mathlib documentation

data.equiv.basic

Equivalence between types #

In this file we define two types:

Then we define

Tags #

equivalence, congruence, bijective map

source
@[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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@[protected, 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
source
theorem equiv.coe_fn_injective {α : Sort u} {β : Sort v} :
function.injective (λ (e : α ≃ β) (x : α), ⇑e x)

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

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@[protected, simp, norm_cast]
theorem equiv.coe_inj {α : Sort u} {β : Sort v} {e₁ e₂ : α ≃ β} :
⇑e₁ = ⇑e₂ ↔ 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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@[protected]
theorem equiv.congr_arg {α : Sort u} {β : Sort v} {f : α ≃ β} {x x' : α} :
x = x' → ⇑f x = ⇑f x'
source
@[protected]
theorem equiv.congr_fun {α : Sort u} {β : Sort v} {f g : α ≃ β} (h : f = g) (x : α) :
⇑f x = ⇑g x
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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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@[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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@[protected]
theorem equiv.perm.congr_arg {α : Sort u} {f : equiv.perm α} {x x' : α} :
x = x' → ⇑f x = ⇑f x'
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@[protected]
theorem equiv.perm.congr_fun {α : Sort u} {f g : equiv.perm α} (h : 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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@[protected]
def equiv.refl (α : Sort u_1) :
α ≃ α

Any type is equivalent to itself.

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@[protected, instance]
def equiv.inhabited' {α : Sort u} :
inhabited (α ≃ α)
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@[protected]
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.simps.symm_apply {α : Sort u} {β : Sort v} (e : α ≃ β) :
β → α

See Note [custom simps projection]

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@[simp]
theorem function.involutive.to_equiv_apply {α : Sort u} (f : α → α) (h : function.involutive f) (ᾰ : α) :
⇑(function.involutive.to_equiv f h) ᾰ = f ᾰ
source
@[simp]
theorem function.involutive.to_equiv_symm_apply {α : Sort u} (f : α → α) (h : function.involutive f) (ᾰ : α) :
⇑((function.involutive.to_equiv f h).symm) ᾰ = f ᾰ
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@[protected]
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.

Equations
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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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@[protected]
theorem equiv.injective {α : Sort u} {β : Sort v} (e : α ≃ β) :
function.injective ⇑e
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@[protected]
theorem equiv.surjective {α : Sort u} {β : Sort v} (e : α ≃ β) :
function.surjective ⇑e
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@[protected]
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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@[protected]
theorem equiv.subsingleton {α : Sort u} {β : Sort v} (e : α ≃ β) [subsingleton β] :
subsingleton α
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@[protected]
theorem equiv.subsingleton.symm {α : Sort u} {β : Sort v} (e : α ≃ β) [subsingleton α] :
subsingleton β
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theorem equiv.subsingleton_iff {α : Sort u} {β : Sort v} (e : α ≃ β) :
subsingleton α ↔ subsingleton β
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@[protected, instance]
def equiv.equiv_subsingleton_cod {α : Sort u} {β : Sort v} [subsingleton β] :
subsingleton (α ≃ β)
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@[protected, instance]
def equiv.equiv_subsingleton_dom {α : Sort u} {β : Sort v} [subsingleton α] :
subsingleton (α ≃ β)
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@[protected, instance]
def equiv.perm_subsingleton {α : Sort u} [subsingleton α] :
subsingleton (equiv.perm α)
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theorem equiv.perm.subsingleton_eq_refl {α : Sort u} [subsingleton α] (e : equiv.perm α) :
e = equiv.refl α
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@[protected]
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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@[protected]
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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@[protected]
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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@[protected]
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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@[simp]
theorem equiv.perm.coe_subsingleton {α : Type u_1} [subsingleton α] (e : equiv.perm α) :
⇑e = 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, nolint]
theorem equiv.symm_symm_apply {α : Sort u} {β : Sort v} (f : α ≃ β) (b : α) :
⇑(f.symm.symm) b = ⇑f b
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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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@[simp]
theorem equiv.cast_symm {α β : Sort u_1} (h : α = β) :
(equiv.cast h).symm = equiv.cast _
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@[simp]
theorem equiv.cast_refl {α : Sort u_1} (h : α = α := rfl) :
equiv.cast h = equiv.refl α
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@[simp]
theorem equiv.cast_trans {α β γ : Sort u_1} (h : α = β) (h2 : β = γ) :
(equiv.cast h).trans (equiv.cast h2) = equiv.cast _
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theorem equiv.cast_eq_iff_heq {α β : Sort u_1} (h : α = β) {a : α} {b : β} :
⇑(equiv.cast h) a = b ↔ a == b
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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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@[simp]
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
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@[simp]
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
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@[simp]
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
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@[simp]
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
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@[simp]
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
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@[simp]
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
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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.

