Documentation

Mathlib.Logic.Embedding.Basic

Injective functions #

structure Function.Embedding (α : Sort u_1) (β : Sort u_2) :
Sort (max (max 1 u_1) u_2)

α ↪ β is a bundled injective function.

Instances For

    An embedding, a.k.a. a bundled injective function.

    Equations
    Instances For
      instance Function.instFunLikeEmbedding {α : Sort u} {β : Sort v} :
      FunLike (α β) α β
      Equations
      • Function.instFunLikeEmbedding = { coe := Function.Embedding.toFun, coe_injective' := }
      Equations
      • =
      theorem Function.exists_surjective_iff {α : Sort u_1} {β : Sort u_2} :
      (∃ (f : αβ), Function.Surjective f) Nonempty (αβ) Nonempty (β α)
      instance Function.instCanLiftForAllEmbeddingCoeInstFunLikeEmbeddingInjective {α : Sort u_1} {β : Sort u_2} :
      CanLift (αβ) (α β) DFunLike.coe Function.Injective
      Equations
      • =
      def Equiv.toEmbedding {α : Sort u} {β : Sort v} (f : α β) :
      α β

      Convert an α ≃ β to α ↪ β.

      This is also available as a coercion Equiv.coeEmbedding. The explicit Equiv.toEmbedding version is preferred though, since the coercion can have issues inferring the type of the resulting embedding. For example:

      -- Works:
      example (s : Finset (Fin 3)) (f : Equiv.Perm (Fin 3)) : s.map f.toEmbedding = s.map f := by simp
      -- Error, `f` has type `Fin 3 ≃ Fin 3` but is expected to have type `Fin 3 ↪ ?m_1 : Type ?`
      example (s : Finset (Fin 3)) (f : Equiv.Perm (Fin 3)) : s.map f = s.map f.toEmbedding := by simp
      
      Equations
      Instances For
        @[simp]
        theorem Equiv.coe_toEmbedding {α : Sort u} {β : Sort v} (f : α β) :
        (Equiv.toEmbedding f) = f
        theorem Equiv.toEmbedding_apply {α : Sort u} {β : Sort v} (f : α β) (a : α) :
        theorem Equiv.toEmbedding_injective {α : Sort u} {β : Sort v} :
        Function.Injective Equiv.toEmbedding
        instance Equiv.coeEmbedding {α : Sort u} {β : Sort v} :
        Coe (α β) (α β)
        Equations
        • Equiv.coeEmbedding = { coe := Equiv.toEmbedding }
        @[reducible]
        instance Equiv.Perm.coeEmbedding {α : Sort u} :
        Coe (Equiv.Perm α) (α α)
        Equations
        • Equiv.Perm.coeEmbedding = Equiv.coeEmbedding
        theorem Function.Embedding.coe_injective {α : Sort u_1} {β : Sort u_2} :
        Function.Injective fun (f : α β) => f
        theorem Function.Embedding.ext {α : Sort u_1} {β : Sort u_2} {f : α β} {g : α β} (h : ∀ (x : α), f x = g x) :
        f = g
        theorem Function.Embedding.ext_iff {α : Sort u_1} {β : Sort u_2} {f : α β} {g : α β} :
        (∀ (x : α), f x = g x) f = g
        @[simp]
        theorem Function.Embedding.toFun_eq_coe {α : Sort u_1} {β : Sort u_2} (f : α β) :
        f.toFun = f
        @[simp]
        theorem Function.Embedding.coeFn_mk {α : Sort u_1} {β : Sort u_2} (f : αβ) (i : Function.Injective f) :
        { toFun := f, inj' := i } = f
        @[simp]
        theorem Function.Embedding.mk_coe {α : Type u_1} {β : Type u_2} (f : α β) (inj : Function.Injective f) :
        { toFun := f, inj' := inj } = f
        theorem Function.Embedding.injective {α : Sort u_1} {β : Sort u_2} (f : α β) :
        theorem Function.Embedding.apply_eq_iff_eq {α : Sort u_1} {β : Sort u_2} (f : α β) (x : α) (y : α) :
        f x = f y x = y
        @[simp]
        theorem Function.Embedding.refl_apply (α : Sort u_1) (a : α) :
        def Function.Embedding.refl (α : Sort u_1) :
        α α

        The identity map as a Function.Embedding.

