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Mathlib.Logic.Function.Defs

General operations on functions #

theorem Function.flip_def {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {f : αβφ} :
flip f = fun (b : β) (a : α) => f a b
@[reducible, inline]
def Function.dcomp {α : Sort u₁} {β : αSort u₂} {φ : {x : α} → β xSort u₃} (f : {x : α} → (y : β x) → φ y) (g : (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.

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    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.

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      @[reducible, inline]
      abbrev Function.onFun {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} (f : ββφ) (g : αβ) :
      ααφ

      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.

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        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.

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          @[reducible, inline]
          abbrev Function.swap {α : Sort u₁} {β : Sort u₂} {φ : αβSort u₃} (f : (x : α) → (y : β) → φ x y) (y : β) (x : α) :
          φ x y

          For a two-argument function f, swap f is the same function but taking the arguments in the reverse order. swap f y x = f x y.

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            theorem Function.swap_def {α : Sort u₁} {β : Sort u₂} {φ : αβSort u₃} (f : (x : α) → (y : β) → φ x y) :
            swap f = fun (y : β) (x : α) => f x y
            theorem Function.comp_assoc {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {δ : Sort u₄} (f : φδ) (g : βφ) (h : αβ) :
            (f g) h = f g h
            def Function.Bijective {α : Sort u₁} {β : Sort u₂} (f : αβ) :

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

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              theorem Function.Bijective.comp {α : Sort u₁} {β : Sort u₂} {φ : Sort u₃} {g : βφ} {f : αβ} :
              Bijective gBijective fBijective (g f)
              theorem Function.Injective.beq_eq {α : Type u_1} {β : Type u_2} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {f : αβ} (I : Injective f) {a b : α} :
              (f a == f b) = (a == b)
              def Function.IsFixedPt {α : Type u₁} (f : αα) (x : α) :

              A point x is a fixed point of f : α → α if f x = x.

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                def Pi.map {ι : Sort u_1} {α : ιSort u_2} {β : ιSort u_3} (f : (i : ι) → α iβ i) :
                ((i : ι) → α i)(i : ι) → β i

                Sends a dependent function a : ∀ i, α i to a dependent function Pi.map f a : ∀ i, β i by applying f i to i-th component.

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                  @[simp]
                  theorem Pi.map_apply {ι : Sort u_1} {α : ιSort u_2} {β : ιSort u_3} (f : (i : ι) → α iβ i) (a : (i : ι) → α i) (i : ι) :
                  Pi.map f a i = f i (a i)