Equations
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@[simp]
theorem equiv.equiv_congr_refl {α : Sort u_1} {β : Sort u_2} :
(equiv.refl α).equiv_congr (equiv.refl β) = equiv.refl (α ≃ β)
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@[simp]
theorem equiv.equiv_congr_symm {α : Sort u} {β : Sort v} {γ : Sort w} {δ : Sort u_1} (ab : α ≃ β) (cd : γ ≃ δ) :
(ab.equiv_congr cd).symm = ab.symm.equiv_congr cd.symm
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@[simp]
theorem equiv.equiv_congr_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) :
(ab.equiv_congr de).trans (bc.equiv_congr ef) = (ab.trans bc).equiv_congr (de.trans ef)
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@[simp]
theorem equiv.equiv_congr_refl_left {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} (bg : β ≃ γ) (e : α ≃ β) :
⇑((equiv.refl α).equiv_congr bg) e = e.trans bg
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@[simp]
theorem equiv.equiv_congr_refl_right {α : Sort u_1} {β : Sort u_2} (ab e : α ≃ β) :
⇑(ab.equiv_congr (equiv.refl β)) e = ab.symm.trans e
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@[simp]
theorem equiv.equiv_congr_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) :
⇑(⇑(ab.equiv_congr cd) e) x = ⇑cd (⇑e (⇑(ab.symm) x))
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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.perm_congr_def {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : equiv.perm α') :
⇑(e.perm_congr) p = (e.symm.trans p).trans e
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@[simp]
theorem equiv.perm_congr_refl {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') :
⇑(e.perm_congr) (equiv.refl α') = equiv.refl β'
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@[simp]
theorem equiv.perm_congr_symm {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') :
e.perm_congr.symm = e.symm.perm_congr
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@[simp]
theorem equiv.perm_congr_apply {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : equiv.perm α') (x : β') :
⇑(⇑(e.perm_congr) p) x = ⇑e (⇑p (⇑(e.symm) x))
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theorem equiv.perm_congr_symm_apply {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p : equiv.perm β') (x : α') :
⇑(⇑(e.perm_congr.symm) p) x = ⇑(e.symm) (⇑p (⇑e x))
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theorem equiv.perm_congr_trans {α' : Type u_1} {β' : Type u_2} (e : α' ≃ β') (p p' : equiv.perm α') :
equiv.trans (⇑(e.perm_congr) p) (⇑(e.perm_congr) p') = ⇑(e.perm_congr) (equiv.trans p p')
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@[protected]
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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@[protected]
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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@[simp]
theorem equiv.symm_image_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) :
⇑(e.symm) '' (⇑e '' s) = s
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theorem equiv.eq_image_iff_symm_image_eq {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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@[simp]
theorem equiv.image_symm_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set β) :
⇑e '' (⇑(e.symm) '' s) = s
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@[simp]
theorem equiv.image_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set β) :
⇑e '' (⇑e ⁻¹' s) = s
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@[simp]
theorem equiv.preimage_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) :
⇑e ⁻¹' (⇑e '' s) = s
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@[protected]
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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@[simp]
theorem equiv.symm_preimage_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set β) :
⇑(e.symm) ⁻¹' (⇑e ⁻¹' s) = s
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@[simp]
theorem equiv.preimage_symm_preimage {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) :
⇑e ⁻¹' (⇑(e.symm) ⁻¹' s) = s
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@[simp]
theorem equiv.preimage_subset {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s t : set β) :
⇑e ⁻¹' s ⊆ ⇑e ⁻¹' t ↔ s ⊆ t
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@[simp]
theorem equiv.image_subset {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s t : set α) :
⇑e '' s ⊆ ⇑e '' t ↔ s ⊆ t
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@[simp]
theorem equiv.image_eq_iff_eq {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s t : set α) :
⇑e '' s = ⇑e '' t ↔ s = t
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theorem equiv.preimage_eq_iff_eq_image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set β) (t : set α) :
⇑e ⁻¹' s = t ↔ s = ⇑e '' t
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theorem equiv.eq_preimage_iff_image_eq {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) (t : set β) :
s = ⇑e ⁻¹' t ↔ ⇑e '' s = t
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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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@[simp]
theorem equiv.ulift_symm_apply {α : Type v} :
⇑(equiv.ulift.symm) = ulift.up
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@[protected]
def equiv.ulift {α : Type v} :
ulift α ≃ α

ulift α is equivalent to α.

Equations
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@[simp]
theorem equiv.ulift_apply {α : Type v} :
⇑equiv.ulift = ulift.down
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@[simp]
theorem equiv.plift_apply {α : Sort u} :
⇑equiv.plift = plift.down
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@[protected]
def equiv.plift {α : Sort u} :
plift α ≃ α

plift α is equivalent to α.

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@[simp]
theorem equiv.plift_symm_apply {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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@[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) (ᾰ : α₂) :
⇑(e₁.arrow_congr e₂) f ᾰ = (⇑e₂ ∘ f ∘ ⇑(e₁.symm)) ᾰ
source
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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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)
source
@[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')
source
@[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
source
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} (hα : α₁ ≃ α₂) (hβ : β₁ ≃ β₂) (f : α₁ → β₁) (ᾰ : α₂) :
⇑(hα.arrow_congr' hβ) f ᾰ = ⇑hβ (f (⇑(hα.symm) ᾰ))
source
@[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)
source
@[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')
source
@[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
source
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.