        Equations
        Instances For
          @[simp]
          theorem Function.Embedding.trans_apply {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (f : α β) (g : β γ) :
          ∀ (a : α), (Function.Embedding.trans f g) a = g (f a)
          def Function.Embedding.trans {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (f : α β) (g : β γ) :
          α γ

          Composition of f : α ↪ β and g : β ↪ γ.

          Equations
          Instances For
            @[simp]
            theorem Function.Embedding.congr_apply {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort x} (e₁ : α β) (e₂ : γ δ) (f : α γ) :
            (Function.Embedding.congr e₁ e₂ f) = (Function.Embedding.trans f (Equiv.toEmbedding e₂)) e₁.symm
            def Function.Embedding.congr {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort x} (e₁ : α β) (e₂ : γ δ) (f : α γ) :
            β δ

            Transfer an embedding along a pair of equivalences.

            Equations
            Instances For
              noncomputable def Function.Embedding.ofSurjective {α : Sort u_1} {β : Sort u_2} (f : βα) (hf : Function.Surjective f) :
              α β

              A right inverse surjInv of a surjective function as an Embedding.

              Equations
              Instances For
                noncomputable def Function.Embedding.equivOfSurjective {α : Sort u_1} {β : Sort u_2} (f : α β) (hf : Function.Surjective f) :
                α β

                Convert a surjective Embedding to an Equiv

                Equations
                Instances For
                  def Function.Embedding.ofIsEmpty {α : Sort u_1} {β : Sort u_2} [IsEmpty α] :
                  α β

                  There is always an embedding from an empty type.

                  Equations
                  • Function.Embedding.ofIsEmpty = { toFun := fun (a : α) => isEmptyElim a, inj' := }
                  Instances For
                    def Function.Embedding.setValue {α : Sort u_1} {β : Sort u_2} (f : α β) (a : α) (b : β) [(a' : α) → Decidable (a' = a)] [(a' : α) → Decidable (f a' = b)] :
                    α β

                    Change the value of an embedding f at one point. If the prescribed image is already occupied by some f a', then swap the values at these two points.

                    Equations
                    Instances For
                      @[simp]
                      theorem Function.Embedding.setValue_eq {α : Sort u_1} {β : Sort u_2} (f : α β) (a : α) (b : β) [(a' : α) → Decidable (a' = a)] [(a' : α) → Decidable (f a' = b)] :
                      @[simp]
                      theorem Function.Embedding.setValue_eq_iff {α : Sort u_1} {β : Sort u_2} (f : α β) {a : α} {a' : α} {b : β} [(a' : α) → Decidable (a' = a)] [(a' : α) → Decidable (f a' = b)] :
                      @[simp]
                      theorem Function.Embedding.some_apply {α : Type u_1} :
                      Function.Embedding.some = some
                      def Function.Embedding.some {α : Type u_1} :
                      α Option α

                      Embedding into Option α using some.

                      Equations
                      • Function.Embedding.some = { toFun := some, inj' := }
                      Instances For
                        @[simp]
                        theorem Function.Embedding.optionMap_apply {α : Type u_1} {β : Type u_2} (f : α β) :
                        def Function.Embedding.optionMap {α : Type u_1} {β : Type u_2} (f : α β) :

                        A version of Option.map for Function.Embeddings.

                        Equations
                        Instances For
                          def Function.Embedding.subtype {α : Sort u_1} (p : αProp) :

                          Embedding of a Subtype.

                          Equations
                          Instances For
                            @[simp]
                            theorem Function.Embedding.coe_subtype {α : Sort u_1} (p : αProp) :
                            (Function.Embedding.subtype p) = Subtype.val
                            noncomputable def Function.Embedding.quotientOut (α : Sort u_1) [s : Setoid α] :

                            Quotient.out as an embedding.

