Equivalence between types

In this file we define two types:

• Equiv α β a.k.a. α ≃ β: a bijective map α → β bundled with its inverse map; we use this (and not equality!) to express that various Types or Sorts are equivalent.

• Equiv.Perm α: the group of permutations α ≃ α. More lemmas about Equiv.Perm can be found in GroupTheory.Perm.

Then we define

• canonical isomorphisms between various types: e.g.,

  • Equiv.refl α is the identity map interpreted as α ≃ α;

• operations on equivalences: e.g.,

  • Equiv.symm e : β ≃ α is the inverse of e : α ≃ β;

  • Equiv.trans e₁ e₂ : α ≃ γ is the composition of e₁ : α ≃ β and e₂ : β ≃ γ (note the order of the arguments!);

• definitions that transfer some instances along an equivalence. By convention, we transfer instances from right to left.

  • Equiv.inhabited takes e : α ≃ β and [Inhabited β] and returns Inhabited α;
  • Equiv.unique takes e : α ≃ β and [Unique β] and returns Unique α;
  • Equiv.decidableEq takes e : α ≃ β and [DecidableEq β] and returns DecidableEq α.

  More definitions of this kind can be found in other files. E.g., Data.Equiv.TransferInstance does it for many algebraic type classes like Group, Module, etc.

Many more such isomorphisms and operations are defined in Logic.Equiv.Basic.

Tags

equivalence, congruence, bijective map

structure Equiv (α : Sort u_1) (β : Sort u_2) : Sort (max (max 1 u_1) u_2)

• toFun : α → β
• invFun : β → α
• left_inv : Function.LeftInverse s.invFun s.toFun
• right_inv : Function.RightInverse s.invFun s.toFun

α ≃ β is the type of functions from α → β with a two-sided inverse.

def «term_≃_» : Lean.TrailingParserDescr

def EquivLike.toEquiv {α : Sort u} {β : Sort v} {F : Sort u_1} [EquivLike F α β] (f : F) : α ≃ β

Turn an element of a type F satisfying EquivLike F α β into an actual Equiv. This is declared as the default coercion from F to α ≃ β.

instance instCoeTCEquiv {α : Sort u} {β : Sort v} {F : Sort u_1} [EquivLike F α β] : CoeTC F (α ≃ β)

Any type satisfying EquivLike can be cast into Equiv via EquivLike.toEquiv.

@[reducible]

def Equiv.Perm (α : Sort u_1) : Sort (max 1 u_1)

Perm α is the type of bijections from α to itself.

instance Equiv.instEquivLikeEquiv {α : Sort u} {β : Sort v} : EquivLike (α ≃ β) α β

instance Equiv.instFunLikeEquiv {α : Sort u} {β : Sort v} : FunLike (α ≃ β) α fun x => β

Helper instance when inference gets stuck on following the normal chain EquivLike → EmbeddingLike → FunLike → CoeFun.

@[simp]

theorem Equiv.coe_fn_mk {α : Sort u} {β : Sort v} (f : α → β) (g : β → α) (l : Function.LeftInverse g f) (r : Function.RightInverse g f) : ↑{ toFun := f, invFun := g, left_inv := l, right_inv := r } = f

theorem Equiv.coe_fn_injective {α : Sort u} {β : Sort v} : Function.Injective fun e => ↑e

The map (r ≃ s) → (r → s) is injective.

theorem Equiv.coe_inj {α : Sort u} {β : Sort v} {e₁ : α ≃ β} {e₂ : α ≃ β} : ↑e₁ = ↑e₂ ↔ e₁ = e₂

theorem Equiv.ext {α : Sort u} {β : Sort v} {f : α ≃ β} {g : α ≃ β} (H : ∀ (x : α), ↑f x = ↑g x) : f = g

theorem Equiv.congr_arg {α : Sort u} {β : Sort v} {f : α ≃ β} {x : α} {x' : α} : x = x' → ↑f x = ↑f x'

theorem Equiv.congr_fun {α : Sort u} {β : Sort v} {f : α ≃ β} {g : α ≃ β} (h : f = g) (x : α) : ↑f x = ↑g x

theorem Equiv.ext_iff {α : Sort u} {β : Sort v} {f : α ≃ β} {g : α ≃ β} : f = g ↔ ∀ (x : α), ↑f x = ↑g x

theorem Equiv.Perm.ext {α : Sort u} {σ : Equiv.Perm α} {τ : Equiv.Perm α} (H : ∀ (x : α), ↑σ x = ↑τ x) : σ = τ

theorem Equiv.Perm.congr_arg {α : Sort u} {f : Equiv.Perm α} {x : α} {x' : α} : x = x' → ↑f x = ↑f x'

theorem Equiv.Perm.congr_fun {α : Sort u} {f : Equiv.Perm α} {g : Equiv.Perm α} (h : f = g) (x : α) : ↑f x = ↑g x

theorem Equiv.Perm.ext_iff {α : Sort u} {σ : Equiv.Perm α} {τ : Equiv.Perm α} : σ = τ ↔ ∀ (x : α), ↑σ x = ↑τ x

def Equiv.refl (α : Sort u_1) : α ≃ α

Any type is equivalent to itself.

instance Equiv.inhabited' {α : Sort u} : Inhabited (α ≃ α)

def Equiv.symm {α : Sort u} {β : Sort v} (e : α ≃ β) : β ≃ α

Inverse of an equivalence e : α ≃ β.

def Equiv.Simps.symm_apply {α : Sort u} {β : Sort v} (e : α ≃ β) : β → α

See Note [custom simps projection]

theorem Equiv.left_inv' {α : Sort u} {β : Sort v} (e : α ≃ β) : Function.LeftInverse ↑e.symm ↑e

theorem Equiv.right_inv' {α : Sort u} {β : Sort v} (e : α ≃ β) : Function.RightInverse ↑e.symm ↑e

def Equiv.trans {α : Sort u} {β : Sort v} {γ : Sort w} (e₁ : α ≃ β) (e₂ : β ≃ γ) : α ≃ γ

Composition of equivalences e₁ : α ≃ β and e₂ : β ≃ γ.