Equations
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@[simp]
theorem equiv.conj_apply {α : Sort u} {β : Sort v} (e : α ≃ β) (f : α → α) (ᾰ : β) :
⇑(e.conj) f ᾰ = ⇑e (f (⇑(e.symm) ᾰ))
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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
source
@[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
source
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
source
theorem equiv.semiconj_conj {α₁ : Type u_1} {β₁ : Type u_2} (e : α₁ ≃ β₁) (f : α₁ → α₁) :
function.semiconj ⇑e f (⇑(e.conj) f)
source
theorem equiv.semiconj₂_conj {α₁ : Type u_1} {β₁ : Type u_2} (e : α₁ ≃ β₁) (f : α₁ → α₁ → α₁) :
function.semiconj₂ ⇑e f (⇑(e.arrow_congr e.conj) f)
source
@[protected, instance]
def equiv.arrow_congr.is_associative {α₁ : Type u_1} {β₁ : Type u_2} (e : α₁ ≃ β₁) (f : α₁ → α₁ → α₁) [is_associative α₁ f] :
is_associative β₁ (⇑(e.arrow_congr (e.arrow_congr e)) f)
source
@[protected, instance]
def equiv.arrow_congr.is_idempotent {α₁ : Type u_1} {β₁ : Type u_2} (e : α₁ ≃ β₁) (f : α₁ → α₁ → α₁) [is_idempotent α₁ f] :
is_idempotent β₁ (⇑(e.arrow_congr (e.arrow_congr e)) f)
source
@[protected, instance]
def equiv.arrow_congr.is_left_cancel {α₁ : Type u_1} {β₁ : Type u_2} (e : α₁ ≃ β₁) (f : α₁ → α₁ → α₁) [is_left_cancel α₁ f] :
is_left_cancel β₁ (⇑(e.arrow_congr (e.arrow_congr e)) f)
source
@[protected, instance]
def equiv.arrow_congr.is_right_cancel {α₁ : Type u_1} {β₁ : Type u_2} (e : α₁ ≃ β₁) (f : α₁ → α₁ → α₁) [is_right_cancel α₁ f] :
is_right_cancel β₁ (⇑(e.arrow_congr (e.arrow_congr e)) f)
source
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.true_arrow_equiv (α : Sort u_1) :
(true → α) ≃ α

The sort of maps from true 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.

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

Equations
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@[simp]
theorem equiv.prod_comm_apply (α : Type u_1) (β : Type u_2) (ᾰ : α × β) :
⇑(equiv.prod_comm α β) ᾰ = ᾰ.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
source
@[simp]
theorem equiv.prod_assoc_symm_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)
source
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.

Equations
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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)
source
theorem equiv.prod_assoc_preimage {α : Type u_1} {β : Type u_2} {γ : Type u_3} {s : set α} {t : set β} {u : set γ} :
⇑(equiv.prod_assoc α β γ) ⁻¹' s.prod (t.prod u) = (s.prod t).prod u
source
@[simp]
theorem equiv.curry_symm_apply (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
⇑((equiv.curry α β γ).symm) = function.uncurry
source
def equiv.curry (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
(α × β → γ) ≃ (α → β → γ)

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

Equations
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@[simp]
theorem equiv.curry_apply (α : Type u_1) (β : Type u_2) (γ : Type u_3) :
⇑(equiv.curry α β γ) = function.curry
source
def equiv.prod_punit (α : Type u_1) :
α × punit ≃ α

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

Equations
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@[simp]
theorem equiv.prod_punit_apply (α : Type u_1) (p : α × punit) :
⇑(equiv.prod_punit α) p = p.fst
source
@[simp]
theorem equiv.prod_punit_symm_apply (α : Type u_1) (a : α) :
⇑((equiv.prod_punit α).symm) a = (a, punit.star)
source
def equiv.punit_prod (α : Type u_1) :
punit × α ≃ α

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

Equations
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@[simp]
theorem equiv.punit_prod_symm_apply (α : Type u_1) (ᾰ : α) :
⇑((equiv.punit_prod α).symm) ᾰ = (punit.star, ᾰ)
source
@[simp]
theorem equiv.punit_prod_apply (α : Type u_1) (ᾰ : punit × α) :
⇑(equiv.punit_prod α) ᾰ = ᾰ.snd
source
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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@[simp]
theorem equiv.sum_congr_apply {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (ea : α₁ ≃ α₂) (eb : β₁ ≃ β₂) (ᾰ : α₁ ⊕ β₁) :
⇑(ea.sum_congr eb) ᾰ = sum.map ⇑ea ⇑eb ᾰ
source
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'.

Equations
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@[simp]
theorem equiv.sum_congr_trans {α₁ : Type u_1} {α₂ : Type u_2} {β₁ : Type u_3} {β₂ : Type u_4} {γ₁ : Type u_5} {γ₂ : Type u_6} (e : α₁ ≃ β₁) (f : α₂ ≃ β₂) (g : β₁ ≃ γ₁) (h : β₂ ≃ γ₂) :
(e.sum_congr f).trans (g.sum_congr h) = (e.trans g).sum_congr (f.trans h)
source
@[simp]
theorem equiv.sum_congr_symm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} (e : α ≃ β) (f : γ ≃ δ) :
(e.sum_congr f).symm = e.symm.sum_congr f.symm
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@[simp]
theorem equiv.sum_congr_refl {α : Type u_1} {β : Type u_2} :
(equiv.refl α).sum_congr (equiv.refl β) = equiv.refl (α ⊕ β)
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def equiv.perm.sum_congr {α : Type u_1} {β : Type u_2} (ea : equiv.perm α) (eb : equiv.perm β) :
equiv.perm (α ⊕ β)

Combine a permutation of α and of β into a permutation of α ⊕ β.