                            Equations
                            Instances For
                              @[simp]
                              theorem Function.Embedding.coe_quotientOut (α : Sort u_1) [Setoid α] :
                              (Function.Embedding.quotientOut α) = Quotient.out
                              def Function.Embedding.punit {β : Sort u_1} (b : β) :

                              Choosing an element b : β gives an embedding of PUnit into β.

                              Equations
                              Instances For
                                @[simp]
                                theorem Function.Embedding.sectl_apply (α : Type u_1) {β : Type u_2} (b : β) (a : α) :
                                (Function.Embedding.sectl α b) a = (a, b)
                                def Function.Embedding.sectl (α : Type u_1) {β : Type u_2} (b : β) :
                                α α × β

                                Fixing an element b : β gives an embedding α ↪ α × β.

                                Equations
                                Instances For
                                  @[simp]
                                  theorem Function.Embedding.sectr_apply {α : Type u_1} (a : α) (β : Type u_2) (b : β) :
                                  (Function.Embedding.sectr a β) b = (a, b)
                                  def Function.Embedding.sectr {α : Type u_1} (a : α) (β : Type u_2) :
                                  β α × β

                                  Fixing an element a : α gives an embedding β ↪ α × β.

                                  Equations
                                  Instances For
                                    def Function.Embedding.prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e₁ : α β) (e₂ : γ δ) :
                                    α × γ β × δ

                                    If e₁ and e₂ are embeddings, then so is Prod.map e₁ e₂ : (a, b) ↦ (e₁ a, e₂ b).

                                    Equations
                                    Instances For
                                      @[simp]
                                      theorem Function.Embedding.coe_prodMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e₁ : α β) (e₂ : γ δ) :
                                      (Function.Embedding.prodMap e₁ e₂) = Prod.map e₁ e₂
                                      def Function.Embedding.pprodMap {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {δ : Sort u_4} (e₁ : α β) (e₂ : γ δ) :
                                      PProd α γ PProd β δ

                                      If e₁ and e₂ are embeddings, then so is fun ⟨a, b⟩ ↦ ⟨e₁ a, e₂ b⟩ : PProd α γ → PProd β δ.

                                      Equations
                                      Instances For
                                        def Function.Embedding.sumMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e₁ : α β) (e₂ : γ δ) :
                                        α γ β δ

                                        If e₁ and e₂ are embeddings, then so is Sum.map e₁ e₂.

                                        Equations
                                        Instances For
                                          @[simp]
                                          theorem Function.Embedding.coe_sumMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e₁ : α β) (e₂ : γ δ) :
                                          (Function.Embedding.sumMap e₁ e₂) = Sum.map e₁ e₂
                                          @[simp]
                                          theorem Function.Embedding.inl_apply {α : Type u_1} {β : Type u_2} (val : α) :
                                          Function.Embedding.inl val = Sum.inl val
                                          def Function.Embedding.inl {α : Type u_1} {β : Type u_2} :
                                          α α β

                                          The embedding of α into the sum α ⊕ β.

                                          Equations
                                          • Function.Embedding.inl = { toFun := Sum.inl, inj' := }
                                          Instances For
                                            @[simp]
                                            theorem Function.Embedding.inr_apply {α : Type u_1} {β : Type u_2} (val : β) :
                                            Function.Embedding.inr val = Sum.inr val
                                            def Function.Embedding.inr {α : Type u_1} {β : Type u_2} :
                                            β α β

                                            The embedding of β into the sum α ⊕ β.

                                            Equations
                                            • Function.Embedding.inr = { toFun := Sum.inr, inj' := }
                                            Instances For
                                              @[simp]
                                              theorem Function.Embedding.sigmaMk_apply {α : Type u_1} {β : αType u_3} (a : α) (snd : β a) :
                                              (Function.Embedding.sigmaMk a) snd = { fst := a, snd := snd }
                                              def Function.Embedding.sigmaMk {α : Type u_1} {β : αType u_3} (a : α) :
                                              β a (x : α) × β x

                                              Sigma.mk as a Function.Embedding.