@[simp]

theorem Equiv.instTransSortSortSortEquivEquivEquiv_trans : ∀ {a : Sort u_2} {b : Sort u_1} {c : Sort u_3} (e₁ : a ≃ b) (e₂ : b ≃ c), Trans.trans e₁ e₂ = e₁.trans e₂

instance Equiv.instTransSortSortSortEquivEquivEquiv : Trans Equiv Equiv Equiv

@[simp]

theorem Equiv.toFun_as_coe {α : Sort u} {β : Sort v} (e : α ≃ β) : e.toFun = ↑e

@[simp]

theorem Equiv.toFun_as_coe_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (x : α) : Equiv.toFun e x = ↑e x

@[simp]

theorem Equiv.invFun_as_coe {α : Sort u} {β : Sort v} (e : α ≃ β) : e.invFun = ↑e.symm

theorem Equiv.injective {α : Sort u} {β : Sort v} (e : α ≃ β) : Function.Injective ↑e

theorem Equiv.surjective {α : Sort u} {β : Sort v} (e : α ≃ β) : Function.Surjective ↑e

theorem Equiv.bijective {α : Sort u} {β : Sort v} (e : α ≃ β) : Function.Bijective ↑e

theorem Equiv.subsingleton {α : Sort u} {β : Sort v} (e : α ≃ β) [Subsingleton β] : Subsingleton α

theorem Equiv.subsingleton.symm {α : Sort u} {β : Sort v} (e : α ≃ β) [Subsingleton α] : Subsingleton β

theorem Equiv.subsingleton_congr {α : Sort u} {β : Sort v} (e : α ≃ β) : Subsingleton α ↔ Subsingleton β

instance Equiv.equiv_subsingleton_cod {α : Sort u} {β : Sort v} [Subsingleton β] : Subsingleton (α ≃ β)

instance Equiv.equiv_subsingleton_dom {α : Sort u} {β : Sort v} [Subsingleton α] : Subsingleton (α ≃ β)

instance Equiv.permUnique {α : Sort u} [Subsingleton α] : Unique (Equiv.Perm α)

theorem Equiv.Perm.subsingleton_eq_refl {α : Sort u} [Subsingleton α] (e : Equiv.Perm α) : e = Equiv.refl α

def Equiv.decidableEq {α : Sort u} {β : Sort v} (e : α ≃ β) [DecidableEq β] : DecidableEq α

Transfer DecidableEq across an equivalence.

theorem Equiv.nonempty_congr {α : Sort u} {β : Sort v} (e : α ≃ β) : Nonempty α ↔ Nonempty β

theorem Equiv.nonempty {α : Sort u} {β : Sort v} (e : α ≃ β) [Nonempty β] : Nonempty α

def Equiv.inhabited {α : Sort u} {β : Sort v} [Inhabited β] (e : α ≃ β) : Inhabited α

If α ≃ β and β is inhabited, then so is α.

def Equiv.unique {α : Sort u} {β : Sort v} [Unique β] (e : α ≃ β) : Unique α

If α ≃ β and β is a singleton type, then so is α.

def Equiv.cast {α : Sort u_1} {β : Sort u_1} (h : α = β) : α ≃ β

Equivalence between equal types.

@[simp]

theorem Equiv.coe_fn_symm_mk {α : Sort u} {β : Sort v} (f : α → β) (g : β → α) (l : Function.LeftInverse g f) (r : Function.RightInverse g f) : ↑{ toFun := f, invFun := g, left_inv := l, right_inv := r }.symm = g

@[simp]

theorem Equiv.coe_refl {α : Sort u} : ↑(Equiv.refl α) = id

theorem Equiv.Perm.coe_subsingleton {α : Type u_1} [Subsingleton α] (e : Equiv.Perm α) : ↑e = id

This cannot be a simp lemmas as it incorrectly matches against e : α ≃ synonym α, when synonym α is semireducible. This makes a mess of Multiplicative.ofAdd etc.

@[simp]

theorem Equiv.refl_apply {α : Sort u} (x : α) : ↑(Equiv.refl α) x = x

@[simp]

theorem Equiv.coe_trans {α : Sort u} {β : Sort v} {γ : Sort w} (f : α ≃ β) (g : β ≃ γ) : ↑(f.trans g) = ↑g ∘ ↑f

@[simp]

theorem Equiv.trans_apply {α : Sort u} {β : Sort v} {γ : Sort w} (f : α ≃ β) (g : β ≃ γ) (a : α) : ↑(f.trans g) a = ↑g (↑f a)

@[simp]

theorem Equiv.apply_symm_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (x : β) : ↑e (↑e.symm x) = x

@[simp]

theorem Equiv.symm_apply_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (x : α) : ↑e.symm (↑e x) = x

@[simp]

theorem Equiv.symm_comp_self {α : Sort u} {β : Sort v} (e : α ≃ β) : ↑e.symm ∘ ↑e = id

@[simp]

theorem Equiv.self_comp_symm {α : Sort u} {β : Sort v} (e : α ≃ β) : ↑e ∘ ↑e.symm = id

@[simp]

theorem Equiv.symm_trans_apply {α : Sort u} {β : Sort v} {γ : Sort w} (f : α ≃ β) (g : β ≃ γ) (a : γ) : ↑(f.trans g).symm a = ↑f.symm (↑g.symm a)

