mathlib documentation

data.equiv.basic

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@[nolint]
structure equiv (α : Sort u_1) (β : Sort u_2) :
Sort (max 1 (imax u_1 u_2) (imax u_2 u_1))

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

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def function.involutive.to_equiv {α : Sort u} (f : α → α) (h : function.involutive f) :
α ≃ α

Convert an involutive function f to an equivalence with to_fun = inv_fun = f.

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def equiv.perm (α : Sort u_1) :
Sort (max 1 u_1)

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

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@[instance]
def equiv.has_coe_to_fun {α : Sort u} {β : Sort v} :
has_coe_to_fun (α ≃ β)

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@[simp]
theorem equiv.coe_fn_mk {α : Sort u} {β : Sort v} (f : α → β) (g : β → α) (l : function.left_inverse g f) (r : function.right_inverse g f) :
⇑{to_fun := f, inv_fun := g, left_inv := l, right_inv := r} = f

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theorem equiv.coe_fn_injective {α : Sort u} {β : Sort v} ⦃e₁ e₂ : α ≃ β⦄ (a : ⇑e₁ = ⇑e₂) :
e₁ = e₂

The map coe_fn : (r ≃ s) → (r → s) is injective. We can't use function.injective here but mimic its signature by using ⦃e₁ e₂⦄.

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@[ext]
theorem equiv.ext {α : Sort u} {β : Sort v} {f g : α ≃ β} (H : ∀ (x : α), ⇑f x = ⇑g x) :
f = g

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@[ext]
theorem equiv.perm.ext {α : Sort u} {σ τ : equiv.perm α} (H : ∀ (x : α), ⇑σ x = ⇑τ x) :
σ = τ

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

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theorem equiv.perm.ext_iff {α : Sort u} {σ τ : equiv.perm α} :
σ = τ ↔ ∀ (x : α), ⇑σ x = ⇑τ x

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def equiv.refl (α : Sort u_1) :
α ≃ α

Any type is equivalent to itself.

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

Inverse of an equivalence e : α ≃ β.

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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.

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@[simp]
theorem equiv.to_fun_as_coe {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.to_fun = ⇑e

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@[simp]
theorem equiv.inv_fun_as_coe {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.inv_fun = ⇑(e.symm)

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theorem equiv.injective {α : Sort u} {β : Sort v} (e : α ≃ β) :
function.injective ⇑e

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theorem equiv.surjective {α : Sort u} {β : Sort v} (e : α ≃ β) :
function.surjective ⇑e

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theorem equiv.bijective {α : Sort u} {β : Sort v} (f : α ≃ β) :
function.bijective ⇑f

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@[simp]
theorem equiv.range_eq_univ {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
set.range ⇑e = set.univ

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

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def equiv.decidable_eq {α : Sort u} {β : Sort v} (e : α ≃ β) [decidable_eq β] :
decidable_eq α

Transfer decidable_eq across an equivalence.

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theorem equiv.nonempty_iff_nonempty {α : Sort u} {β : Sort v} (e : α ≃ β) :
nonempty α ↔ nonempty β

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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.

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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.

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

Equivalence between equal types.

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@[simp]
theorem equiv.coe_fn_symm_mk {α : Sort u} {β : Sort v} (f : α → β) (g : β → α) (l : function.left_inverse g f) (r : function.right_inverse g f) :
⇑({to_fun := f, inv_fun := g, left_inv := l, right_inv := r}.symm) = g

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@[simp]
theorem equiv.coe_refl {α : Sort u} :
⇑(equiv.refl α) = id

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theorem equiv.refl_apply {α : Sort u} (x : α) :
⇑(equiv.refl α) x = x

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@[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

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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)

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@[simp]
theorem equiv.apply_symm_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (x : β) :
⇑e (⇑(e.symm) x) = x

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@[simp]
theorem equiv.symm_apply_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (x : α) :
⇑(e.symm) (⇑e x) = x

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@[simp]
theorem equiv.symm_comp_self {α : Sort u} {β : Sort v} (e : α ≃ β) :
⇑(e.symm) ∘ ⇑e = id

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@[simp]
theorem equiv.self_comp_symm {α : Sort u} {β : Sort v} (e : α ≃ β) :
⇑e ∘ ⇑(e.symm) = id

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@[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)

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@[simp]
theorem equiv.apply_eq_iff_eq {α : Sort u} {β : Sort v} (f : α ≃ β) (x y : α) :
⇑f x = ⇑f y ↔ x = y

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theorem equiv.apply_eq_iff_eq_symm_apply {α : Sort u_1} {β : Sort u_2} (f : α ≃ β) (x : α) (y : β) :
⇑f x = y ↔ x = ⇑(f.symm) y

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@[simp]
theorem equiv.cast_apply {α β : Sort u_1} (h : α = β) (x : α) :
⇑(equiv.cast h) x = cast h x

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theorem equiv.symm_apply_eq {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) {x : β} {y : α} :
⇑(e.symm) x = y ↔ x = ⇑e y

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theorem equiv.eq_symm_apply {α : Sort u_1} {β : Sort u_2} (e : α ≃ β) {x : β} {y : α} :
y = ⇑(e.symm) x ↔ ⇑e y = x

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@[simp]
theorem equiv.symm_symm {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.symm.symm = e

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@[simp]
theorem equiv.trans_refl {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.trans (equiv.refl β) = e

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@[simp]
theorem equiv.refl_symm {α : Sort u} :
(equiv.refl α).symm = equiv.refl α

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@[simp]
theorem equiv.refl_trans {α : Sort u} {β : Sort v} (e : α ≃ β) :
(equiv.refl α).trans e = e

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@[simp]
theorem equiv.symm_trans {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.symm.trans e = equiv.refl β

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@[simp]
theorem equiv.trans_symm {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.trans e.symm = equiv.refl α

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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)

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theorem equiv.left_inverse_symm {α : Sort u} {β : Sort v} (f : α ≃ β) :
function.left_inverse ⇑(f.symm) ⇑f

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theorem equiv.right_inverse_symm {α : Sort u} {β : Sort v} (f : α ≃ β) :
function.right_inverse ⇑(f.symm) ⇑f

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def equiv.equiv_congr {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.

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def equiv.perm_congr {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.

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theorem equiv.image_eq_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) :
⇑e '' s = ⇑(e.symm) ⁻¹' s

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theorem equiv.subset_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) (t : set β) :
t ⊆ ⇑e '' s ↔ ⇑(e.symm) '' t ⊆ s

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theorem equiv.symm_image_image {α : Type u_1} {β : Type u_2} (f : α ≃ β) (s : set α) :
⇑(f.symm) '' (⇑f '' s) = s

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theorem equiv.image_compl {α : Type u_1} {β : Type u_2} (f : α ≃ β) (s : set α) :
⇑f '' sᶜ = (⇑f '' s)ᶜ

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@[instance]
def equiv.perm.perm_group {α : Type u} :
group (equiv.perm α)

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@[simp]
theorem equiv.perm.mul_apply {α : Type u} (f g : equiv.perm α) (x : α) :
(⇑f * g) x = ⇑f (⇑g x)

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@[simp]
theorem equiv.perm.one_apply {α : Type u} (x : α) :
⇑1 x = x

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@[simp]
theorem equiv.perm.inv_apply_self {α : Type u} (f : equiv.perm α) (x : α) :
⇑f⁻¹ (⇑f x) = x

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@[simp]
theorem equiv.perm.apply_inv_self {α : Type u} (f : equiv.perm α) (x : α) :
⇑f (⇑f⁻¹ x) = x

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theorem equiv.perm.one_def {α : Type u} :
1 = equiv.refl α

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theorem equiv.perm.mul_def {α : Type u} (f g : equiv.perm α) :
f * g = equiv.trans g f

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theorem equiv.perm.inv_def {α : Type u} (f : equiv.perm α) :
f⁻¹ = equiv.symm f

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def equiv.equiv_empty {α : Sort u} (h : α → false) :
α ≃ empty

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

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def equiv.false_equiv_empty  :
false ≃ empty

false is equivalent to empty.