Equations
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@[simp]
theorem equiv.perm.sum_congr_apply {α : Type u_1} {β : Type u_2} (ea : equiv.perm α) (eb : equiv.perm β) (x : α ⊕ β) :
⇑(ea.sum_congr eb) x = sum.map ⇑ea ⇑eb x
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@[simp]
theorem equiv.perm.sum_congr_trans {α : Type u_1} {β : Type u_2} (e : equiv.perm α) (f : equiv.perm β) (g : equiv.perm α) (h : equiv.perm β) :
equiv.trans (e.sum_congr f) (g.sum_congr h) = equiv.perm.sum_congr (equiv.trans e g) (equiv.trans f h)
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@[simp]
theorem equiv.perm.sum_congr_symm {α : Type u_1} {β : Type u_2} (e : equiv.perm α) (f : equiv.perm β) :
equiv.symm (e.sum_congr f) = equiv.perm.sum_congr (equiv.symm e) (equiv.symm f)
source
@[simp]
theorem equiv.perm.sum_congr_refl {α : Type u_1} {β : Type u_2} :
equiv.perm.sum_congr (equiv.refl α) (equiv.refl β) = equiv.refl (α ⊕ β)
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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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@[simp]
theorem equiv.sum_comm_apply (α : Type u_1) (β : Type u_2) (ᾰ : α ⊕ β) :
⇑(equiv.sum_comm α β) ᾰ = ᾰ.swap
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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.

Equations
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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.

Equations
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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
source
@[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)
source
@[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.

Equations
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@[simp]
theorem equiv.sum_empty_symm_apply (α : Type u_1) (val : α) :
⇑((equiv.sum_empty α).symm) val = sum.inl val
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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.

Equations
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@[simp]
theorem equiv.empty_sum_symm_apply (α : Type u_1) (ᾰ : α) :
⇑((equiv.empty_sum α).symm) ᾰ = sum.inr ᾰ
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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.

Equations
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@[simp]
theorem equiv.sum_pempty_symm_apply (α : Type u_1) (val : α) :
⇑((equiv.sum_pempty α).symm) val = sum.inl val
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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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@[simp]
theorem equiv.pempty_sum_symm_apply (α : Type u_1) (ᾰ : α) :
⇑((equiv.pempty_sum α).symm) ᾰ = sum.inr ᾰ
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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.

Equations
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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 punit.star
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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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@[simp]
theorem equiv.option_equiv_sum_punit_coe {α : Type u_1} (a : α) :
⇑(equiv.option_equiv_sum_punit α) ↑a = sum.inl a
source
@[simp]
theorem equiv.option_equiv_sum_punit_symm_inl {α : Type u_1} (a : α) :
⇑((equiv.option_equiv_sum_punit α).symm) (sum.inl a) = ↑a
source
@[simp]
theorem equiv.option_equiv_sum_punit_symm_inr {α : Type u_1} (a : punit) :
⇑((equiv.option_equiv_sum_punit α).symm) (sum.inr a) = none
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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) :
α ⊕ β ≃ Σ (b : bool), cond b α β

α ⊕ β is equivalent to a sigma-type over bool. Note that this definition assumes α and β to be types from the same universe, so it cannot by used directly to transfer theorems about sigma types to theorems about sum types. In many cases one can use ulift to work around this difficulty.

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

Equations
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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)
source
@[simp]
theorem equiv.sigma_preimage_equiv_symm_apply_snd_coe {α : Type u_1} {β : Type u_2} (f : α → β) (x : α) :
↑((⇑((equiv.sigma_preimage_equiv f).symm) x).snd) = x
source
def equiv.set_prod_equiv_sigma {α : Type u_1} {β : Type u_2} (s : set (α × β)) :
↥s ≃ Σ (x : α), ↥{y : β | (x, y) ∈ s}

A set s in α × β is equivalent to the sigma-type Σ x, {y | (x, y) ∈ s}.

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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
source
@[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
source
@[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⟩
source
@[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⟩
source
def equiv.subtype_congr {α : Type u_1} {p q : α → Prop} [decidable_pred p] [decidable_pred q] (e : {x // p x} ≃ {x // q x}) (f : {x // ¬p x} ≃ {x // ¬q x}) :
equiv.perm α

Combines an equiv between two subtypes with an equiv between their complements to form a permutation.

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def equiv.perm.subtype_congr {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) :
equiv.perm ε

Combining permutations on ε that permute only inside or outside the subtype split induced by p : ε → Prop constructs a permutation on ε.