@[simp]

theorem Equiv.symm_symm_apply {α : Sort u} {β : Sort v} (f : α ≃ β) (b : α) : ↑f.symm.symm b = ↑f b

theorem Equiv.apply_eq_iff_eq {α : Sort u} {β : Sort v} (f : α ≃ β) {x : α} {y : α} : ↑f x = ↑f y ↔ x = y

theorem Equiv.apply_eq_iff_eq_symm_apply {α : Sort u} {β : Sort v} {x : α} {y : (fun x => β) x} (f : α ≃ β) : ↑f x = y ↔ x = ↑f.symm y

@[simp]

theorem Equiv.cast_apply {α : Sort u_1} {β : Sort u_1} (h : α = β) (x : α) : ↑(Equiv.cast h) x = cast h x

@[simp]

theorem Equiv.cast_symm {α : Sort u_1} {β : Sort u_1} (h : α = β) : (Equiv.cast h).symm = Equiv.cast (_ : β = α)

@[simp]

theorem Equiv.cast_refl {α : Sort u_1} (h : optParam (α = α) (_ : α = α)) : Equiv.cast h = Equiv.refl α

@[simp]

theorem Equiv.cast_trans {α : Sort u_1} {β : Sort u_1} {γ : Sort u_1} (h : α = β) (h2 : β = γ) : (Equiv.cast h).trans (Equiv.cast h2) = Equiv.cast (_ : α = γ)

theorem Equiv.cast_eq_iff_heq {α : Sort u_1} {β : Sort u_1} (h : α = β) {a : α} {b : β} : ↑(Equiv.cast h) a = b ↔ HEq a b

theorem Equiv.symm_apply_eq {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) {x : β} {y : (fun x => α) x} : ↑e.symm x = y ↔ x = ↑e y

theorem Equiv.eq_symm_apply {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) {x : β} {y : (fun x => α) x} : y = ↑e.symm x ↔ ↑e y = x

@[simp]

theorem Equiv.symm_symm {α : Sort u} {β : Sort v} (e : α ≃ β) : e.symm.symm = e

@[simp]

theorem Equiv.trans_refl {α : Sort u} {β : Sort v} (e : α ≃ β) : e.trans (Equiv.refl β) = e

@[simp]

theorem Equiv.refl_symm {α : Sort u} : (Equiv.refl α).symm = Equiv.refl α

@[simp]

theorem Equiv.refl_trans {α : Sort u} {β : Sort v} (e : α ≃ β) : (Equiv.refl α).trans e = e

@[simp]

theorem Equiv.symm_trans_self {α : Sort u} {β : Sort v} (e : α ≃ β) : e.symm.trans e = Equiv.refl β

@[simp]

theorem Equiv.self_trans_symm {α : Sort u} {β : Sort v} (e : α ≃ β) : e.trans e.symm = Equiv.refl α

theorem Equiv.trans_assoc {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort u_1} (ab : α ≃ β) (bc : β ≃ γ) (cd : γ ≃ δ) : (ab.trans bc).trans cd = ab.trans (bc.trans cd)

theorem Equiv.leftInverse_symm {α : Sort u} {β : Sort v} (f : α ≃ β) : Function.LeftInverse ↑f.symm ↑f

theorem Equiv.rightInverse_symm {α : Sort u} {β : Sort v} (f : α ≃ β) : Function.RightInverse ↑f.symm ↑f

theorem Equiv.injective_comp {α : Sort u} {β : Sort v} {γ : Sort w} (e : α ≃ β) (f : β → γ) : Function.Injective (f ∘ ↑e) ↔ Function.Injective f

theorem Equiv.comp_injective {α : Sort u} {β : Sort v} {γ : Sort w} (f : α → β) (e : β ≃ γ) : Function.Injective (↑e ∘ f) ↔ Function.Injective f

theorem Equiv.surjective_comp {α : Sort u} {β : Sort v} {γ : Sort w} (e : α ≃ β) (f : β → γ) : Function.Surjective (f ∘ ↑e) ↔ Function.Surjective f

theorem Equiv.comp_surjective {α : Sort u} {β : Sort v} {γ : Sort w} (f : α → β) (e : β ≃ γ) : Function.Surjective (↑e ∘ f) ↔ Function.Surjective f

theorem Equiv.bijective_comp {α : Sort u} {β : Sort v} {γ : Sort w} (e : α ≃ β) (f : β → γ) : Function.Bijective (f ∘ ↑e) ↔ Function.Bijective f

theorem Equiv.comp_bijective {α : Sort u} {β : Sort v} {γ : Sort w} (f : α → β) (e : β ≃ γ) : Function.Bijective (↑e ∘ f) ↔ Function.Bijective f

def Equiv.equivCongr {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort u_1} (ab : α ≃ β) (cd : γ ≃ δ) : α ≃ γ ≃ (β ≃ δ)

If α is equivalent to β and γ is equivalent to δ, then the type of equivalences α ≃ γ is equivalent to the type of equivalences β ≃ δ.

@[simp]

theorem Equiv.equivCongr_refl {α : Sort u_1} {β : Sort u_2} : Equiv.equivCongr (Equiv.refl α) (Equiv.refl β) = Equiv.refl (α ≃ β)

@[simp]

theorem Equiv.equivCongr_symm {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort u_1} (ab : α ≃ β) (cd : γ ≃ δ) : (Equiv.equivCongr ab cd).symm = Equiv.equivCongr ab.symm cd.symm

@[simp]

theorem Equiv.equivCongr_trans {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort u_1} {ε : Sort u_2} {ζ : Sort u_3} (ab : α ≃ β) (de : δ ≃ ε) (bc : β ≃ γ) (ef : ε ≃ ζ) : (Equiv.equivCongr ab de).trans (Equiv.equivCongr bc ef) = Equiv.equivCongr (ab.trans bc) (de.trans ef)