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def equiv.equiv_pempty {α : Sort v'} (h : α → false) :
α ≃ pempty

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

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def equiv.false_equiv_pempty  :
false ≃ pempty

false is equivalent to pempty.

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def equiv.empty_equiv_pempty  :
empty ≃ pempty

empty is equivalent to pempty.

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def equiv.pempty_equiv_pempty  :
pempty ≃ pempty

pempty types from any two universes are equivalent.

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def equiv.empty_of_not_nonempty {α : Sort u_1} (h : ¬nonempty α) :
α ≃ empty

If α is not nonempty, then it is equivalent to empty.

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def equiv.pempty_of_not_nonempty {α : Sort u_1} (h : ¬nonempty α) :
α ≃ pempty

If α is not nonempty, then it is equivalent to pempty.

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def equiv.prop_equiv_punit {p : Prop} (h : p) :
p ≃ punit

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

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def equiv.true_equiv_punit  :
true ≃ punit

true is equivalent to punit.

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def equiv.ulift {α : Type v} :
ulift α ≃ α

ulift α is equivalent to α.

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@[simp]
theorem equiv.coe_ulift {α : Type v} :
⇑equiv.ulift = ulift.down

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@[simp]
theorem equiv.coe_ulift_symm {α : Type v} :
⇑(equiv.ulift.symm) = ulift.up

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def equiv.plift {α : Sort u} :
plift α ≃ α

plift α is equivalent to α.

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@[simp]
theorem equiv.coe_plift {α : Sort u} :
⇑equiv.plift = plift.down

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@[simp]
theorem equiv.coe_plift_symm {α : Sort u} :
⇑(equiv.plift.symm) = plift.up

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def equiv.of_iff {P Q : Prop} (h : P ↔ Q) :
P ≃ Q

equivalence of propositions is the same as iff

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def equiv.arrow_congr {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.

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@[simp]
theorem equiv.arrow_congr_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) (x : α₂) :
⇑(e₁.arrow_congr e₂) f x = ⇑e₂ (f (⇑(e₁.symm) x))

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theorem equiv.arrow_congr_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) :
⇑(ea.arrow_congr ec) (g ∘ f) = ⇑(eb.arrow_congr ec) g ∘ ⇑(ea.arrow_congr eb) f

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@[simp]
theorem equiv.arrow_congr_refl {α : Sort u_1} {β : Sort u_2} :
(equiv.refl α).arrow_congr (equiv.refl β) = equiv.refl (α → β)

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@[simp]
theorem equiv.arrow_congr_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) :
(e₁.trans e₂).arrow_congr (e₁'.trans e₂') = (e₁.arrow_congr e₁').trans (e₂.arrow_congr e₂')

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@[simp]
theorem equiv.arrow_congr_symm {α₁ : Sort u_1} {β₁ : Sort u_2} {α₂ : Sort u_3} {β₂ : Sort u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) :
(e₁.arrow_congr e₂).symm = e₁.symm.arrow_congr e₂.symm

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def equiv.arrow_congr' {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.arrow_congr in Type, rather than Sort.

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

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@[simp]
theorem equiv.arrow_congr'_apply {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) (f : α₁ → β₁) (x : α₂) :
⇑(e₁.arrow_congr' e₂) f x = ⇑e₂ (f (⇑(e₁.symm) x))

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@[simp]
theorem equiv.arrow_congr'_refl {α : Type u_1} {β : Type u_2} :
(equiv.refl α).arrow_congr' (equiv.refl β) = equiv.refl (α → β)

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@[simp]
theorem equiv.arrow_congr'_trans {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : 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) :
(e₁.trans e₂).arrow_congr' (e₁'.trans e₂') = (e₁.arrow_congr' e₁').trans (e₂.arrow_congr' e₂')

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@[simp]
theorem equiv.arrow_congr'_symm {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) :
(e₁.arrow_congr' e₂).symm = e₁.symm.arrow_congr' e₂.symm

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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.

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@[simp]
theorem equiv.conj_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (f : α → α) (x : β) :
⇑(e.conj) f x = ⇑e (f (⇑(e.symm) x))

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@[simp]
theorem equiv.conj_refl {α : Sort u} :
(equiv.refl α).conj = equiv.refl (α → α)

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@[simp]
theorem equiv.conj_symm {α : Sort u} {β : Sort v} (e : α ≃ β) :
e.conj.symm = e.symm.conj

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@[simp]
theorem equiv.conj_trans {α : Sort u} {β : Sort v} {γ : Sort w} (e₁ : α ≃ β) (e₂ : β ≃ γ) :
(e₁.trans e₂).conj = e₁.conj.trans e₂.conj

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

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def equiv.punit_equiv_punit  :
punit ≃ punit

punit sorts in any two universes are equivalent.

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def equiv.arrow_punit_equiv_punit (α : Sort u_1) :
(α → punit) ≃ punit

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

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def equiv.punit_arrow_equiv (α : Sort u_1) :
(punit → α) ≃ α

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

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def equiv.empty_arrow_equiv_punit (α : Sort u_1) :
(empty → α) ≃ punit

The sort of maps from empty is equivalent to punit.

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def equiv.pempty_arrow_equiv_punit (α : Sort u_1) :
(pempty → α) ≃ punit

The sort of maps from pempty is equivalent to punit.

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def equiv.false_arrow_equiv_punit (α : Sort u_1) :
(false → α) ≃ punit

The sort of maps from false is equivalent to punit.

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def equiv.prod_congr {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) :
α₁ × β₁ ≃ α₂ × β₂

Product of two equivalences. If α₁ ≃ α₂ and β₁ ≃ β₂, then α₁ × β₁ ≃ α₂ × β₂.

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@[simp]
theorem equiv.coe_prod_congr {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) :
⇑(e₁.prod_congr e₂) = prod.map ⇑e₁ ⇑e₂

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@[simp]
theorem equiv.prod_congr_symm {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) :
(e₁.prod_congr e₂).symm = e₁.symm.prod_congr e₂.symm

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def equiv.prod_comm (α : Type u_1) (β : Type u_2) :
α × β ≃ β × α

Type product is commutative up to an equivalence: α × β ≃ β × α.

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@[simp]
theorem equiv.coe_prod_comm (α : Type u_1) (β : Type u_2) :
⇑(equiv.prod_comm α β) = prod.swap

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@[simp]
theorem equiv.prod_comm_symm (α : Type u_1) (β : Type u_2) :
(equiv.prod_comm α β).symm = equiv.prod_comm β α

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def equiv.prod_assoc (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(α × β) × γ ≃ α × β × γ

Type product is associative up to an equivalence.

