ f = id.

def Function.HasLeftInverse {α : Sort u₁} {β : Sort u₂} (f : α → β) : Prop

HasLeftInverse f means that f has an unspecified left inverse.

def Function.RightInverse {α : Sort u₁} {β : Sort u₂} (g : β → α) (f : α → β) : Prop

RightInverse g f means that g is a right inverse to f. That is, f ∘ g = id.

def Function.HasRightInverse {α : Sort u₁} {β : Sort u₂} (f : α → β) : Prop

HasRightInverse f means that f has an unspecified right inverse.

theorem Function.LeftInverse.injective {α : Sort u₁} {β : Sort u₂} {g : β → α} {f : α → β} : Function.LeftInverse g f → Function.Injective f

theorem Function.HasLeftInverse.injective {α : Sort u₁} {β : Sort u₂} {f : α → β} : Function.HasLeftInverse f → Function.Injective f

theorem Function.rightInverse_of_injective_of_leftInverse {α : Sort u₁} {β : Sort u₂} {f : α → β} {g : β → α} (injf : Function.Injective f) (lfg : Function.LeftInverse f g) : Function.RightInverse f g

theorem Function.RightInverse.surjective {α : Sort u₁} {β : Sort u₂} {f : α → β} {g : β → α} (h : Function.RightInverse g f) : Function.Surjective f

theorem Function.HasRightInverse.surjective {α : Sort u₁} {β : Sort u₂} {f : α → β} : Function.HasRightInverse f → Function.Surjective f

theorem Function.leftInverse_of_surjective_of_rightInverse {α : Sort u₁} {β : Sort u₂} {f : α → β} {g : β → α} (surjf : Function.Surjective f) (rfg : Function.RightInverse f g) : Function.LeftInverse f g

theorem Function.injective_id {α : Sort u₁} : Function.Injective id

theorem Function.surjective_id {α : Sort u₁} : Function.Surjective id

theorem Function.bijective_id {α : Sort u₁} : Function.Bijective id

@[inline]

def Function.curry {α : Type u₁} {β : Type u₂} {φ : Type u₃} : (α × β → φ) → α → β → φ

Interpret a function on α × β as a function with two arguments.

@[inline]

def Function.uncurry {α : Type u₁} {β : Type u₂} {φ : Type u₃} : (α → β → φ) → α × β → φ

Interpret a function with two arguments as a function on α × β

@[simp]

theorem Function.curry_uncurry {α : Type u₁} {β : Type u₂} {φ : Type u₃} (f : α → β → φ) : Function.curry (Function.uncurry f) = f

@[simp]

theorem Function.uncurry_curry {α : Type u₁} {β : Type u₂} {φ : Type u₃} (f : α × β → φ) : Function.uncurry (Function.curry f) = f

theorem Function.LeftInverse.id {α : Type u₁} {β : Type u₂} {g : β → α} {f : α → β} (h : Function.LeftInverse g f) : g ∘ f = id

theorem Function.RightInverse.id {α : Type u₁} {β : Type u₂} {g : β → α} {f : α → β} (h : Function.RightInverse g f) : f ∘ g = id