@[simp]

theorem Equiv.equivCongr_refl_left {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (bg : β ≃ γ) (e : α ≃ β) : ↑(Equiv.equivCongr (Equiv.refl α) bg) e = e.trans bg

@[simp]

theorem Equiv.equivCongr_refl_right {α : Sort u_1} {β : Sort u_2} (ab : α ≃ β) (e : α ≃ β) : ↑(Equiv.equivCongr ab (Equiv.refl β)) e = ab.symm.trans e

@[simp]

theorem Equiv.equivCongr_apply_apply {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort u_1} (ab : α ≃ β) (cd : γ ≃ δ) (e : α ≃ γ) (x : β) : ↑(↑(Equiv.equivCongr ab cd) e) x = ↑cd (↑e (↑ab.symm x))

def Equiv.permCongr {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') : Equiv.Perm α' ≃ Equiv.Perm β'

If α is equivalent to β, then Perm α is equivalent to Perm β.

theorem Equiv.permCongr_def {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : Equiv.Perm α') : ↑(Equiv.permCongr e) p = (e.symm.trans p).trans e

@[simp]

theorem Equiv.permCongr_refl {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') : ↑(Equiv.permCongr e) (Equiv.refl α') = Equiv.refl β'

@[simp]

theorem Equiv.permCongr_symm {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') : (Equiv.permCongr e).symm = Equiv.permCongr e.symm

@[simp]

theorem Equiv.permCongr_apply {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : Equiv.Perm α') (x : β') : ↑(↑(Equiv.permCongr e) p) x = ↑e (↑p (↑e.symm x))

theorem Equiv.permCongr_symm_apply {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : Equiv.Perm β') (x : α') : ↑(↑(Equiv.permCongr e).symm p) x = ↑e.symm (↑p (↑e x))

theorem Equiv.permCongr_trans {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : Equiv.Perm α') (p' : Equiv.Perm α') : (↑(Equiv.permCongr e) p).trans (↑(Equiv.permCongr e) p') = ↑(Equiv.permCongr e) (p.trans p')

def Equiv.equivOfIsEmpty (α : Sort u_1) (β : Sort u_2) [IsEmpty α] [IsEmpty β] : α ≃ β

Two empty types are equivalent.

def Equiv.equivEmpty (α : Sort u) [IsEmpty α] : α ≃ Empty

If α is an empty type, then it is equivalent to the Empty type.

def Equiv.equivPEmpty (α : Sort v) [IsEmpty α] : α ≃ PEmpty.{u}

If α is an empty type, then it is equivalent to the PEmpty type in any universe.

def Equiv.equivEmptyEquiv (α : Sort u) : α ≃ Empty ≃ IsEmpty α

α is equivalent to an empty type iff α is empty.

def Equiv.propEquivPEmpty {p : Prop} (h : ¬p) : p ≃ PEmpty.{u_1}

The Sort of proofs of a false proposition is equivalent to PEmpty.

def Equiv.equivOfUnique (α : Sort u) (β : Sort v) [Unique α] [Unique β] : α ≃ β

If both α and β have a unique element, then α ≃ β.

def Equiv.equivPUnit (α : Sort u) [Unique α] : α ≃ PUnit.{v}

If α has a unique element, then it is equivalent to any PUnit.

def Equiv.propEquivPUnit {p : Prop} (h : p) : p ≃ PUnit.{0}

The Sort of proofs of a true proposition is equivalent to PUnit.

@[simp]

theorem Equiv.ulift_apply {α : Type v} : ↑Equiv.ulift = ULift.down

def Equiv.ulift {α : Type v} : ULift.{u, v} α ≃ α

ULift α is equivalent to α.

@[simp]

theorem Equiv.plift_apply {α : Sort u} : ↑Equiv.plift = PLift.down

def Equiv.plift {α : Sort u} : PLift α ≃ α

PLift α is equivalent to α.

def Equiv.ofIff {P : Prop} {Q : Prop} (h : P ↔ Q) : P ≃ Q

equivalence of propositions is the same as iff

@[simp]

theorem Equiv.arrowCongr_apply {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) (f : α₁ → β₁) : ∀ (a : α₂), ↑(Equiv.arrowCongr e₁ e₂) f a = (↑e₂ ∘ f ∘ ↑e₁.symm) a

def Equiv.arrowCongr {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) : (α₁ → β₁) ≃ (α₂ → β₂)

If α₁ is equivalent to α₂ and β₁ is equivalent to β₂, then the type of maps α₁ → β₁ is equivalent to the type of maps α₂ → β₂.

theorem Equiv.arrowCongr_comp {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.arrowCongr ea ec) (g ∘ f) = ↑(Equiv.arrowCongr eb ec) g ∘ ↑(Equiv.arrowCongr ea eb) f

@[simp]

theorem Equiv.arrowCongr_refl {α : Sort u_1} {β : Sort u_2} : Equiv.arrowCongr (Equiv.refl α) (Equiv.refl β) = Equiv.refl (α → β)

@[simp]

theorem Equiv.arrowCongr_trans {α₁ : Sort u_1} {α₂ : Sort u_2} {β₁ : 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.arrowCongr (e₁.trans e₂) (e₁'.trans e₂') = (Equiv.arrowCongr e₁ e₁').trans (Equiv.arrowCongr e₂ e₂')

@[simp]

theorem Equiv.arrowCongr_symm {α₁ : Sort u_1} {α₂ : Sort u_2} {β₁ : Sort u_3} {β₂ : Sort u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) : (Equiv.arrowCongr e₁ e₂).symm = Equiv.arrowCongr e₁.symm e₂.symm

@[simp]

theorem Equiv.arrowCongr'_apply {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (hα : α₁ ≃ α₂) (hβ : β₁ ≃ β₂) (f : α₁ → β₁) : ∀ (a : α₂), ↑(Equiv.arrowCongr' hα hβ) f a = ↑hβ (f (↑hα.symm a))

def Equiv.arrowCongr' {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (hα : α₁ ≃ α₂) (hβ : β₁ ≃ β₂) : (α₁ → β₁) ≃ (α₂ → β₂)

A version of Equiv.arrowCongr in Type, rather than Sort.