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@[simp]
theorem equiv.prod_assoc_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : (α × β) × γ) :
⇑(equiv.prod_assoc α β γ) p = (p.fst.fst, p.fst.snd, p.snd)

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@[simp]
theorem equiv.prod_assoc_sym_apply {α : Type u_1} {β : Type u_2} {γ : Type u_3} (p : α × β × γ) :
⇑((equiv.prod_assoc α β γ).symm) p = ((p.fst, p.snd.fst), p.snd.snd)

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def equiv.prod_punit (α : Type u_1) :
α × punit ≃ α

punit is a right identity for type product up to an equivalence.

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@[simp]
theorem equiv.prod_punit_apply {α : Type u_1} (a : α × punit) :
⇑(equiv.prod_punit α) a = a.fst

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def equiv.punit_prod (α : Type u_1) :
punit × α ≃ α

punit is a left identity for type product up to an equivalence.

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@[simp]
theorem equiv.punit_prod_apply {α : Type u_1} (a : punit × α) :
⇑(equiv.punit_prod α) a = a.snd

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def equiv.prod_empty (α : Type u_1) :
α × empty ≃ empty

empty type is a right absorbing element for type product up to an equivalence.

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def equiv.empty_prod (α : Type u_1) :
empty × α ≃ empty

empty type is a left absorbing element for type product up to an equivalence.

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def equiv.prod_pempty (α : Type u_1) :
α × pempty ≃ pempty

pempty type is a right absorbing element for type product up to an equivalence.

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def equiv.pempty_prod (α : Type u_1) :
pempty × α ≃ pempty

pempty type is a left absorbing element for type product up to an equivalence.

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def equiv.psum_equiv_sum (α : Type u_1) (β : Type u_2) :
psum α β ≃ α ⊕ β

psum is equivalent to sum.

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def equiv.sum_congr {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (ea : α₁ ≃ α₂) (eb : β₁ ≃ β₂) :
α₁ ⊕ β₁ ≃ α₂ ⊕ β₂

If α ≃ α' and β ≃ β', then α ⊕ β ≃ α' ⊕ β'.

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@[simp]
theorem equiv.sum_congr_apply {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : β₁ ≃ β₂) (a : α₁ ⊕ β₁) :
⇑(e₁.sum_congr e₂) a = sum.map ⇑e₁ ⇑e₂ a

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@[simp]
theorem equiv.sum_congr_symm {α β γ δ : Type u} (e : α ≃ β) (f : γ ≃ δ) :
(e.sum_congr f).symm = e.symm.sum_congr f.symm

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def equiv.bool_equiv_punit_sum_punit  :
bool ≃ punit ⊕ punit

bool is equivalent the sum of two punits.

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def equiv.Prop_equiv_bool  :
Prop ≃ bool

Prop is noncomputably equivalent to bool.

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def equiv.sum_comm (α : Type u_1) (β : Type u_2) :
α ⊕ β ≃ β ⊕ α

Sum of types is commutative up to an equivalence.

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@[simp]
theorem equiv.sum_comm_apply (α : Type u_1) (β : Type u_2) (a : α ⊕ β) :
⇑(equiv.sum_comm α β) a = a.swap

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@[simp]
theorem equiv.sum_comm_symm (α : Type u_1) (β : Type u_2) :
(equiv.sum_comm α β).symm = equiv.sum_comm β α

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def equiv.sum_assoc (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(α ⊕ β) ⊕ γ ≃ α ⊕ β ⊕ γ

Sum of types is associative up to an equivalence.

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@[simp]
theorem equiv.sum_assoc_apply_in1 {α : Type u_1} {β : Type u_2} {γ : Type u_3} (a : α) :
⇑(equiv.sum_assoc α β γ) (sum.inl (sum.inl a)) = sum.inl a

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@[simp]
theorem equiv.sum_assoc_apply_in2 {α : Type u_1} {β : Type u_2} {γ : Type u_3} (b : β) :
⇑(equiv.sum_assoc α β γ) (sum.inl (sum.inr b)) = sum.inr (sum.inl b)

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@[simp]
theorem equiv.sum_assoc_apply_in3 {α : Type u_1} {β : Type u_2} {γ : Type u_3} (c : γ) :
⇑(equiv.sum_assoc α β γ) (sum.inr c) = sum.inr (sum.inr c)

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def equiv.sum_empty (α : Type u_1) :
α ⊕ empty ≃ α

Sum with empty is equivalent to the original type.

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@[simp]
theorem equiv.sum_empty_apply_inl {α : Type u_1} (a : α) :
⇑(equiv.sum_empty α) (sum.inl a) = a

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def equiv.empty_sum (α : Type u_1) :
empty ⊕ α ≃ α

The sum of empty with any Sort* is equivalent to the right summand.

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@[simp]
theorem equiv.empty_sum_apply_inr {α : Type u_1} (a : α) :
⇑(equiv.empty_sum α) (sum.inr a) = a

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def equiv.sum_pempty (α : Type u_1) :
α ⊕ pempty ≃ α

Sum with pempty is equivalent to the original type.

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@[simp]
theorem equiv.sum_pempty_apply_inl {α : Type u_1} (a : α) :
⇑(equiv.sum_pempty α) (sum.inl a) = a

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def equiv.pempty_sum (α : Type u_1) :
pempty ⊕ α ≃ α

The sum of pempty with any Sort* is equivalent to the right summand.

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@[simp]
theorem equiv.pempty_sum_apply_inr {α : Type u_1} (a : α) :
⇑(equiv.pempty_sum α) (sum.inr a) = a

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def equiv.option_equiv_sum_punit (α : Type u_1) :
option α ≃ α ⊕ punit

option α is equivalent to α ⊕ punit

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@[simp]
theorem equiv.option_equiv_sum_punit_none {α : Type u_1} :
⇑(equiv.option_equiv_sum_punit α) none = sum.inr ()

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@[simp]
theorem equiv.option_equiv_sum_punit_some {α : Type u_1} (a : α) :
⇑(equiv.option_equiv_sum_punit α) (some a) = sum.inl a

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def equiv.option_is_some_equiv (α : Type u_1) :
{x // ↥(x.is_some)} ≃ α

The set of x : option α such that is_some x is equivalent to α.

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def equiv.sum_equiv_sigma_bool (α β : Type u_1) :
α ⊕ β ≃ Σ (b : bool), cond b α β

α ⊕ β is equivalent to a sigma-type over bool.

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def equiv.sigma_preimage_equiv {α : Type u_1} {β : Type u_2} (f : α → β) :
(Σ (y : β), {x // f x = y}) ≃ α

sigma_preimage_equiv f for f : α → β is the natural equivalence between the type of all fibres of f and the total space α.

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@[simp]
theorem equiv.sigma_preimage_equiv_apply {α : Type u_1} {β : Type u_2} (f : α → β) (x : Σ (y : β), {x // f x = y}) :
⇑(equiv.sigma_preimage_equiv f) x = x.snd.val

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@[simp]
theorem equiv.sigma_preimage_equiv_symm_apply_fst {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) :
(⇑((equiv.sigma_preimage_equiv f).symm) a).fst = f a

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@[simp]
theorem equiv.sigma_preimage_equiv_symm_apply_snd_fst {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) :
(⇑((equiv.sigma_preimage_equiv f).symm) a).snd.val = a

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def equiv.sum_compl {α : Type u_1} (p : α → Prop) [decidable_pred p] :
{a // p a} ⊕ {a // ¬p a} ≃ α

For any predicate p on α, the sum of the two subtypes {a // p a} and its complement {a // ¬ p a} is naturally equivalent to α.

