General operations on functions

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

Equations

• Function.comp_right f g b a = f b (g a)

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

Equations

• Function.comp_left f g a b = f (g a) b

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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→ β → φ→ φ and g : α → β→ β, produce a function α → α → φ→ α → φ→ 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) x y = f (g x) (g y)

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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

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

Equations

• Function.combine f op g x y = op (f x y) (g x y)

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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

Equations

• Function.swap f y x = f x y

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

Equations

• Function.app f x = f x

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def Function.term_On_ : Lean.TrailingParserDescr

Given functions f : β → β → φ→ β → φ→ φ and g : α → β→ β, produce a function α → α → φ→ α → φ→ 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

• Function.term_On_ = Lean.ParserDescr.trailingNode `Function.term_On_ 2 2 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " on ") (Lean.ParserDescr.cat `term 3))

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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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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→ β is called injective if f x = f y implies x = y.

Equations

• Function.Injective f = ∀ ⦃a₁ a₂ : α⦄, f a₁ = f a₂ → a₁ = a₂

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

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

Equations

• Function.Surjective f = ∀ (b : β), ∃ a, f a = b

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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.

Equations

• Function.Bijective f = (Function.Injective f ∧ Function.Surjective f)

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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.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∘ f = id.

Equations

• Function.LeftInverse g f = ∀ (x : α), g (f x) = x

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

HasLeftInverse f means that f has an unspecified left inverse.

Equations

• Function.HasLeftInverse f = ∃ finv, Function.LeftInverse finv f

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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∘ g = id.

Equations

• Function.RightInverse g f = Function.LeftInverse f g

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

hasRightInverse f means that f has an unspecified right inverse.

Equations

• Function.HasRightInverse f = ∃ finv, Function.RightInverse finv f

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

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

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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

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

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

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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

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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× β as a function with two arguments.

Equations

• Function.curry f a b = f (a, b)

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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× β

Equations

• Function.uncurry f a = f a.fst a.snd

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

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