The equiv_rw tactic is not able to use the default Sort level Equiv.arrowCongr, because Lean's universe rules will not unify ?l_1 with imax (1 ?m_1).

@[simp]

theorem Equiv.arrowCongr'_refl {α : Type u_1} {β : Type u_2} : Equiv.arrowCongr' (Equiv.refl α) (Equiv.refl β) = Equiv.refl (α → β)

@[simp]

theorem Equiv.arrowCongr'_trans {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {α₃ : Type u_5} {β₃ : Type u_6} (e₁ : α₁ ≃ α₂) (e₁' : β₁ ≃ β₂) (e₂ : α₂ ≃ α₃) (e₂' : β₂ ≃ β₃) : Equiv.arrowCongr' (e₁.trans e₂) (e₁'.trans e₂') = (Equiv.arrowCongr' e₁ e₁').trans (Equiv.arrowCongr' e₂ e₂')

@[simp]

theorem Equiv.arrowCongr'_symm {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) : (Equiv.arrowCongr' e₁ e₂).symm = Equiv.arrowCongr' e₁.symm e₂.symm

@[simp]

theorem Equiv.conj_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (f : α → α) : ∀ (a : β), ↑(Equiv.conj e) f a = ↑e (f (↑e.symm a))

def Equiv.conj {α : Sort u} {β : Sort v} (e : α ≃ β) : (α → α) ≃ (β → β)

Conjugate a map f : α → α by an equivalence α ≃ β.

@[simp]

theorem Equiv.conj_refl {α : Sort u} : Equiv.conj (Equiv.refl α) = Equiv.refl (α → α)

@[simp]

theorem Equiv.conj_symm {α : Sort u} {β : Sort v} (e : α ≃ β) : (Equiv.conj e).symm = Equiv.conj e.symm

@[simp]

theorem Equiv.conj_trans {α : Sort u} {β : Sort v} {γ : Sort w} (e₁ : α ≃ β) (e₂ : β ≃ γ) : Equiv.conj (e₁.trans e₂) = (Equiv.conj e₁).trans (Equiv.conj e₂)

theorem Equiv.conj_comp {α : Sort u} {β : Sort v} (e : α ≃ β) (f₁ : α → α) (f₂ : α → α) : ↑(Equiv.conj e) (f₁ ∘ f₂) = ↑(Equiv.conj e) f₁ ∘ ↑(Equiv.conj e) f₂

theorem Equiv.eq_comp_symm {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (e : α ≃ β) (f : β → γ) (g : α → γ) : f = g ∘ ↑e.symm ↔ f ∘ ↑e = g

theorem Equiv.comp_symm_eq {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (e : α ≃ β) (f : β → γ) (g : α → γ) : g ∘ ↑e.symm = f ↔ g = f ∘ ↑e

theorem Equiv.eq_symm_comp {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (e : α ≃ β) (f : γ → α) (g : γ → β) : f = ↑e.symm ∘ g ↔ ↑e ∘ f = g

theorem Equiv.symm_comp_eq {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (e : α ≃ β) (f : γ → α) (g : γ → β) : ↑e.symm ∘ g = f ↔ g = ↑e ∘ f

def Equiv.punitEquivPUnit : PUnit.{v} ≃ PUnit.{w}

PUnit sorts in any two universes are equivalent.

noncomputable def Equiv.propEquivBool : Prop ≃ Bool

Prop is noncomputably equivalent to Bool.

def Equiv.arrowPUnitEquivPUnit (α : Sort u_1) : (α → PUnit.{v}) ≃ PUnit.{w}

The sort of maps to PUnit.{v} is equivalent to PUnit.{w}.

@[simp]

theorem Equiv.piSubsingleton_symm_apply {α : Sort u} (β : α → Sort u_1) [Subsingleton α] (a : α) (x : β a) (b : α) : ↑(Equiv.piSubsingleton β a).symm x b = cast (_ : β a = β b) x

@[simp]

theorem Equiv.piSubsingleton_apply {α : Sort u} (β : α → Sort u_1) [Subsingleton α] (a : α) (f : (x : α) → β x) : ↑(Equiv.piSubsingleton β a) f = Function.eval a f

def Equiv.piSubsingleton {α : Sort u} (β : α → Sort u_1) [Subsingleton α] (a : α) : ((a' : α) → β a') ≃ β a

If α is Subsingleton and a : α, then the type of dependent functions Π (i : α), β i is equivalent to β a.

@[simp]

theorem Equiv.funUnique_apply (α : Sort u) (β : Sort u_1) [Unique α] : ↑(Equiv.funUnique α β) = fun f => f default

def Equiv.funUnique (α : Sort u) (β : Sort u_1) [Unique α] : (α → β) ≃ β

If α has a unique term, then the type of function α → β is equivalent to β.

def Equiv.punitArrowEquiv (α : Sort u_1) : (PUnit.{u} → α) ≃ α

The sort of maps from PUnit is equivalent to the codomain.

def Equiv.trueArrowEquiv (α : Sort u_1) : (True → α) ≃ α

The sort of maps from True is equivalent to the codomain.

def Equiv.arrowPUnitOfIsEmpty (α : Sort u_1) (β : Sort u_2) [IsEmpty α] : (α → β) ≃ PUnit.{u}

The sort of maps from a type that IsEmpty is equivalent to PUnit.

def Equiv.emptyArrowEquivPUnit (α : Sort u_1) : (Empty → α) ≃ PUnit.{u}

The sort of maps from Empty is equivalent to PUnit.

def Equiv.pemptyArrowEquivPUnit (α : Sort u_1) : (PEmpty.{u_2} → α) ≃ PUnit.{u}

The sort of maps from PEmpty is equivalent to PUnit.

def Equiv.falseArrowEquivPUnit (α : Sort u_1) : (False → α) ≃ PUnit.{u}

The sort of maps from False is equivalent to PUnit.