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@[simp]
theorem equiv.sum_compl_apply_inl {α : Type u_1} (p : α → Prop) [decidable_pred p] (x : {a // p a}) :
⇑(equiv.sum_compl p) (sum.inl x) = ↑x

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@[simp]
theorem equiv.sum_compl_apply_inr {α : Type u_1} (p : α → Prop) [decidable_pred p] (x : {a // ¬p a}) :
⇑(equiv.sum_compl p) (sum.inr x) = ↑x

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@[simp]
theorem equiv.sum_compl_apply_symm_of_pos {α : Type u_1} (p : α → Prop) [decidable_pred p] (a : α) (h : p a) :
⇑((equiv.sum_compl p).symm) a = sum.inl ⟨a, h⟩

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@[simp]
theorem equiv.sum_compl_apply_symm_of_neg {α : Type u_1} (p : α → Prop) [decidable_pred p] (a : α) (h : ¬p a) :
⇑((equiv.sum_compl p).symm) a = sum.inr ⟨a, h⟩

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def equiv.subtype_preimage {α : Sort u} {β : Sort v} (p : α → Prop) [decidable_pred p] (x₀ : {a // p a} → β) :
{x // x ∘ coe = x₀} ≃ ({a // ¬p a} → β)

For a fixed function x₀ : {a // p a} → β defined on a subtype of α, the subtype of functions x : α → β that agree with x₀ on the subtype {a // p a} is naturally equivalent to the type of functions {a // ¬ p a} → β.

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@[simp]
theorem equiv.subtype_preimage_apply {α : Sort u} {β : Sort v} (p : α → Prop) [decidable_pred p] (x₀ : {a // p a} → β) (x : {x // x ∘ coe = x₀}) :
⇑(equiv.subtype_preimage p x₀) x = λ (a : {a // ¬p a}), ↑x ↑a

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@[simp]
theorem equiv.subtype_preimage_symm_apply_coe {α : Sort u} {β : Sort v} (p : α → Prop) [decidable_pred p] (x₀ : {a // p a} → β) (x : {a // ¬p a} → β) :
↑(⇑((equiv.subtype_preimage p x₀).symm) x) = λ (a : α), dite (p a) (λ (h : p a), x₀ ⟨a, h⟩) (λ (h : ¬p a), x ⟨a, h⟩)

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theorem equiv.subtype_preimage_symm_apply_coe_pos {α : Sort u} {β : Sort v} (p : α → Prop) [decidable_pred p] (x₀ : {a // p a} → β) (x : {a // ¬p a} → β) (a : α) (h : p a) :
↑(⇑((equiv.subtype_preimage p x₀).symm) x) a = x₀ ⟨a, h⟩

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theorem equiv.subtype_preimage_symm_apply_coe_neg {α : Sort u} {β : Sort v} (p : α → Prop) [decidable_pred p] (x₀ : {a // p a} → β) (x : {a // ¬p a} → β) (a : α) (h : ¬p a) :
↑(⇑((equiv.subtype_preimage p x₀).symm) x) a = x ⟨a, h⟩

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def equiv.fun_unique (α : Sort u) (β : Sort v) [unique α] :
(α → β) ≃ β

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

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@[simp]
theorem equiv.fun_unique_apply {α : Sort u} {β : Sort v} [unique α] (f : α → β) :
⇑(equiv.fun_unique α β) f = f (default α)

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@[simp]
theorem equiv.fun_unique_symm_apply {α : Sort u} {β : Sort v} [unique α] (b : β) (a : α) :
⇑((equiv.fun_unique α β).symm) b a = b

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def equiv.Pi_congr_right {α : Sort u_1} {β₁ : α → Sort u_2} {β₂ : α → Sort u_3} (F : Π (a : α), β₁ a ≃ β₂ a) :
(Π (a : α), β₁ a) ≃ Π (a : α), β₂ a

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

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def equiv.Pi_curry {α : Type u_1} {β : α → Type u_2} (γ : Π (a : α), β a → Sort u_3) :
(Π (x : Σ (i : α), β i), γ x.fst x.snd) ≃ Π (a : α) (b : β a), γ a b

Dependent curry equivalence: the type of dependent functions on Σ i, β i is equivalent to the type of dependent functions of two arguments (i.e., functions to the space of functions).

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def equiv.psigma_equiv_sigma {α : Type u_1} (β : α → Type u_2) :
(Σ' (i : α), β i) ≃ Σ (i : α), β i

A psigma-type is equivalent to the corresponding sigma-type.

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@[simp]
theorem equiv.psigma_equiv_sigma_apply {α : Type u_1} (β : α → Type u_2) (x : Σ' (i : α), β i) :
⇑(equiv.psigma_equiv_sigma β) x = ⟨x.fst, x.snd⟩

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@[simp]
theorem equiv.psigma_equiv_sigma_symm_apply {α : Type u_1} (β : α → Type u_2) (x : Σ (i : α), β i) :
⇑((equiv.psigma_equiv_sigma β).symm) x = ⟨x.fst, x.snd⟩

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def equiv.sigma_congr_right {α : Type u_1} {β₁ : α → Type u_2} {β₂ : α → Type u_3} (F : Π (a : α), β₁ a ≃ β₂ a) :
(Σ (a : α), β₁ a) ≃ Σ (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.

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@[simp]
theorem equiv.sigma_congr_right_apply {α : Type u_1} {β₁ : α → Type u_2} {β₂ : α → Type u_3} (F : Π (a : α), β₁ a ≃ β₂ a) (x : Σ (a : α), (λ (a : α), β₁ a) a) :
⇑(equiv.sigma_congr_right F) x = ⟨x.fst, ⇑(F x.fst) x.snd⟩

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@[simp]
theorem equiv.sigma_congr_right_symm_apply {α : Type u_1} {β₁ : α → Type u_2} {β₂ : α → Type u_3} (F : Π (a : α), β₁ a ≃ β₂ a) (x : Σ (a : α), β₂ a) :
⇑((equiv.sigma_congr_right F).symm) x = ⟨x.fst, ⇑((F x.fst).symm) x.snd⟩

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def equiv.sigma_congr_left {α₁ : Type u_1} {α₂ : Type u_2} {β : α₂ → Type u_3} (e : α₁ ≃ α₂) :
(Σ (a : α₁), β (⇑e a)) ≃ Σ (a : α₂), β a

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

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@[simp]
theorem equiv.sigma_congr_left_apply {α₁ : Type u_1} {α₂ : Type u_2} {β : α₂ → Type u_3} (e : α₁ ≃ α₂) (x : Σ (a : α₁), β (⇑e a)) :
⇑(e.sigma_congr_left) x = ⟨⇑e x.fst, x.snd⟩

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def equiv.sigma_congr_left' {α₁ : Type u_1} {α₂ : Type u_2} {β : α₁ → Type u_3} (f : α₁ ≃ α₂) :
(Σ (a : α₁), β a) ≃ Σ (a : α₂), β (⇑(f.symm) a)

Transporting a sigma type through an equivalence of the base

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def equiv.sigma_congr {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : α₁ → Type u_3} {β₂ : α₂ → Type u_4} (f : α₁ ≃ α₂) (F : Π (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

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def equiv.sigma_equiv_prod (α : Type u_1) (β : Type u_2) :
(Σ (_x : α), β) ≃ α × β

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

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@[simp]
theorem equiv.sigma_equiv_prod_apply {α : Type u_1} {β : Type u_2} (x : Σ (_x : α), β) :
⇑(equiv.sigma_equiv_prod α β) x = (x.fst, x.snd)

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@[simp]
theorem equiv.sigma_equiv_prod_symm_apply {α : Type u_1} {β : Type u_2} (x : α × β) :
⇑((equiv.sigma_equiv_prod α β).symm) x = ⟨x.fst, x.snd⟩

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def equiv.sigma_equiv_prod_of_equiv {α : Type u_1} {β : Type u_2} {β₁ : α → Type u_3} (F : Π (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.

