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.

• toFun : α → β

  An embedding as a function. Use coercion instead.

• inj' : Function.Injective self.toFun

  An embedding is an injective function. Use Function.Embedding.injective instead.

def Function.«term_↪_» : Lean.TrailingParserDescr

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

Equations

• Function.«term_↪_» = Lean.ParserDescr.trailingNode `Function.term_↪_ 25 26 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ↪ ") (Lean.ParserDescr.cat `term 25))

instance Function.instFunLikeEmbedding {α : Sort u} {β : Sort v} : FunLike (α ↪ β) α β

Equations

• Function.instFunLikeEmbedding = { coe := Function.Embedding.toFun, coe_injective' := ⋯ }

instance Function.instEmbeddingLikeEmbeddingInstFunLikeEmbedding {α : Sort u} {β : Sort v} : EmbeddingLike (α ↪ β) α β

Equations

• ⋯ = ⋯

theorem Function.exists_surjective_iff {α : Sort u_1} {β : Sort u_2} : (∃ (f : α → β), Function.Surjective f) ↔ Nonempty (α → β) ∧ Nonempty (β ↪ α)

instance Function.instCanLiftForAllEmbeddingCoeInstFunLikeEmbeddingInjective {α : Sort u_1} {β : Sort u_2} : CanLift (α → β) (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

• Equiv.toEmbedding f = { toFun := ⇑f, inj' := ⋯ }

@[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) : (Equiv.toEmbedding f) a = f a

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

instance Function.Embedding.instUniqueEmbedding {α : Sort u_1} {β : Sort u_2} [IsEmpty α] : Unique (α ↪ β)

Equations

• Function.Embedding.instUniqueEmbedding = { toInhabited := { default := { toFun := fun (a : α) => isEmptyElim a, inj' := ⋯ } }, uniq := ⋯ }

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 : α → β) (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) : Function.Injective ⇑f

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) : (Function.Embedding.refl α) a = a

def Function.Embedding.refl (α : Sort u_1) : α ↪ α

The identity map as a Function.Embedding.

Equations

• Function.Embedding.refl α = { toFun := id, inj' := ⋯ }

@[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

• Function.Embedding.trans f g = { toFun := ⇑g ∘ ⇑f, inj' := ⋯ }

instance Function.Embedding.instTransSortSortSortEmbeddingEmbeddingEmbedding : Trans Function.Embedding Function.Embedding Function.Embedding

Equations

• Function.Embedding.instTransSortSortSortEmbeddingEmbeddingEmbedding = { trans := fun {a : Sort u_1} {b : Sort u_2} {c : Sort u_3} => Function.Embedding.trans }

@[simp]

theorem Function.Embedding.equiv_toEmbedding_trans_symm_toEmbedding {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) : Function.Embedding.trans (Equiv.toEmbedding e) (Equiv.toEmbedding e.symm) = Function.Embedding.refl α

@[simp]

theorem Function.Embedding.equiv_symm_toEmbedding_trans_toEmbedding {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) : Function.Embedding.trans (Equiv.toEmbedding e.symm) (Equiv.toEmbedding e) = Function.Embedding.refl β

@[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

• Function.Embedding.congr e₁ e₂ f = Function.Embedding.trans (Equiv.toEmbedding e₁.symm) (Function.Embedding.trans f (Equiv.toEmbedding e₂))

noncomputable def Function.Embedding.ofSurjective {α : Sort u_1} {β : Sort u_2} (f : β → α) (hf : Function.Surjective f) : α ↪ β

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

Equations

• Function.Embedding.ofSurjective f hf = { toFun := Function.surjInv hf, inj' := ⋯ }

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

• Function.Embedding.equivOfSurjective f hf = Equiv.ofBijective ⇑f ⋯

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' := ⋯ }

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

• Function.Embedding.setValue f a b = { toFun := fun (a' : α) => if a' = a then b else if f a' = b then f a else f a', inj' := ⋯ }

@[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)] : (Function.Embedding.setValue f a b) 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)] : (Function.Embedding.setValue f a b) a' = b ↔ a' = a

@[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' := ⋯ }

@[simp]

theorem Function.Embedding.optionMap_apply {α : Type u_1} {β : Type u_2} (f : α ↪ β) : ⇑(Function.Embedding.optionMap f) = Option.map ⇑f

def Function.Embedding.optionMap {α : Type u_1} {β : Type u_2} (f : α ↪ β) : Option α ↪ Option β

A version of Option.map for Function.Embeddings.

Equations

• Function.Embedding.optionMap f = { toFun := Option.map ⇑f, inj' := ⋯ }

def Function.Embedding.subtype {α : Sort u_1} (p : α → Prop) : Subtype p ↪ α

Embedding of a Subtype.

Equations

• Function.Embedding.subtype p = { toFun := Subtype.val, inj' := ⋯ }

@[simp]

theorem Function.Embedding.coe_subtype {α : Sort u_1} (p : α → Prop) : ⇑(Function.Embedding.subtype p) = Subtype.val

noncomputable def Function.Embedding.quotientOut (α : Sort u_1) [s : Setoid α] : Quotient s ↪ α

Quotient.out as an embedding.