@[simp]

theorem Equiv.psigmaEquivSigma_symm_apply {α : Type u_2} (β : α → Type u_1) (a : (i : α) × β i) : ↑(Equiv.psigmaEquivSigma β).symm a = { fst := a.fst, snd := a.snd }

@[simp]

theorem Equiv.psigmaEquivSigma_apply {α : Type u_2} (β : α → Type u_1) (a : (i : α) ×' β i) : ↑(Equiv.psigmaEquivSigma β) a = { fst := a.fst, snd := a.snd }

def Equiv.psigmaEquivSigma {α : Type u_2} (β : α → Type u_1) : (i : α) ×' β i ≃ (i : α) × β i

A PSigma-type is equivalent to the corresponding Sigma-type.

@[simp]

theorem Equiv.psigmaEquivSigmaPLift_symm_apply {α : Sort u_2} (β : α → Sort u_1) (a : (i : PLift α) × PLift (β i.down)) : ↑(Equiv.psigmaEquivSigmaPLift β).symm a = { fst := a.fst.down, snd := a.snd.down }

@[simp]

theorem Equiv.psigmaEquivSigmaPLift_apply {α : Sort u_2} (β : α → Sort u_1) (a : (i : α) ×' β i) : ↑(Equiv.psigmaEquivSigmaPLift β) a = { fst := { down := a.fst }, snd := { down := a.snd } }

def Equiv.psigmaEquivSigmaPLift {α : Sort u_2} (β : α → Sort u_1) : (i : α) ×' β i ≃ (i : PLift α) × PLift (β i.down)

A PSigma-type is equivalent to the corresponding Sigma-type.

@[simp]

theorem Equiv.psigmaCongrRight_apply {α : Sort u} {β₁ : α → Sort u_1} {β₂ : α → Sort u_2} (F : (a : α) → β₁ a ≃ β₂ a) (a : (a : α) ×' β₁ a) : ↑(Equiv.psigmaCongrRight F) a = { fst := a.fst, snd := ↑(F a.fst) a.snd }

def Equiv.psigmaCongrRight {α : Sort u} {β₁ : α → Sort u_1} {β₂ : α → Sort u_2} (F : (a : α) → β₁ a ≃ β₂ a) : (a : α) ×' β₁ a ≃ (a : α) ×' β₂ a

A family of equivalences Π a, β₁ a ≃ β₂ a generates an equivalence between Σ' a, β₁ a and Σ' a, β₂ a.

theorem Equiv.psigmaCongrRight_trans {α : Sort u_4} {β₁ : α → Sort u_1} {β₂ : α → Sort u_2} {β₃ : α → Sort u_3} (F : (a : α) → β₁ a ≃ β₂ a) (G : (a : α) → β₂ a ≃ β₃ a) : (Equiv.psigmaCongrRight F).trans (Equiv.psigmaCongrRight G) = Equiv.psigmaCongrRight fun a => (F a).trans (G a)

theorem Equiv.psigmaCongrRight_symm {α : Sort u_3} {β₁ : α → Sort u_1} {β₂ : α → Sort u_2} (F : (a : α) → β₁ a ≃ β₂ a) : (Equiv.psigmaCongrRight F).symm = Equiv.psigmaCongrRight fun a => (F a).symm

theorem Equiv.psigmaCongrRight_refl {α : Sort u_2} {β : α → Sort u_1} : (Equiv.psigmaCongrRight fun a => Equiv.refl (β a)) = Equiv.refl ((a : α) ×' β a)

@[simp]

theorem Equiv.sigmaCongrRight_apply {α : Type u_3} {β₁ : α → Type u_1} {β₂ : α → Type u_2} (F : (a : α) → β₁ a ≃ β₂ a) (a : (a : α) × β₁ a) : ↑(Equiv.sigmaCongrRight F) a = { fst := a.fst, snd := ↑(F a.fst) a.snd }

def Equiv.sigmaCongrRight {α : Type u_3} {β₁ : α → Type u_1} {β₂ : α → Type u_2} (F : (a : α) → β₁ a ≃ β₂ a) : (a : α) × β₁ a ≃ (a : α) × β₂ a

A family of equivalences Π a, β₁ a ≃ β₂ a generates an equivalence between Σ a, β₁ a and Σ a, β₂ a.

theorem Equiv.sigmaCongrRight_trans {α : Type u_4} {β₁ : α → Type u_1} {β₂ : α → Type u_2} {β₃ : α → Type u_3} (F : (a : α) → β₁ a ≃ β₂ a) (G : (a : α) → β₂ a ≃ β₃ a) : (Equiv.sigmaCongrRight F).trans (Equiv.sigmaCongrRight G) = Equiv.sigmaCongrRight fun a => (F a).trans (G a)

theorem Equiv.sigmaCongrRight_symm {α : Type u_3} {β₁ : α → Type u_1} {β₂ : α → Type u_2} (F : (a : α) → β₁ a ≃ β₂ a) : (Equiv.sigmaCongrRight F).symm = Equiv.sigmaCongrRight fun a => (F a).symm

theorem Equiv.sigmaCongrRight_refl {α : Type u_2} {β : α → Type u_1} : (Equiv.sigmaCongrRight fun a => Equiv.refl (β a)) = Equiv.refl ((a : α) × β a)

def Equiv.psigmaEquivSubtype {α : Type v} (P : α → Prop) : (i : α) ×' P i ≃ Subtype P

A PSigma with Prop fibers is equivalent to the subtype.

def Equiv.sigmaPLiftEquivSubtype {α : Type v} (P : α → Prop) : (i : α) × PLift (P i) ≃ Subtype P

A Sigma with PLift fibers is equivalent to the subtype.

def Equiv.sigmaULiftPLiftEquivSubtype {α : Type v} (P : α → Prop) : (i : α) × ULift.{u_1, 0} (PLift (P i)) ≃ Subtype P

A Sigma with fun i ↦ ULift (PLift (P i)) fibers is equivalent to { x // P x }. Variant of sigmaPLiftEquivSubtype.