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def equiv.arrow_prod_equiv_prod_arrow (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(γ → α × β) ≃ (γ → α) × (γ → β)

The type of functions to a product α × β is equivalent to the type of pairs of functions γ → α and γ → β.

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def equiv.arrow_arrow_equiv_prod_arrow (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(α → β → γ) ≃ (α × β → γ)

Functions α → β → γ are equivalent to functions on α × β.

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def equiv.sum_arrow_equiv_prod_arrow (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(α ⊕ β → γ) ≃ (α → γ) × (β → γ)

The type of functions on a sum type α ⊕ β is equivalent to the type of pairs of functions on α and on β.

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def equiv.sum_prod_distrib (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(α ⊕ β) × γ ≃ α × γ ⊕ β × γ

Type product is right distributive with respect to type sum up to an equivalence.

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@[simp]
theorem equiv.sum_prod_distrib_apply_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (a : α) (c : γ) :
⇑(equiv.sum_prod_distrib α β γ) (sum.inl a, c) = sum.inl (a, c)

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@[simp]
theorem equiv.sum_prod_distrib_apply_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (b : β) (c : γ) :
⇑(equiv.sum_prod_distrib α β γ) (sum.inr b, c) = sum.inr (b, c)

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def equiv.prod_sum_distrib (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
α × (β ⊕ γ) ≃ α × β ⊕ α × γ

Type product is left distributive with respect to type sum up to an equivalence.

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@[simp]
theorem equiv.prod_sum_distrib_apply_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} (a : α) (b : β) :
⇑(equiv.prod_sum_distrib α β γ) (a, sum.inl b) = sum.inl (a, b)

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@[simp]
theorem equiv.prod_sum_distrib_apply_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} (a : α) (c : γ) :
⇑(equiv.prod_sum_distrib α β γ) (a, sum.inr c) = sum.inr (a, c)

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def equiv.sigma_prod_distrib {ι : Type u_1} (α : ι → Type u_2) (β : Type u_3) :
(Σ (i : ι), α i) × β ≃ Σ (i : ι), α i × β

The product of an indexed sum of types (formally, a sigma-type Σ i, α i) by a type β is equivalent to the sum of products Σ i, (α i × β).

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def equiv.bool_prod_equiv_sum (α : Type u) :
bool × α ≃ α ⊕ α

The product bool × α is equivalent to α ⊕ α.

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def equiv.nat_equiv_nat_sum_punit  :
ℕ ≃ ℕ ⊕ punit

The set of natural numbers is equivalent to ℕ ⊕ punit.

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def equiv.nat_sum_punit_equiv_nat  :
ℕ ⊕ punit ≃ ℕ

ℕ ⊕ punit is equivalent to ℕ.

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def equiv.int_equiv_nat_sum_nat  :
ℤ ≃ ℕ ⊕ ℕ

The type of integer numbers is equivalent to ℕ ⊕ ℕ.

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def equiv.list_equiv_of_equiv {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
list α ≃ list β

An equivalence between α and β generates an equivalence between list α and list β.

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def equiv.fin_equiv_subtype (n : ℕ) :
fin n ≃ {m // m < n}

fin n is equivalent to {m // m < n}.

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def equiv.unique_congr {α : Sort u} {β : Sort v} (e : α ≃ β) :
unique α ≃ unique β

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

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def equiv.subtype_congr {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (e : α ≃ β) (h : ∀ (a : α), p a ↔ q (⇑e a)) :
{a // p a} ≃ {b // q b}

If α is equivalent to β and the predicates p : α → Prop and q : β → Prop are equivalent at corresponding points, then {a // p a} is equivalent to {b // q b}.

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def equiv.subtype_congr_right {α : Sort u} {p q : α → Prop} (e : ∀ (x : α), p x ↔ q x) :
{x // p x} ≃ {x // q x}

If two predicates p and q are pointwise equivalent, then {x // p x} is equivalent to {x // q x}.

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@[simp]
theorem equiv.subtype_congr_right_mk {α : Sort u} {p q : α → Prop} (e : ∀ (x : α), p x ↔ q x) {x : α} (h : p x) :
⇑(equiv.subtype_congr_right e) ⟨x, h⟩ = ⟨x, _⟩

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def equiv.subtype_equiv_of_subtype {α : Sort u} {β : Sort v} {p : β → Prop} (e : α ≃ β) :
{a // p (⇑e a)} ≃ {b // p b}

If α ≃ β, then for any predicate p : β → Prop the subtype {a // p (e a)} is equivalent to the subtype {b // p b}.

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def equiv.subtype_equiv_of_subtype' {α : Sort u} {β : Sort v} {p : α → Prop} (e : α ≃ β) :
{a // p a} ≃ {b // p (⇑(e.symm) b)}

If α ≃ β, then for any predicate p : α → Prop the subtype {a // p a} is equivalent to the subtype {b // p (e.symm b)}. This version is used by equiv_rw.

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def equiv.subtype_congr_prop {α : Type u_1} {p q : α → Prop} (h : p = q) :
subtype p ≃ subtype q

If two predicates are equal, then the corresponding subtypes are equivalent.

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def equiv.set_congr {α : Type u_1} {s t : set α} (h : s = t) :
↥s ≃ ↥t

The subtypes corresponding to equal sets are equivalent.

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def equiv.subtype_subtype_equiv_subtype_exists {α : Type u} (p : α → Prop) (q : subtype p → Prop) :
subtype q ≃ {a // ∃ (h : p a), q ⟨a, h⟩}

A subtype of a subtype is equivalent to the subtype of elements satisfying both predicates. This version allows the “inner” predicate to depend on h : p a.

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def equiv.subtype_subtype_equiv_subtype_inter {α : Type u} (p q : α → Prop) :
{x // q x.val} ≃ {x // p x ∧ q x}

A subtype of a subtype is equivalent to the subtype of elements satisfying both predicates.

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def equiv.subtype_subtype_equiv_subtype {α : Type u} {p q : α → Prop} (h : ∀ {x : α}, q x → p x) :
{x // q x.val} ≃ subtype q

If the outer subtype has more restrictive predicate than the inner one, then we can drop the latter.

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def equiv.subtype_univ_equiv {α : Type u} {p : α → Prop} (h : ∀ (x : α), p x) :
subtype p ≃ α

If a proposition holds for all elements, then the subtype is equivalent to the original type.

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def equiv.subtype_sigma_equiv {α : Type u} (p : α → Type v) (q : α → Prop) :
{y // q y.fst} ≃ Σ (x : subtype q), p x.val

A subtype of a sigma-type is a sigma-type over a subtype.

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def equiv.sigma_subtype_equiv_of_subset {α : Type u} (p : α → Type v) (q : α → Prop) (h : ∀ (x : α), p x → q x) :
(Σ (x : subtype q), p ↑x) ≃ Σ (x : α), p x

A sigma type over a subtype is equivalent to the sigma set over the original type, if the fiber is empty outside of the subset

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def equiv.sigma_subtype_preimage_equiv {α : Type u} {β : Type v} (f : α → β) (p : β → Prop) (h : ∀ (x : α), p (f x)) :
(Σ (y : subtype p), {x // f x = ↑y}) ≃ α

If a predicate p : β → Prop is true on the range of a map f : α → β, then Σ y : {y // p y}, {x // f x = y} is equivalent to α.

