mathlib3 documentation

core / init.function

General operations on functions #

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@[reducible]
def function.comp {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} (f : β → φ) (g : α → β) :
α → φ

Composition of functions: (f ∘ g) x = f (g x).

Equations
  • f ∘ g = λ (x : α), f (g x)
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@[reducible]
def function.dcomp {α : Sort u₁} {β : α → Sort u₂} {φ : Π {x : α}, β x → Sort u₃} (f : Π {x : α} (y : β x), φ y) (g : Π (x : α), β x) (x : α) :
φ (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.

Equations
  • f ∘' g = λ (x : α), f (g x)
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@[reducible]
def function.comp_right {α : Sort u₁} {β : Sort u₂} (f : β → β → β) (g : α → β) :
β → α → β
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@[reducible]
def function.comp_left {α : Sort u₁} {β : Sort u₂} (f : β → β → β) (g : α → β) :
α → β → β
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@[reducible]
def function.on_fun {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.

Equations
  • (f on g) = λ (x y : α), f (g x) (g y)
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@[reducible]
def function.combine {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {δ : Sort u₄} {ζ : Sort u₁} (f : α → β → φ) (op : φ → δ → ζ) (g : α → β → δ) :
α → β → ζ
Equations
  • (f -[op]- g) = λ (x : α) (y : β), op (f x y) (g x y)
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@[reducible]
def function.const {α : Sort u₁} (β : Sort u₂) (a : α) :
β → α

Constant λ _, a.

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@[reducible]
def function.swap {α : Sort u₁} {β : Sort u₂} {φ : α → β → Sort u₃} (f : Π (x : α) (y : β), φ x y) (y : β) (x : α) :
φ x y
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@[reducible]
def function.app {α : Sort u₁} {β : α → Sort u₂} (f : Π (x : α), β x) (x : α) :
β x
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theorem function.left_id {α : Sort u₁} {β : Sort u₂} (f : α → β) :
id ∘ f = f
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theorem function.right_id {α : Sort u₁} {β : Sort u₂} (f : α → β) :
f ∘ id = f
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@[simp]
theorem function.comp_app {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} (f : β → φ) (g : α → β) (a : α) :
(f ∘ g) a = f (g a)
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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
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@[simp]
theorem function.comp.left_id {α : Sort u₁} {β : Sort u₂} (f : α → β) :
id ∘ f = f
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@[simp]
theorem function.comp.right_id {α : Sort u₁} {β : Sort u₂} (f : α → β) :
f ∘ id = f
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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)
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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.

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Instances for function.injective
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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)
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@[reducible]
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.

Equations
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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)
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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.

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Instances for function.bijective
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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)
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def function.left_inverse {α : Sort u₁} {β : Sort u₂} (g : β → α) (f : α → β) :
Prop

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

Equations
Instances for function.left_inverse
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def function.has_left_inverse {α : Sort u₁} {β : Sort u₂} (f : α → β) :
Prop

has_left_inverse f means that f has an unspecified left inverse.

Equations
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def function.right_inverse {α : Sort u₁} {β : Sort u₂} (g : β → α) (f : α → β) :
Prop

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

Equations
Instances for function.right_inverse
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def function.has_right_inverse {α : Sort u₁} {β : Sort u₂} (f : α → β) :
Prop

has_right_inverse f means that f has an unspecified right inverse.

Equations
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theorem function.left_inverse.injective {α : Sort u₁} {β : Sort u₂} {g : β → α} {f : α → β} :
function.left_inverse g f → function.injective f
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theorem function.has_left_inverse.injective {α : Sort u₁} {β : Sort u₂} {f : α → β} :
function.has_left_inverse f → function.injective f
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theorem function.right_inverse_of_injective_of_left_inverse {α : Sort u₁} {β : Sort u₂} {f : α → β} {g : β → α} (injf : function.injective f) (lfg : function.left_inverse f g) :
function.right_inverse f g
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theorem function.right_inverse.surjective {α : Sort u₁} {β : Sort u₂} {f : α → β} {g : β → α} (h : function.right_inverse g f) :
function.surjective f
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theorem function.has_right_inverse.surjective {α : Sort u₁} {β : Sort u₂} {f : α → β} :
function.has_right_inverse f → function.surjective f
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theorem function.left_inverse_of_surjective_of_right_inverse {α : Sort u₁} {β : Sort u₂} {f : α → β} {g : β → α} (surjf : function.surjective f) (rfg : function.right_inverse f g) :
function.left_inverse f g
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theorem function.injective_id {α : Sort u₁} :
function.injective id
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theorem function.surjective_id {α : Sort u₁} :
function.surjective id
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theorem function.bijective_id {α : Sort u₁} :
function.bijective id
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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.

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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

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@[simp]
theorem function.curry_uncurry {α : Type u₁} {β : Type u₂} {φ : Type u₃} (f : α → β → φ) :
function.curry (function.uncurry f) = f
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@[simp]
theorem function.uncurry_curry {α : Type u₁} {β : Type u₂} {φ : Type u₃} (f : α × β → φ) :
function.uncurry (function.curry f) = f
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@[protected]
theorem function.left_inverse.id {α : Type u₁} {β : Type u₂} {g : β → α} {f : α → β} (h : function.left_inverse g f) :
g ∘ f = id
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@[protected]
theorem function.right_inverse.id {α : Type u₁} {β : Type u₂} {g : β → α} {f : α → β} (h : function.right_inverse g f) :
f ∘ g = id