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theorem equiv.perm.subtype_congr.apply {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) (a : ε) :
⇑(ep.subtype_congr en) a = dite (p a) (λ (h : p a), ↑(⇑ep ⟨a, h⟩)) (λ (h : ¬p a), ↑(⇑en ⟨a, h⟩))
source
@[simp]
theorem equiv.perm.subtype_congr.left_apply {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) {a : ε} (h : p a) :
⇑(ep.subtype_congr en) a = ↑(⇑ep ⟨a, h⟩)
source
@[simp]
theorem equiv.perm.subtype_congr.left_apply_subtype {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) (a : {a // p a}) :
⇑(ep.subtype_congr en) ↑a = ↑(⇑ep a)
source
@[simp]
theorem equiv.perm.subtype_congr.right_apply {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) {a : ε} (h : ¬p a) :
⇑(ep.subtype_congr en) a = ↑(⇑en ⟨a, h⟩)
source
@[simp]
theorem equiv.perm.subtype_congr.right_apply_subtype {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) (a : {a // ¬p a}) :
⇑(ep.subtype_congr en) ↑a = ↑(⇑en a)
source
@[simp]
theorem equiv.perm.subtype_congr.refl {ε : Type u_1} {p : ε → Prop} [decidable_pred p] :
equiv.perm.subtype_congr (equiv.refl {a // p a}) (equiv.refl {a // ¬p a}) = equiv.refl ε
source
@[simp]
theorem equiv.perm.subtype_congr.symm {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep : equiv.perm {a // p a}) (en : equiv.perm {a // ¬p a}) :
equiv.symm (ep.subtype_congr en) = equiv.perm.subtype_congr (equiv.symm ep) (equiv.symm en)
source
@[simp]
theorem equiv.perm.subtype_congr.trans {ε : Type u_1} {p : ε → Prop} [decidable_pred p] (ep ep' : equiv.perm {a // p a}) (en en' : equiv.perm {a // ¬p a}) :
equiv.trans (ep.subtype_congr en) (ep'.subtype_congr en') = equiv.perm.subtype_congr (equiv.trans ep ep') (equiv.trans en en')
source
@[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) (a : α) :
↑(⇑((equiv.subtype_preimage p x₀).symm) x) a = dite (p a) (λ (h : p a), x₀ ⟨a, h⟩) (λ (h : ¬p a), x ⟨a, h⟩)
source
@[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}) (a : {a // ¬p a}) :
⇑(equiv.subtype_preimage p x₀) x a = ↑x ↑a
source
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.

Equations
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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⟩
source
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.

Equations
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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
source
@[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)
source
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 → Type 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).

This is sigma.curry and sigma.uncurry together as an equiv.

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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.

Equations
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@[simp]
theorem equiv.psigma_equiv_sigma_symm_apply {α : Type u_1} (β : α → Type u_2) (a : Σ (i : α), β i) :
⇑((equiv.psigma_equiv_sigma β).symm) a = ⟨a.fst, a.snd⟩
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@[simp]
theorem equiv.psigma_equiv_sigma_apply {α : Type u_1} (β : α → Type u_2) (a : Σ' (i : α), β i) :
⇑(equiv.psigma_equiv_sigma β) a = ⟨a.fst, a.snd⟩
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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) (a : Σ (a : α), β₁ a) :
⇑(equiv.sigma_congr_right F) a = ⟨a.fst, ⇑(F a.fst) a.snd⟩
source
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.

Equations
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@[simp]
theorem equiv.sigma_congr_right_trans {α : Type u_1} {β₁ : α → Type u_2} {β₂ : α → Type u_3} {β₃ : α → Type u_4} (F : Π (a : α), β₁ a ≃ β₂ a) (G : Π (a : α), β₂ a ≃ β₃ a) :
(equiv.sigma_congr_right F).trans (equiv.sigma_congr_right G) = equiv.sigma_congr_right (λ (a : α), (F a).trans (G a))
source
@[simp]
theorem equiv.sigma_congr_right_symm {α : Type u_1} {β₁ : α → Type u_2} {β₂ : α → Type u_3} (F : Π (a : α), β₁ a ≃ β₂ a) :
(equiv.sigma_congr_right F).symm = equiv.sigma_congr_right (λ (a : α), (F a).symm)
source
@[simp]
theorem equiv.sigma_congr_right_refl {α : Type u_1} {β : α → Type u_2} :
equiv.sigma_congr_right (λ (a : α), equiv.refl (β a)) = equiv.refl (Σ (a : α), β a)
source
def equiv.perm.sigma_congr_right {α : Type u_1} {β : α → Type u_2} (F : Π (a : α), equiv.perm (β a)) :
equiv.perm (Σ (a : α), β a)

A family of permutations Π a, perm (β a) generates a permuation perm (Σ a, β₁ a).

Equations
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@[simp]
theorem equiv.perm.sigma_congr_right_trans {α : Type u_1} {β : α → Type u_2} (F G : Π (a : α), equiv.perm (β a)) :
equiv.trans (equiv.perm.sigma_congr_right F) (equiv.perm.sigma_congr_right G) = equiv.perm.sigma_congr_right (λ (a : α), equiv.trans (F a) (G a))
source
@[simp]
theorem equiv.perm.sigma_congr_right_symm {α : Type u_1} {β : α → Type u_2} (F : Π (a : α), equiv.perm (β a)) :
equiv.symm (equiv.perm.sigma_congr_right F) = equiv.perm.sigma_congr_right (λ (a : α), equiv.symm (F a))
source
@[simp]
theorem equiv.perm.sigma_congr_right_refl {α : Type u_1} {β : α → Type u_2} :
equiv.perm.sigma_congr_right (λ (a : α), equiv.refl (β a)) = equiv.refl (Σ (a : α), β a)
source
@[simp]
theorem equiv.sigma_congr_left_apply {α₁ : Type u_1} {α₂ : Type u_2} {β : α₂ → Type u_3} (e : α₁ ≃ α₂) (a : Σ (a : α₁), β (⇑e a)) :
⇑(e.sigma_congr_left) a = ⟨⇑e a.fst, a.snd⟩
source
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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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.