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def equiv.sigma_subtype_preimage_equiv_subtype {α : Type u} {β : Type v} (f : α → β) {p : α → Prop} {q : β → Prop} (h : ∀ (x : α), p x ↔ q (f x)) :
(Σ (y : subtype q), {x // f x = ↑y}) ≃ subtype p

If for each x we have p x ↔ q (f x), then Σ y : {y // q y}, f ⁻¹' {y} is equivalent to {x // p x}.

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def equiv.pi_equiv_subtype_sigma (ι : Type u_1) (π : ι → Type u_2) :
(Π (i : ι), «π» i) ≃ ↥{f : ι → (Σ (i : ι), «π» i) | ∀ (i : ι), (f i).fst = i}

The pi-type Π i, π i is equivalent to the type of sections f : ι → Σ i, π i of the sigma type such that for all i we have (f i).fst = i.

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def equiv.subtype_pi_equiv_pi {α : Sort u} {β : α → Sort v} {p : Π (a : α), β a → Prop} :
{f // ∀ (a : α), p a (f a)} ≃ Π (a : α), {b // p a b}

The set of functions f : Π a, β a such that for all a we have p a (f a) is equivalent to the set of functions Π a, {b : β a // p a b}.

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def equiv.subtype_prod_equiv_prod {α : Type u} {β : Type v} {p : α → Prop} {q : β → Prop} :
{c // p c.fst ∧ q c.snd} ≃ {a // p a} × {b // q b}

A subtype of a product defined by componentwise conditions is equivalent to a product of subtypes.

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def equiv.subtype_equiv_codomain {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) :
{g // g ∘ coe = f} ≃ Y

The type of all functions X → Y with prescribed values for all x' ≠ x is equivalent to the codomain Y.

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@[simp]
theorem equiv.coe_subtype_equiv_codomain {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) :
⇑(equiv.subtype_equiv_codomain f) = λ (g : {g // g ∘ coe = f}), ↑g x

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@[simp]
theorem equiv.subtype_equiv_codomain_apply {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) (g : {g // g ∘ coe = f}) :
⇑(equiv.subtype_equiv_codomain f) g = ↑g x

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theorem equiv.coe_subtype_equiv_codomain_symm {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) :
⇑((equiv.subtype_equiv_codomain f).symm) = λ (y : Y), ⟨λ (x' : X), dite (x' ≠ x) (λ (h : x' ≠ x), f ⟨x', h⟩) (λ (h : ¬x' ≠ x), y), _⟩

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@[simp]
theorem equiv.subtype_equiv_codomain_symm_apply {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) (y : Y) (x' : X) :
↑(⇑((equiv.subtype_equiv_codomain f).symm) y) x' = dite (x' ≠ x) (λ (h : x' ≠ x), f ⟨x', h⟩) (λ (h : ¬x' ≠ x), y)

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@[simp]
theorem equiv.subtype_equiv_codomain_symm_apply_eq {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) (y : Y) :
↑(⇑((equiv.subtype_equiv_codomain f).symm) y) x = y

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theorem equiv.subtype_equiv_codomain_symm_apply_ne {X : Type u_1} {Y : Type u_2} [decidable_eq X] {x : X} (f : {x' // x' ≠ x} → Y) (y : Y) (x' : X) (h : x' ≠ x) :
↑(⇑((equiv.subtype_equiv_codomain f).symm) y) x' = f ⟨x', h⟩

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def equiv.set.univ (α : Type u_1) :
↥set.univ ≃ α

univ α is equivalent to α.

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@[simp]
theorem equiv.set.univ_apply {α : Type u} (x : ↥set.univ) :
⇑(equiv.set.univ α) x = ↑x

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@[simp]
theorem equiv.set.univ_symm_apply {α : Type u} (x : α) :
⇑((equiv.set.univ α).symm) x = ⟨x, trivial⟩

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def equiv.set.empty (α : Type u_1) :
↥∅ ≃ empty

An empty set is equivalent to the empty type.

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def equiv.set.pempty (α : Type u_1) :
↥∅ ≃ pempty

An empty set is equivalent to a pempty type.

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def equiv.set.union' {α : Type u_1} {s t : set α} (p : α → Prop) [decidable_pred p] (hs : ∀ (x : α), x ∈ s → p x) (ht : ∀ (x : α), x ∈ t → ¬p x) :
↥(s ∪ t) ≃ ↥s ⊕ ↥t

If sets s and t are separated by a decidable predicate, then s ∪ t is equivalent to s ⊕ t.

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def equiv.set.union {α : Type u_1} {s t : set α} [decidable_pred (λ (x : α), x ∈ s)] (H : s ∩ t ⊆ ∅) :
↥(s ∪ t) ≃ ↥s ⊕ ↥t

If sets s and t are disjoint, then s ∪ t is equivalent to s ⊕ t.

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theorem equiv.set.union_apply_left {α : Type u_1} {s t : set α} [decidable_pred (λ (x : α), x ∈ s)] (H : s ∩ t ⊆ ∅) {a : ↥(s ∪ t)} (ha : ↑a ∈ s) :
⇑(equiv.set.union H) a = sum.inl ⟨↑a, ha⟩

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theorem equiv.set.union_apply_right {α : Type u_1} {s t : set α} [decidable_pred (λ (x : α), x ∈ s)] (H : s ∩ t ⊆ ∅) {a : ↥(s ∪ t)} (ha : ↑a ∈ t) :
⇑(equiv.set.union H) a = sum.inr ⟨↑a, ha⟩

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def equiv.set.singleton {α : Type u_1} (a : α) :
↥{a} ≃ punit

A singleton set is equivalent to a punit type.

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def equiv.set.of_eq {α : Type u} {s t : set α} (h : s = t) :
↥s ≃ ↥t

Equal sets are equivalent.

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@[simp]
theorem equiv.set.of_eq_apply {α : Type u} {s t : set α} (h : s = t) (a : ↥s) :
⇑(equiv.set.of_eq h) a = ⟨↑a, _⟩

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@[simp]
theorem equiv.set.of_eq_symm_apply {α : Type u} {s t : set α} (h : s = t) (a : ↥t) :
⇑((equiv.set.of_eq h).symm) a = ⟨↑a, _⟩

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def equiv.set.insert {α : Type u} {s : set α} [decidable_pred s] {a : α} (H : a ∉ s) :
↥(insert a s) ≃ ↥s ⊕ punit

If a ∉ s, then insert a s is equivalent to s ⊕ punit.

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def equiv.set.sum_compl {α : Type u_1} (s : set α) [decidable_pred s] :
↥s ⊕ ↥sᶜ ≃ α

If s : set α is a set with decidable membership, then s ⊕ sᶜ is equivalent to α.

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@[simp]
theorem equiv.set.sum_compl_apply_inl {α : Type u} (s : set α) [decidable_pred s] (x : ↥s) :
⇑(equiv.set.sum_compl s) (sum.inl x) = ↑x

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@[simp]
theorem equiv.set.sum_compl_apply_inr {α : Type u} (s : set α) [decidable_pred s] (x : ↥sᶜ) :
⇑(equiv.set.sum_compl s) (sum.inr x) = ↑x

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theorem equiv.set.sum_compl_symm_apply_of_mem {α : Type u} {s : set α} [decidable_pred s] {x : α} (hx : x ∈ s) :
⇑((equiv.set.sum_compl s).symm) x = sum.inl ⟨x, hx⟩

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theorem equiv.set.sum_compl_symm_apply_of_not_mem {α : Type u} {s : set α} [decidable_pred s] {x : α} (hx : x ∉ s) :
⇑((equiv.set.sum_compl s).symm) x = sum.inr ⟨x, hx⟩

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def equiv.set.sum_diff_subset {α : Type u_1} {s t : set α} (h : s ⊆ t) [decidable_pred s] :
↥s ⊕ ↥(t \ s) ≃ ↥t

sum_diff_subset s t is the natural equivalence between s ⊕ (t \ s) and t, where s and t are two sets.