@[reducible]

def Equiv.Perm.sigmaCongrRight {α : Type u_1} {β : α → Type u_2} (F : (a : α) → Equiv.Perm (β a)) : Equiv.Perm ((a : α) × β a)

A family of permutations Π a, Perm (β a) generates a permutation Perm (Σ a, β₁ a).

@[simp]

theorem Equiv.Perm.sigmaCongrRight_trans {α : Type u_1} {β : α → Type u_2} (F : (a : α) → Equiv.Perm (β a)) (G : (a : α) → Equiv.Perm (β a)) : (Equiv.Perm.sigmaCongrRight F).trans (Equiv.Perm.sigmaCongrRight G) = Equiv.Perm.sigmaCongrRight fun a => (F a).trans (G a)

@[simp]

theorem Equiv.Perm.sigmaCongrRight_symm {α : Type u_1} {β : α → Type u_2} (F : (a : α) → Equiv.Perm (β a)) : (Equiv.Perm.sigmaCongrRight F).symm = Equiv.Perm.sigmaCongrRight fun a => (F a).symm

@[simp]

theorem Equiv.Perm.sigmaCongrRight_refl {α : Type u_1} {β : α → Type u_2} : (Equiv.Perm.sigmaCongrRight fun a => Equiv.refl (β a)) = Equiv.refl ((a : α) × β a)

@[simp]

theorem Equiv.sigmaCongrLeft_apply {α₂ : Type u_1} {α₁ : Type u_2} {β : α₂ → Type u_3} (e : α₁ ≃ α₂) (a : (a : α₁) × β (↑e a)) : ↑(Equiv.sigmaCongrLeft e) a = { fst := ↑e a.fst, snd := a.snd }

def Equiv.sigmaCongrLeft {α₂ : Type u_1} {α₁ : Type u_2} {β : α₂ → Type u_3} (e : α₁ ≃ α₂) : (a : α₁) × β (↑e a) ≃ (a : α₂) × β a

An equivalence f : α₁ ≃ α₂ generates an equivalence between Σ a, β (f a) and Σ a, β a.

def Equiv.sigmaCongrLeft' {α₁ : Type u_1} {α₂ : Type u_2} {β : α₁ → Type u_3} (f : α₁ ≃ α₂) : (a : α₁) × β a ≃ (a : α₂) × β (↑f.symm a)

Transporting a sigma type through an equivalence of the base

def Equiv.sigmaCongr {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : α₁ → Type u_3} {β₂ : α₂ → Type u_4} (f : α₁ ≃ α₂) (F : (a : α₁) → β₁ a ≃ β₂ (↑f a)) : Sigma β₁ ≃ Sigma β₂

Transporting a sigma type through an equivalence of the base and a family of equivalences of matching fibers

@[simp]

theorem Equiv.sigmaEquivProd_apply (α : Type u_1) (β : Type u_2) (a : (_ : α) × β) : ↑(Equiv.sigmaEquivProd α β) a = (a.fst, a.snd)

@[simp]

theorem Equiv.sigmaEquivProd_symm_apply (α : Type u_1) (β : Type u_2) (a : α × β) : ↑(Equiv.sigmaEquivProd α β).symm a = { fst := a.fst, snd := a.snd }

def Equiv.sigmaEquivProd (α : Type u_1) (β : Type u_2) : (_ : α) × β ≃ α × β

Sigma type with a constant fiber is equivalent to the product.

def Equiv.sigmaEquivProdOfEquiv {α : Type u_1} {β : Type u_2} {β₁ : α → Type u_3} (F : (a : α) → β₁ a ≃ β) : Sigma β₁ ≃ α × β

If each fiber of a Sigma type is equivalent to a fixed type, then the sigma type is equivalent to the product.

def Equiv.sigmaAssoc {α : Type u_1} {β : α → Type u_2} (γ : (a : α) → β a → Type u_3) : (ab : (a : α) × β a) × γ ab.fst ab.snd ≃ (a : α) × (b : β a) × γ a b

Dependent product of types is associative up to an equivalence.

theorem Equiv.exists_unique_congr {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (f : α ≃ β) (h : ∀ {x : α}, p x ↔ q (↑f x)) : (∃! x, p x) ↔ ∃! y, q y

theorem Equiv.exists_unique_congr_left' {α : Sort u} {β : Sort v} {p : α → Prop} (f : α ≃ β) : (∃! x, p x) ↔ ∃! y, p (↑f.symm y)

theorem Equiv.exists_unique_congr_left {α : Sort u} {β : Sort v} {p : β → Prop} (f : α ≃ β) : (∃! x, p (↑f x)) ↔ ∃! y, p y

theorem Equiv.forall_congr {α : Sort u} {β : Sort v} {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

theorem Equiv.forall_congr' {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (f : α ≃ β) (h : ∀ {x : β}, p (↑f.symm x) ↔ q x) : ((x : α) → p x) ↔ (y : β) → q y

theorem Equiv.forall₂_congr {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} {p : α₁ → β₁ → Prop} {q : α₂ → β₂ → Prop} (eα : α₁ ≃ α₂) (eβ : β₁ ≃ β₂) (h : ∀ {x : α₁} {y : β₁}, p x y ↔ q (↑eα x) (↑eβ y)) : ((x : α₁) → (y : β₁) → p x y) ↔ (x : α₂) → (y : β₂) → q x y