Equations
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@[simp]
theorem equiv.sigma_equiv_prod_apply (α : Type u_1) (β : Type u_2) (a : Σ (_x : α), β) :
⇑(equiv.sigma_equiv_prod α β) a = (a.fst, a.snd)
source
@[simp]
theorem equiv.sigma_equiv_prod_symm_apply (α : Type u_1) (β : Type u_2) (a : α × β) :
⇑((equiv.sigma_equiv_prod α β).symm) a = ⟨a.fst, a.snd⟩
source
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.prod_congr_left {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) :
β₁ × α₁ ≃ β₂ × α₁

A family of equivalences Π (a : α₁), β₁ ≃ β₂ generates an equivalence between β₁ × α₁ and β₂ × α₁.

Equations
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@[simp]
theorem equiv.prod_congr_left_apply {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) (b : β₁) (a : α₁) :
⇑(equiv.prod_congr_left e) (b, a) = (⇑(e a) b, a)
source
theorem equiv.prod_congr_refl_right {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : β₁ ≃ β₂) :
e.prod_congr (equiv.refl α₁) = equiv.prod_congr_left (λ (_x : α₁), e)
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def equiv.prod_congr_right {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) :
α₁ × β₁ ≃ α₁ × β₂

A family of equivalences Π (a : α₁), β₁ ≃ β₂ generates an equivalence between α₁ × β₁ and α₁ × β₂.

Equations
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@[simp]
theorem equiv.prod_congr_right_apply {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) (a : α₁) (b : β₁) :
⇑(equiv.prod_congr_right e) (a, b) = (a, ⇑(e a) b)
source
theorem equiv.prod_congr_refl_left {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : β₁ ≃ β₂) :
(equiv.refl α₁).prod_congr e = equiv.prod_congr_right (λ (_x : α₁), e)
source
@[simp]
theorem equiv.prod_congr_left_trans_prod_comm {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) :
(equiv.prod_congr_left e).trans (equiv.prod_comm β₂ α₁) = (equiv.prod_comm β₁ α₁).trans (equiv.prod_congr_right e)
source
@[simp]
theorem equiv.prod_congr_right_trans_prod_comm {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) :
(equiv.prod_congr_right e).trans (equiv.prod_comm α₁ β₂) = (equiv.prod_comm α₁ β₁).trans (equiv.prod_congr_left e)
source
theorem equiv.sigma_congr_right_sigma_equiv_prod {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) :
(equiv.sigma_congr_right e).trans (equiv.sigma_equiv_prod α₁ β₂) = (equiv.sigma_equiv_prod α₁ β₁).trans (equiv.prod_congr_right e)
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theorem equiv.sigma_equiv_prod_sigma_congr_right {α₁ : Type u_1} {β₁ : Type u_2} {β₂ : Type u_3} (e : α₁ → β₁ ≃ β₂) :
(equiv.sigma_equiv_prod α₁ β₁).symm.trans (equiv.sigma_congr_right e) = (equiv.prod_congr_right e).trans (equiv.sigma_equiv_prod α₁ β₂).symm
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@[simp]
theorem equiv.prod_shear_symm_apply {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : α₁ → β₁ ≃ β₂) :
⇑((e₁.prod_shear e₂).symm) = λ (y : α₂ × β₂), (⇑(e₁.symm) y.fst, ⇑((e₂ (⇑(e₁.symm) y.fst)).symm) y.snd)
source
@[simp]
theorem equiv.prod_shear_apply {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : α₁ → β₁ ≃ β₂) :
⇑(e₁.prod_shear e₂) = λ (x : α₁ × β₁), (⇑e₁ x.fst, ⇑(e₂ x.fst) x.snd)
source
def equiv.prod_shear {α₁ : Type u_1} {β₁ : Type u_2} {α₂ : Type u_3} {β₂ : Type u_4} (e₁ : α₁ ≃ α₂) (e₂ : α₁ → β₁ ≃ β₂) :
α₁ × β₁ ≃ α₂ × β₂

A variation on equiv.prod_congr where the equivalence in the second component can depend on the first component. A typical example is a shear mapping, explaining the name of this declaration.

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def equiv.perm.prod_extend_right {α₁ : Type u_1} {β₁ : Type u_2} [decidable_eq α₁] (a : α₁) (e : equiv.perm β₁) :
equiv.perm (α₁ × β₁)

prod_extend_right a e extends e : perm β to perm (α × β) by sending (a, b) to (a, e b) and keeping the other (a', b) fixed.

Equations
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@[simp]
theorem equiv.perm.prod_extend_right_apply_eq {α₁ : Type u_1} {β₁ : Type u_2} [decidable_eq α₁] (a : α₁) (e : equiv.perm β₁) (b : β₁) :
⇑(equiv.perm.prod_extend_right a e) (a, b) = (a, ⇑e b)
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theorem equiv.perm.prod_extend_right_apply_ne {α₁ : Type u_1} {β₁ : Type u_2} [decidable_eq α₁] (e : equiv.perm β₁) {a a' : α₁} (h : a' ≠ a) (b : β₁) :
⇑(equiv.perm.prod_extend_right a e) (a', b) = (a', b)
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theorem equiv.perm.eq_of_prod_extend_right_ne {α₁ : Type u_1} {β₁ : Type u_2} [decidable_eq α₁] {e : equiv.perm β₁} {a a' : α₁} {b : β₁} (h : ⇑(equiv.perm.prod_extend_right a e) (a', b) ≠ (a', b)) :
a' = a
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@[simp]
theorem equiv.perm.fst_prod_extend_right {α₁ : Type u_1} {β₁ : Type u_2} [decidable_eq α₁] (a : α₁) (e : equiv.perm β₁) (ab : α₁ × β₁) :
(⇑(equiv.perm.prod_extend_right a e) ab).fst = ab.fst
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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.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.