                                              Equations
                                              Instances For
                                                @[simp]
                                                theorem Function.Embedding.sigmaMap_apply {α : Type u_1} {α' : Type u_2} {β : αType u_3} {β' : α'Type u_4} (f : α α') (g : (a : α) → β a β' (f a)) (x : (a : α) × β a) :
                                                (Function.Embedding.sigmaMap f g) x = Sigma.map (f) (fun (a : α) => (g a)) x
                                                def Function.Embedding.sigmaMap {α : Type u_1} {α' : Type u_2} {β : αType u_3} {β' : α'Type u_4} (f : α α') (g : (a : α) → β a β' (f a)) :
                                                (a : α) × β a (a' : α') × β' a'

                                                If f : α ↪ α' is an embedding and g : Π a, β α ↪ β' (f α) is a family of embeddings, then Sigma.map f g is an embedding.

                                                Equations
                                                Instances For
                                                  @[simp]
                                                  theorem Function.Embedding.piCongrRight_apply {α : Sort u_1} {β : αSort u_2} {γ : αSort u_3} (e : (a : α) → β a γ a) (f : (a : α) → β a) (a : α) :
                                                  def Function.Embedding.piCongrRight {α : Sort u_1} {β : αSort u_2} {γ : αSort u_3} (e : (a : α) → β a γ a) :
                                                  ((a : α) → β a) (a : α) → γ a

                                                  Define an embedding (Π a : α, β a) ↪ (Π a : α, γ a) from a family of embeddings e : Π a, (β a ↪ γ a). This embedding sends f to fun a ↦ e a (f a).

                                                  Equations
                                                  Instances For
                                                    def Function.Embedding.arrowCongrRight {α : Sort u} {β : Sort v} {γ : Sort w} (e : α β) :
                                                    (γα) γβ

                                                    An embedding e : α ↪ β defines an embedding (γ → α) ↪ (γ → β) that sends each f to e ∘ f.

                                                    Equations
                                                    Instances For
                                                      @[simp]
                                                      theorem Function.Embedding.arrowCongrRight_apply {α : Sort u} {β : Sort v} {γ : Sort w} (e : α β) (f : γα) :
                                                      noncomputable def Function.Embedding.arrowCongrLeft {α : Sort u} {β : Sort v} {γ : Sort w} [Inhabited γ] (e : α β) :
                                                      (αγ) βγ

                                                      An embedding e : α ↪ β defines an embedding (α → γ) ↪ (β → γ) for any inhabited type γ. This embedding sends each f : α → γ to a function g : β → γ such that g ∘ e = f and g y = default whenever y ∉ range e.

                                                      Equations
                                                      Instances For
                                                        def Function.Embedding.subtypeMap {α : Sort u_1} {β : Sort u_2} {p : αProp} {q : βProp} (f : α β) (h : ∀ ⦃x : α⦄, p xq (f x)) :
                                                        { x : α // p x } { y : β // q y }

                                                        Restrict both domain and codomain of an embedding.

                                                        Equations
                                                        Instances For
                                                          theorem Function.Embedding.swap_apply {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (f : α β) (x : α) (y : α) (z : α) :
                                                          (Equiv.swap (f x) (f y)) (f z) = f ((Equiv.swap x y) z)
                                                          theorem Function.Embedding.swap_comp {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (f : α β) (x : α) (y : α) :
                                                          (Equiv.swap (f x) (f y)) f = f (Equiv.swap x y)
                                                          @[simp]
                                                          theorem Equiv.asEmbedding_apply {β : Sort u_1} {α : Sort u_2} {p : βProp} (e : α Subtype p) :
                                                          ∀ (a : α), (Equiv.asEmbedding e) a = (e a)
                                                          def Equiv.asEmbedding {β : Sort u_1} {α : Sort u_2} {p : βProp} (e : α Subtype p) :
                                                          α β

                                                          Given an equivalence to a subtype, produce an embedding to the elements of the corresponding set.

                                                          Equations
                                                          Instances For
                                                            def Equiv.subtypeInjectiveEquivEmbedding (α : Sort u_1) (β : Sort u_2) :
                                                            { f : αβ // Function.Injective f } (α β)

                                                            The type of embeddings α ↪ β is equivalent to the subtype of all injective functions α → β.