theorem Equiv.forall₂_congr' {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} {p : α₁ → β₁ → Prop} {q : α₂ → β₂ → Prop} (eα : α₁ ≃ α₂) (eβ : β₁ ≃ β₂) (h : ∀ {x : α₂} {y : β₂}, p (↑eα.symm x) (↑eβ.symm y) ↔ q x y) : ((x : α₁) → (y : β₁) → p x y) ↔ (x : α₂) → (y : β₂) → q x y

theorem Equiv.forall₃_congr {α₁ : Sort u_1} {β₁ : Sort u_2} {γ₁ : Sort u_3} {α₂ : Sort u_4} {β₂ : Sort u_5} {γ₂ : Sort u_6} {p : α₁ → β₁ → γ₁ → Prop} {q : α₂ → β₂ → γ₂ → Prop} (eα : α₁ ≃ α₂) (eβ : β₁ ≃ β₂) (eγ : γ₁ ≃ γ₂) (h : ∀ {x : α₁} {y : β₁} {z : γ₁}, p x y z ↔ q (↑eα x) (↑eβ y) (↑eγ z)) : ((x : α₁) → (y : β₁) → (z : γ₁) → p x y z) ↔ (x : α₂) → (y : β₂) → (z : γ₂) → q x y z

theorem Equiv.forall₃_congr' {α₁ : Sort u_1} {β₁ : Sort u_2} {γ₁ : Sort u_3} {α₂ : Sort u_4} {β₂ : Sort u_5} {γ₂ : Sort u_6} {p : α₁ → β₁ → γ₁ → Prop} {q : α₂ → β₂ → γ₂ → Prop} (eα : α₁ ≃ α₂) (eβ : β₁ ≃ β₂) (eγ : γ₁ ≃ γ₂) (h : ∀ {x : α₂} {y : β₂} {z : γ₂}, p (↑eα.symm x) (↑eβ.symm y) (↑eγ.symm z) ↔ q x y z) : ((x : α₁) → (y : β₁) → (z : γ₁) → p x y z) ↔ (x : α₂) → (y : β₂) → (z : γ₂) → q x y z

theorem Equiv.forall_congr_left' {α : Sort u} {β : Sort v} {p : α → Prop} (f : α ≃ β) : ((x : α) → p x) ↔ (y : β) → p (↑f.symm y)

theorem Equiv.forall_congr_left {α : Sort u} {β : Sort v} {p : β → Prop} (f : α ≃ β) : ((x : α) → p (↑f x)) ↔ (y : β) → p y

theorem Equiv.exists_congr_left {α : Sort u_1} {β : Sort u_2} (f : α ≃ β) {p : α → Prop} : (∃ a, p a) ↔ ∃ b, p (↑f.symm b)

def Quot.congr {α : Sort u} {β : Sort v} {ra : α → α → Prop} {rb : β → β → Prop} (e : α ≃ β) (eq : ∀ (a₁ a₂ : α), ra a₁ a₂ ↔ rb (↑e a₁) (↑e a₂)) : Quot ra ≃ Quot rb

An equivalence e : α ≃ β generates an equivalence between quotient spaces, if ra a₁ a₂ ↔ rb (e a₁) (e a₂).

@[simp]

theorem Quot.congr_mk {α : Sort u} {β : Sort v} {ra : α → α → Prop} {rb : β → β → Prop} (e : α ≃ β) (eq : ∀ (a₁ a₂ : α), ra a₁ a₂ ↔ rb (↑e a₁) (↑e a₂)) (a : α) : ↑(Quot.congr e eq) (Quot.mk ra a) = Quot.mk rb (↑e a)

def Quot.congrRight {α : Sort u} {r : α → α → Prop} {r' : α → α → Prop} (eq : ∀ (a₁ a₂ : α), r a₁ a₂ ↔ r' a₁ a₂) : Quot r ≃ Quot r'

Quotients are congruent on equivalences under equality of their relation. An alternative is just to use rewriting with eq, but then computational proofs get stuck.

def Quot.congrLeft {α : Sort u} {β : Sort v} {r : α → α → Prop} (e : α ≃ β) : Quot r ≃ Quot fun b b' => r (↑e.symm b) (↑e.symm b')

An equivalence e : α ≃ β generates an equivalence between the quotient space of α by a relation ra and the quotient space of β by the image of this relation under e.

def Quotient.congr {α : Sort u} {β : Sort v} {ra : Setoid α} {rb : Setoid β} (e : α ≃ β) (eq : ∀ (a₁ a₂ : α), Setoid.r a₁ a₂ ↔ Setoid.r (↑e a₁) (↑e a₂)) : Quotient ra ≃ Quotient rb

An equivalence e : α ≃ β generates an equivalence between quotient spaces, if ra a₁ a₂ ↔ rb (e a₁) (e a₂).

@[simp]

theorem Quotient.congr_mk {α : Sort u} {β : Sort v} {ra : Setoid α} {rb : Setoid β} (e : α ≃ β) (eq : ∀ (a₁ a₂ : α), Setoid.r a₁ a₂ ↔ Setoid.r (↑e a₁) (↑e a₂)) (a : α) : ↑(Quotient.congr e eq) (Quotient.mk ra a) = Quotient.mk rb (↑e a)

def Quotient.congrRight {α : Sort u} {r : Setoid α} {r' : Setoid α} (eq : ∀ (a₁ a₂ : α), Setoid.r a₁ a₂ ↔ Setoid.r a₁ a₂) : Quotient r ≃ Quotient r'

Quotients are congruent on equivalences under equality of their relation. An alternative is just to use rewriting with eq, but then computational proofs get stuck.