Equations
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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)
source
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.

Equations
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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)
source
@[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)
source
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.bool_to_equiv_prod (α : Type u) :
(bool → α) ≃ α × α

The function type bool → α is equivalent to α × α.

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@[simp]
theorem equiv.bool_to_equiv_prod_apply {α : Type u} (f : bool → α) :
⇑(equiv.bool_to_equiv_prod α) f = (f ff, f tt)
source
@[simp]
theorem equiv.bool_to_equiv_prod_symm_apply_ff {α : Type u} (p : α × α) :
⇑((equiv.bool_to_equiv_prod α).symm) p ff = p.fst
source
@[simp]
theorem equiv.bool_to_equiv_prod_symm_apply_tt {α : Type u} (p : α × α) :
⇑((equiv.bool_to_equiv_prod α).symm) p tt = p.snd
source
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_equiv {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}. For the statement where α = β, that is, e : perm α, see perm.subtype_perm.

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@[simp]
theorem equiv.subtype_equiv_refl {α : Sort u} {p : α → Prop} (h : (∀ (a : α), p a ↔ p (⇑(equiv.refl α) a)) := _) :
(equiv.refl α).subtype_equiv h = equiv.refl {a // p a}
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@[simp]
theorem equiv.subtype_equiv_symm {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (e : α ≃ β) (h : ∀ (a : α), p a ↔ q (⇑e a)) :
(e.subtype_equiv h).symm = e.symm.subtype_equiv _
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@[simp]
theorem equiv.subtype_equiv_trans {α : Sort u} {β : Sort v} {γ : Sort w} {p : α → Prop} {q : β → Prop} {r : γ → Prop} (e : α ≃ β) (f : β ≃ γ) (h : ∀ (a : α), p a ↔ q (⇑e a)) (h' : ∀ (b : β), q b ↔ r (⇑f b)) :
(e.subtype_equiv h).trans (f.subtype_equiv h') = (e.trans f).subtype_equiv _
source
@[simp]
theorem equiv.subtype_equiv_apply {α : Sort u} {β : Sort v} {p : α → Prop} {q : β → Prop} (e : α ≃ β) (h : ∀ (a : α), p a ↔ q (⇑e a)) (x : {x // p x}) :
⇑(e.subtype_equiv h) x = ⟨⇑e ↑x, _⟩
source
@[simp]
theorem equiv.subtype_equiv_right_symm_apply_coe {α : Sort u} {p q : α → Prop} (e : ∀ (x : α), p x ↔ q x) (y : {b // (λ (x : α), q x) b}) :
↑(⇑((equiv.subtype_equiv_right e).symm) y) = ↑y
source
def equiv.subtype_equiv_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_equiv_right_apply_coe {α : Sort u} {p q : α → Prop} (e : ∀ (x : α), p x ↔ q x) (x : {a // (λ (x : α), p x) a}) :
↑(⇑(equiv.subtype_equiv_right e) x) = ↑x
source
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_equiv_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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@[simp]
theorem equiv.set_congr_apply {α : Type u_1} {s t : set α} (h : s = t) (x : {a // (λ (a : α), (λ (x : α), x ∈ s) a) a}) :
⇑(equiv.set_congr h) x = ⟨↑x, _⟩
source
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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@[simp]
theorem equiv.subtype_subtype_equiv_subtype_exists_apply {α : Type u} (p : α → Prop) (q : subtype p → Prop) (a : subtype q) :
↑(⇑(equiv.subtype_subtype_equiv_subtype_exists p q) a) = ↑a
source
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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@[simp]
theorem equiv.subtype_subtype_equiv_subtype_inter_apply {α : Type u} (p q : α → Prop) (a : {x // q x.val}) :
↑(⇑(equiv.subtype_subtype_equiv_subtype_inter p q) a) = ↑a
source
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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@[simp]
theorem equiv.subtype_subtype_equiv_subtype_apply {α : Type u} {p q : α → Prop} (h : ∀ (x : α), q x → p x) (a : {x // q x.val}) :
↑(⇑(equiv.subtype_subtype_equiv_subtype h) a) = ↑a
source
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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@[simp]
theorem equiv.subtype_univ_equiv_symm_apply {α : Type u} {p : α → Prop} (h : ∀ (x : α), p x) (x : α) :
⇑((equiv.subtype_univ_equiv h).symm) x = ⟨x, _⟩
source
@[simp]
theorem equiv.subtype_univ_equiv_apply {α : Type u} {p : α → Prop} (h : ∀ (x : α), p x) (x : subtype p) :
⇑(equiv.subtype_univ_equiv h) x = ↑x
source
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
source
@[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
source
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), _⟩
source
@[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)
source
@[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
source
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⟩
source
@[simp]
theorem equiv.image_apply_coe {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) (x : ↥s) :
↑(⇑(e.image s) x) = ⇑e x.val
source
def equiv.image {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) :
↥s ≃ ↥(⇑e '' s)

A set is equivalent to its image under an equivalence.