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@[simp]
theorem equiv.set.sum_diff_subset_apply_inl {α : Type u_1} {s t : set α} (h : s ⊆ t) [decidable_pred s] (x : ↥s) :
⇑(equiv.set.sum_diff_subset h) (sum.inl x) = set.inclusion h x

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@[simp]
theorem equiv.set.sum_diff_subset_apply_inr {α : Type u_1} {s t : set α} (h : s ⊆ t) [decidable_pred s] (x : ↥(t \ s)) :
⇑(equiv.set.sum_diff_subset h) (sum.inr x) = set.inclusion _ x

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theorem equiv.set.sum_diff_subset_symm_apply_of_mem {α : Type u_1} {s t : set α} (h : s ⊆ t) [decidable_pred s] {x : ↥t} (hx : x.val ∈ s) :
⇑((equiv.set.sum_diff_subset h).symm) x = sum.inl ⟨↑x, hx⟩

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theorem equiv.set.sum_diff_subset_symm_apply_of_not_mem {α : Type u_1} {s t : set α} (h : s ⊆ t) [decidable_pred s] {x : ↥t} (hx : x.val ∉ s) :
⇑((equiv.set.sum_diff_subset h).symm) x = sum.inr ⟨↑x, _⟩

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def equiv.set.union_sum_inter {α : Type u} (s t : set α) [decidable_pred s] :
↥(s ∪ t) ⊕ ↥(s ∩ t) ≃ ↥s ⊕ ↥t

If s is a set with decidable membership, then the sum of s ∪ t and s ∩ t is equivalent to s ⊕ t.

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def equiv.set.prod {α : Type u_1} {β : Type u_2} (s : set α) (t : set β) :
↥(s.prod t) ≃ ↥s × ↥t

The set product of two sets is equivalent to the type product of their coercions to types.

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def equiv.set.image_of_inj_on {α : Type u_1} {β : Type u_2} (f : α → β) (s : set α) (H : set.inj_on f s) :
↥s ≃ ↥(f '' s)

If a function f is injective on a set s, then s is equivalent to f '' s.

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def equiv.set.image {α : Type u_1} {β : Type u_2} (f : α → β) (s : set α) (H : function.injective f) :
↥s ≃ ↥(f '' s)

If f is an injective function, then s is equivalent to f '' s.

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@[simp]
theorem equiv.set.image_apply {α : Type u_1} {β : Type u_2} (f : α → β) (s : set α) (H : function.injective f) (a : α) (h : a ∈ s) :
⇑(equiv.set.image f s H) ⟨a, h⟩ = ⟨f a, _⟩

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theorem equiv.set.image_symm_preimage {α : Type u_1} {β : Type u_2} {f : α → β} (hf : function.injective f) (u s : set α) :
(λ (x : ↥(f '' s)), ↑(⇑((equiv.set.image f s hf).symm) x)) ⁻¹' u = coe ⁻¹' (f '' u)

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def equiv.set.range {α : Sort u_1} {β : Type u_2} (f : α → β) (H : function.injective f) :
α ≃ ↥(set.range f)

If f : α → β is an injective function, then α is equivalent to the range of f.

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@[simp]
theorem equiv.set.range_apply {α : Sort u_1} {β : Type u_2} (f : α → β) (H : function.injective f) (a : α) :
⇑(equiv.set.range f H) a = ⟨f a, _⟩

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theorem equiv.set.apply_range_symm {α : Sort u_1} {β : Type u_2} (f : α → β) (H : function.injective f) (b : ↥(set.range f)) :
f (⇑((equiv.set.range f H).symm) b) = ↑b

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def equiv.set.congr {α : Type u_1} {β : Type u_2} (e : α ≃ β) :
set α ≃ set β

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

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def equiv.set.sep {α : Type u} (s : set α) (t : α → Prop) :
↥{x ∈ s | t x} ≃ ↥{x : ↥s | t ↑x}

The set {x ∈ s | t x} is equivalent to the set of x : s such that t x.

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def equiv.of_bijective {α : Sort u_1} {β : Type u_2} (f : α → β) (hf : function.bijective f) :
α ≃ β

If f is a bijective function, then its domain is equivalent to its codomain.

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@[simp]
theorem equiv.coe_of_bijective {α : Sort u_1} {β : Type u_2} {f : α → β} (hf : function.bijective f) :
⇑(equiv.of_bijective f hf) = f

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def equiv.of_injective {α : Sort u_1} {β : Type u_2} (f : α → β) (hf : function.injective f) :
α ≃ ↥(set.range f)

If f is an injective function, then its domain is equivalent to its range.

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@[simp]
theorem equiv.of_injective_apply {α : Sort u_1} {β : Type u_2} (f : α → β) (hf : function.injective f) (x : α) :
⇑(equiv.of_injective f hf) x = ⟨f x, _⟩

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def equiv.subtype_quotient_equiv_quotient_subtype {α : Sort u} (p₁ : α → Prop) [s₁ : setoid α] [s₂ : setoid (subtype p₁)] (p₂ : quotient s₁ → Prop) (hp₂ : ∀ (a : α), p₁ a ↔ p₂ ⟦a⟧) (h : ∀ (x y : subtype p₁), setoid.r x y ↔ ↑x ≈ ↑y) :
{x // p₂ x} ≃ quotient s₂

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def equiv.swap_core {α : Sort u} [decidable_eq α] (a b r : α) :
α

A helper function for equiv.swap.

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theorem equiv.swap_core_self {α : Sort u} [decidable_eq α] (r a : α) :
equiv.swap_core a a r = r

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theorem equiv.swap_core_swap_core {α : Sort u} [decidable_eq α] (r a b : α) :
equiv.swap_core a b (equiv.swap_core a b r) = r

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theorem equiv.swap_core_comm {α : Sort u} [decidable_eq α] (r a b : α) :
equiv.swap_core a b r = equiv.swap_core b a r

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def equiv.swap {α : Sort u} [decidable_eq α] (a b : α) :
equiv.perm α

swap a b is the permutation that swaps a and b and leaves other values as is.