                                                            Equations
                                                            • One or more equations did not get rendered due to their size.
                                                            Instances For
                                                              @[simp]
                                                              theorem Equiv.embeddingCongr_apply {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {δ : Sort u_4} (h : α β) (h' : γ δ) (f : α γ) :
                                                              def Equiv.embeddingCongr {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {δ : Sort u_4} (h : α β) (h' : γ δ) :
                                                              (α γ) (β δ)

                                                              If α₁ ≃ α₂ and β₁ ≃ β₂, then the type of embeddings α₁ ↪ β₁ is equivalent to the type of embeddings α₂ ↪ β₂.

                                                              Equations
                                                              • One or more equations did not get rendered due to their size.
                                                              Instances For
                                                                @[simp]
                                                                @[simp]
                                                                theorem Equiv.embeddingCongr_trans {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} {α₃ : Sort u_5} {β₃ : Sort u_6} (e₁ : α₁ α₂) (e₁' : β₁ β₂) (e₂ : α₂ α₃) (e₂' : β₂ β₃) :
                                                                Equiv.embeddingCongr (e₁.trans e₂) (e₁'.trans e₂') = (Equiv.embeddingCongr e₁ e₁').trans (Equiv.embeddingCongr e₂ e₂')
                                                                @[simp]
                                                                theorem Equiv.embeddingCongr_symm {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} (e₁ : α₁ α₂) (e₂ : β₁ β₂) :
                                                                (Equiv.embeddingCongr e₁ e₂).symm = Equiv.embeddingCongr e₁.symm e₂.symm
                                                                theorem Equiv.embeddingCongr_apply_trans {α₁ : Sort u_1} {β₁ : Sort u_2} {γ₁ : Sort u_3} {α₂ : Sort u_4} {β₂ : Sort u_5} {γ₂ : Sort u_6} (ea : α₁ α₂) (eb : β₁ β₂) (ec : γ₁ γ₂) (f : α₁ β₁) (g : β₁ γ₁) :
                                                                @[simp]
                                                                theorem Equiv.trans_toEmbedding {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α β) (f : β γ) :
                                                                def subtypeOrLeftEmbedding {α : Type u_1} (p : αProp) (q : αProp) [DecidablePred p] :
                                                                { x : α // p x q x } { x : α // p x } { x : α // q x }

                                                                A subtype {x // p x ∨ q x} over a disjunction of p q : α → Prop can be injectively split into a sum of subtypes {x // p x} ⊕ {x // q x} such that ¬ p x is sent to the right.

                                                                Equations
                                                                • One or more equations did not get rendered due to their size.
                                                                Instances For
                                                                  theorem subtypeOrLeftEmbedding_apply_left {α : Type u_1} {p : αProp} {q : αProp} [DecidablePred p] (x : { x : α // p x q x }) (hx : p x) :
                                                                  (subtypeOrLeftEmbedding p q) x = Sum.inl { val := x, property := hx }
                                                                  theorem subtypeOrLeftEmbedding_apply_right {α : Type u_1} {p : αProp} {q : αProp} [DecidablePred p] (x : { x : α // p x q x }) (hx : ¬p x) :
                                                                  (subtypeOrLeftEmbedding p q) x = Sum.inr { val := x, property := }
                                                                  @[simp]
                                                                  theorem Subtype.impEmbedding_apply_coe {α : Type u_1} (p : αProp) (q : αProp) (h : ∀ (x : α), p xq x) (x : { x : α // p x }) :
                                                                  ((Subtype.impEmbedding p q h) x) = x
                                                                  def Subtype.impEmbedding {α : Type u_1} (p : αProp) (q : αProp) (h : ∀ (x : α), p xq x) :
                                                                  { x : α // p x } { x : α // q x }

                                                                  A subtype {x // p x} can be injectively sent to into a subtype {x // q x}, if p x → q x for all x : α.

                                                                  Equations
                                                                  • Subtype.impEmbedding p q h = { toFun := fun (x : { x : α // p x }) => { val := x, property := }, inj' := }
                                                                  Instances For