Equations

• Function.Embedding.quotientOut α = { toFun := Quotient.out, inj' := ⋯ }

@[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) : PUnit.{u_2} ↪ β

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

Equations

• Function.Embedding.punit b = { toFun := fun (x : PUnit.{u_2}) => b, inj' := ⋯ }

@[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

• Function.Embedding.sectl α b = { toFun := fun (a : α) => (a, b), inj' := ⋯ }

@[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

• Function.Embedding.sectr a β = { toFun := fun (b : β) => (a, b), inj' := ⋯ }

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

• Function.Embedding.prodMap e₁ e₂ = { toFun := Prod.map ⇑e₁ ⇑e₂, inj' := ⋯ }

@[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

• Function.Embedding.pprodMap e₁ e₂ = { toFun := fun (x : PProd α γ) => { fst := e₁ x.fst, snd := e₂ x.snd }, inj' := ⋯ }

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

• Function.Embedding.sumMap e₁ e₂ = { toFun := Sum.map ⇑e₁ ⇑e₂, inj' := ⋯ }

@[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' := ⋯ }

@[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' := ⋯ }

@[simp]

theorem Function.Embedding.sigmaMk_apply {α : Type u_1} {β : α → Type 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 : α → Type u_3} (a : α) : β a ↪ (x : α) × β x

Sigma.mk as a Function.Embedding.

Equations

• Function.Embedding.sigmaMk a = { toFun := Sigma.mk a, inj' := ⋯ }

@[simp]

theorem Function.Embedding.sigmaMap_apply {α : Type u_1} {α' : Type u_2} {β : α → Type 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 : α → Type 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

• Function.Embedding.sigmaMap f g = { toFun := Sigma.map ⇑f fun (a : α) => ⇑(g a), inj' := ⋯ }

@[simp]

theorem Function.Embedding.piCongrRight_apply {α : Sort u_1} {β : α → Sort u_2} {γ : α → Sort u_3} (e : (a : α) → β a ↪ γ a) (f : (a : α) → β a) (a : α) : (Function.Embedding.piCongrRight e) f a = (e a) (f a)

def Function.Embedding.piCongrRight {α : Sort u_1} {β : α → Sort u_2} {γ : α → Sort 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

• Function.Embedding.piCongrRight e = { toFun := fun (f : (a : α) → β a) (a : α) => (e a) (f a), inj' := ⋯ }

def Function.Embedding.arrowCongrRight {α : Sort u} {β : Sort v} {γ : Sort w} (e : α ↪ β) : (γ → α) ↪ γ → β

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

Equations

• Function.Embedding.arrowCongrRight e = Function.Embedding.piCongrRight fun (x : γ) => e

@[simp]

theorem Function.Embedding.arrowCongrRight_apply {α : Sort u} {β : Sort v} {γ : Sort w} (e : α ↪ β) (f : γ → α) : (Function.Embedding.arrowCongrRight e) f = ⇑e ∘ f

noncomputable def Function.Embedding.arrowCongrLeft {α : Sort u} {β : Sort v} {γ : Sort w} [Inhabited γ] (e : α ↪ β) : (α → γ) ↪ β → γ

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

• Function.Embedding.arrowCongrLeft e = { toFun := fun (f : α → γ) => Function.extend (⇑e) f default, inj' := ⋯ }

def Function.Embedding.subtypeMap {α : Sort u_1} {β : Sort u_2} {p : α → Prop} {q : β → Prop} (f : α ↪ β) (h : ∀ ⦃x : α⦄, p x → q (f x)) : { x : α // p x } ↪ { y : β // q y }

Restrict both domain and codomain of an embedding.

Equations

• Function.Embedding.subtypeMap f h = { toFun := Subtype.map (⇑f) h, inj' := ⋯ }

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 : β → Prop} (e : α ≃ Subtype p) : ∀ (a : α), (Equiv.asEmbedding e) a = ↑(e a)

def Equiv.asEmbedding {β : Sort u_1} {α : Sort u_2} {p : β → Prop} (e : α ≃ Subtype p) : α ↪ β

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

Equations

• Equiv.asEmbedding e = Function.Embedding.trans (Equiv.toEmbedding e) (Function.Embedding.subtype p)

def Equiv.subtypeInjectiveEquivEmbedding (α : Sort u_1) (β : Sort u_2) : { f : α → β // 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.

@[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) : (Equiv.embeddingCongr h h') f = Function.Embedding.congr h h' f

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.

@[simp]

theorem Equiv.embeddingCongr_refl {α : Sort u_1} {β : Sort u_2} : Equiv.embeddingCongr (Equiv.refl α) (Equiv.refl β) = Equiv.refl (α ↪ β)

@[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) : (Equiv.embeddingCongr ea ec) (Function.Embedding.trans f g) = Function.Embedding.trans ((Equiv.embeddingCongr ea eb) f) ((Equiv.embeddingCongr eb ec) g)

@[simp]

theorem Equiv.refl_toEmbedding {α : Type u_1} : Equiv.toEmbedding (Equiv.refl α) = Function.Embedding.refl α

@[simp]

theorem Equiv.trans_toEmbedding {α : Type u_1} {β : Type u_2} {γ : Type u_3} (e : α ≃ β) (f : β ≃ γ) : Equiv.toEmbedding (e.trans f) = Function.Embedding.trans (Equiv.toEmbedding e) (Equiv.toEmbedding f)

def subtypeOrLeftEmbedding {α : Type u_1} (p : α → Prop) (q : α → Prop) [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.

theorem subtypeOrLeftEmbedding_apply_left {α : Type u_1} {p : α → Prop} {q : α → Prop} [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 : α → Prop} {q : α → Prop} [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 : α → Prop) (q : α → Prop) (h : ∀ (x : α), p x → q x) (x : { x : α // p x }) : ↑((Subtype.impEmbedding p q h) x) = ↑x

def Subtype.impEmbedding {α : Type u_1} (p : α → Prop) (q : α → Prop) (h : ∀ (x : α), p x → q 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' := ⋯ }