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theorem equiv.swap_self {α : Sort u} [decidable_eq α] (a : α) :
equiv.swap a a = equiv.refl α

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theorem equiv.swap_comm {α : Sort u} [decidable_eq α] (a b : α) :
equiv.swap a b = equiv.swap b a

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theorem equiv.swap_apply_def {α : Sort u} [decidable_eq α] (a b x : α) :
⇑(equiv.swap a b) x = ite (x = a) b (ite (x = b) a x)

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@[simp]
theorem equiv.swap_apply_left {α : Sort u} [decidable_eq α] (a b : α) :
⇑(equiv.swap a b) a = b

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@[simp]
theorem equiv.swap_apply_right {α : Sort u} [decidable_eq α] (a b : α) :
⇑(equiv.swap a b) b = a

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theorem equiv.swap_apply_of_ne_of_ne {α : Sort u} [decidable_eq α] {a b x : α} (a_1 : x ≠ a) (a_2 : x ≠ b) :
⇑(equiv.swap a b) x = x

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@[simp]
theorem equiv.swap_swap {α : Sort u} [decidable_eq α] (a b : α) :
equiv.trans (equiv.swap a b) (equiv.swap a b) = equiv.refl α

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theorem equiv.swap_comp_apply {α : Sort u} [decidable_eq α] {a b x : α} (π : equiv.perm α) :
⇑(equiv.trans «π» (equiv.swap a b)) x = ite (⇑«π» x = a) b (ite (⇑«π» x = b) a (⇑«π» x))

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@[simp]
theorem equiv.swap_inv {α : Type u_1} [decidable_eq α] (x y : α) :
(equiv.swap x y)⁻¹ = equiv.swap x y

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@[simp]
theorem equiv.symm_trans_swap_trans {α : Sort u} {β : Sort v} [decidable_eq α] [decidable_eq β] (a b : α) (e : α ≃ β) :
(e.symm.trans (equiv.swap a b)).trans e = equiv.swap (⇑e a) (⇑e b)

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@[simp]
theorem equiv.swap_mul_self {α : Type u_1} [decidable_eq α] (i j : α) :
(equiv.swap i j) * equiv.swap i j = 1

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@[simp]
theorem equiv.swap_apply_self {α : Type u_1} [decidable_eq α] (i j a : α) :
⇑(equiv.swap i j) (⇑(equiv.swap i j) a) = a

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def equiv.set_value {α : Sort u} {β : Sort v} [decidable_eq α] (f : α ≃ β) (a : α) (b : β) :
α ≃ β

Augment an equivalence with a prescribed mapping f a = b

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@[simp]
theorem equiv.set_value_eq {α : Sort u} {β : Sort v} [decidable_eq α] (f : α ≃ β) (a : α) (b : β) :
⇑(f.set_value a b) a = b

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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

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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

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theorem equiv.forall₂_congr {α₁ : Sort ua1} {α₂ : Sort ua2} {β₁ : Sort ub1} {β₂ : Sort ub2} {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

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theorem equiv.forall₂_congr' {α₁ : Sort ua1} {α₂ : Sort ua2} {β₁ : Sort ub1} {β₂ : Sort ub2} {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

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theorem equiv.forall₃_congr {α₁ : Sort ua1} {α₂ : Sort ua2} {β₁ : Sort ub1} {β₂ : Sort ub2} {γ₁ : Sort ug1} {γ₂ : Sort ug2} {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

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theorem equiv.forall₃_congr' {α₁ : Sort ua1} {α₂ : Sort ua2} {β₁ : Sort ub1} {β₂ : Sort ub2} {γ₁ : Sort ug1} {γ₂ : Sort ug2} {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

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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)

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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

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def equiv.Pi_congr_left' {α : Sort u} {β : Sort v} (P : α → Sort w) (e : α ≃ β) :
(Π (a : α), P a) ≃ Π (b : β), P (⇑(e.symm) b)

Transport dependent functions through an equivalence of the base space.

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@[simp]
theorem equiv.Pi_congr_left'_apply {α : Sort u} {β : Sort v} (P : α → Sort w) (e : α ≃ β) (f : Π (a : α), P a) (b : β) :
⇑(equiv.Pi_congr_left' P e) f b = f (⇑(e.symm) b)

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@[simp]
theorem equiv.Pi_congr_left'_symm_apply {α : Sort u} {β : Sort v} (P : α → Sort w) (e : α ≃ β) (g : Π (b : β), P (⇑(e.symm) b)) (a : α) :
⇑((equiv.Pi_congr_left' P e).symm) g a = _.mpr (g (⇑e a))

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def equiv.Pi_congr_left {α : Sort u} {β : Sort v} (P : β → Sort w) (e : α ≃ β) :
(Π (a : α), P (⇑e a)) ≃ Π (b : β), P b

Transporting dependent functions through an equivalence of the base, expressed as a "simplification".

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def equiv.Pi_congr {α : Sort u} {β : Sort v} {W : α → Sort w} {Z : β → Sort z} (h₁ : α ≃ β) (h₂ : Π (a : α), W a ≃ Z (⇑h₁ a)) :
(Π (a : α), W a) ≃ Π (b : β), Z b

Transport dependent functions through an equivalence of the base spaces and a family of equivalences of the matching fibers.

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def equiv.Pi_congr' {α : Sort u} {β : Sort v} {W : α → Sort w} {Z : β → Sort z} (h₁ : α ≃ β) (h₂ : Π (b : β), W (⇑(h₁.symm) b) ≃ Z b) :
(Π (a : α), W a) ≃ Π (b : β), Z b

Transport dependent functions through an equivalence of the base spaces and a family of equivalences of the matching fibres.

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@[instance]
def ulift.subsingleton {α : Type u_1} [subsingleton α] :
subsingleton (ulift α)

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@[instance]
def plift.subsingleton {α : Sort u_1} [subsingleton α] :
subsingleton (plift α)

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@[instance]
def ulift.decidable_eq {α : Type u_1} [decidable_eq α] :
decidable_eq (ulift α)

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@[instance]
def plift.decidable_eq {α : Sort u_1} [decidable_eq α] :
decidable_eq (plift α)

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def equiv_of_unique_of_unique {α : Sort u} {β : Sort v} [unique α] [unique β] :
α ≃ β

If both α and β are singletons, then α ≃ β.

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def equiv_punit_of_unique {α : Sort u} [unique α] :
α ≃ punit

If α is a singleton, then it is equivalent to any punit.

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def subsingleton_prod_self_equiv {α : Type u_1} [subsingleton α] :
α × α ≃ α

If α is a subsingleton, then it is equivalent to α × α.

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def equiv_of_subsingleton_of_subsingleton {α : Sort u} {β : Sort v} [subsingleton α] [subsingleton β] (f : α → β) (g : β → α) :
α ≃ β

To give an equivalence between two subsingleton types, it is sufficient to give any two functions between them.

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def unique_unique_equiv {α : Sort u} :
unique (unique α) ≃ unique α

unique (unique α) is equivalent to unique α.

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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).

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def quot.congr_right {α : Sort u} {r 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.

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def quot.congr_left {α : Sort u} {β : Sort v} {r : α → α → Prop} (e : α ≃ β) :
quot r ≃ quot (λ (b b' : β), 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.

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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).

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def quotient.congr_right {α : Sort u} {r 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.

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def set.bij_on.equiv {α : Type u_1} {β : Type u_2} {s : set β} (f : α → β) (h : set.bij_on f set.univ s) :
α ≃ ↥s

If a function is a bijection between univ and a set s in the target type, it induces an equivalence between the original type and the type ↑s.

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theorem dite_comp_equiv_update {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : set α} (e : β ≃ ↥s) (v : β → γ) (w : α → γ) (j : β) (x : γ) [decidable_eq β] [decidable_eq α] [Π (j : α), decidable (j ∈ s)] :
(λ (i : α), dite (i ∈ s) (λ (h : i ∈ s), function.update v j x (⇑(e.symm) ⟨i, h⟩)) (λ (h : i ∉ s), w i)) = function.update (λ (i : α), dite (i ∈ s) (λ (h : i ∈ s), v (⇑(e.symm) ⟨i, h⟩)) (λ (h : i ∉ s), w i)) ↑(⇑e j) x

The composition of an updated function with an equiv on a subset can be expressed as an updated function.