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@[simp]
theorem equiv.image_symm_apply_coe {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) (y : ↥(⇑e '' s)) :
↑(⇑((e.image s).symm) y) = ⇑(e.symm) y.val
source
@[simp]
theorem equiv.set.univ_symm_apply (α : Type u_1) (a : α) :
⇑((equiv.set.univ α).symm) a = ⟨a, trivial⟩
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@[protected]
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_1) (ᾰ : ↥set.univ) :
⇑(equiv.set.univ α) ᾰ = ↑ᾰ
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@[protected]
def equiv.set.empty (α : Type u_1) :
↥∅ ≃ empty

An empty set is equivalent to the empty type.

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@[protected]
def equiv.set.pempty (α : Type u_1) :
↥∅ ≃ pempty

An empty set is equivalent to a pempty type.

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@[protected]
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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@[protected]
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⟩
source
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⟩
source
@[simp]
theorem equiv.set.union_symm_apply_left {α : Type u_1} {s t : set α} [decidable_pred (λ (x : α), x ∈ s)] (H : s ∩ t ⊆ ∅) (a : ↥s) :
⇑((equiv.set.union H).symm) (sum.inl a) = ⟨↑a, _⟩
source
@[simp]
theorem equiv.set.union_symm_apply_right {α : Type u_1} {s t : set α} [decidable_pred (λ (x : α), x ∈ s)] (H : s ∩ t ⊆ ∅) (a : ↥t) :
⇑((equiv.set.union H).symm) (sum.inr a) = ⟨↑a, _⟩
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@[protected]
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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@[simp]
theorem equiv.set.of_eq_apply {α : Type u} {s t : set α} (h : s = t) (x : ↥s) :
⇑(equiv.set.of_eq h) x = ⟨↑x, _⟩
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@[protected]
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_symm_apply {α : Type u} {s t : set α} (h : s = t) (x : ↥t) :
⇑((equiv.set.of_eq h).symm) x = ⟨↑x, _⟩
source
@[protected]
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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@[simp]
theorem equiv.set.insert_symm_apply_inl {α : Type u} {s : set α} [decidable_pred s] {a : α} (H : a ∉ s) (b : ↥s) :
⇑((equiv.set.insert H).symm) (sum.inl b) = ⟨↑b, _⟩
source
@[simp]
theorem equiv.set.insert_symm_apply_inr {α : Type u} {s : set α} [decidable_pred s] {a : α} (H : a ∉ s) (b : punit) :
⇑((equiv.set.insert H).symm) (sum.inr b) = ⟨a, _⟩
source
@[simp]
theorem equiv.set.insert_apply_left {α : Type u} {s : set α} [decidable_pred s] {a : α} (H : a ∉ s) :
⇑(equiv.set.insert H) ⟨a, _⟩ = sum.inr punit.star
source
@[simp]
theorem equiv.set.insert_apply_right {α : Type u} {s : set α} [decidable_pred s] {a : α} (H : a ∉ s) (b : ↥s) :
⇑(equiv.set.insert H) ⟨↑b, _⟩ = sum.inl b
source
@[protected]
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
source
@[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
source
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⟩
source
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⟩
source
@[simp]
theorem equiv.set.sum_compl_symm_apply {α : Type u_1} {s : set α} [decidable_pred s] {x : ↥s} :
⇑((equiv.set.sum_compl s).symm) ↑x = sum.inl x
source
@[simp]
theorem equiv.set.sum_compl_symm_apply_compl {α : Type u_1} {s : set α} [decidable_pred s] {x : ↥sᶜ} :
⇑((equiv.set.sum_compl s).symm) ↑x = sum.inr x
source
@[protected]
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.

Equations
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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
source
@[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
source
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⟩
source
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, _⟩
source
@[protected]
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.

Equations
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@[protected]
def equiv.set.compl {α : Type u} {β : Type v} {s : set α} {t : set β} [decidable_pred s] [decidable_pred t] (e₀ : ↥s ≃ ↥t) :
{e // ∀ (x : ↥s), ⇑e ↑x = ↑(⇑e₀ x)} ≃ (↥sᶜ ≃ ↥tᶜ)

Given an equivalence e₀ between sets s : set α and t : set β, the set of equivalences e : α ≃ β such that e ↑x = ↑(e₀ x) for each x : s is equivalent to the set of equivalences between sᶜ and tᶜ.

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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) (p : ↥s) :
⇑(equiv.set.image f s H) p = ⟨f ↑p, _⟩
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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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theorem equiv.set.congr_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (s : set α) :
⇑(equiv.set.congr e) s = ⇑e '' s
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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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theorem equiv.set.congr_symm_apply {α : Type u_1} {β : Type u_2} (e : α ≃ β) (t : set β) :
⇑((equiv.set.congr e).symm) t = ⇑(e.symm) '' t
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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.set.powerset {α : Type u_1} (S : set α) :
↥𝒫S ≃ set ↥S

The set 𝒫 S := {x | x ⊆ S} is equivalent to the type set S.

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theorem equiv.of_left_inverse_symm_apply {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : nonempty α → β → α) (hf : ∀ (h : nonempty α), function.left_inverse (f_inv h) f) (b : ↥(set.range f)) :
⇑((equiv.of_left_inverse f f_inv hf).symm) b = f_inv _ ↑b
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def equiv.of_left_inverse {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : nonempty α → β → α) (hf : ∀ (h : nonempty α), function.left_inverse (f_inv h) f) :
α ≃ ↥(set.range f)

If f : α → β has a left-inverse when α is nonempty, then α is computably equivalent to the range of f.

While awkward, the nonempty α hypothesis on f_inv and hf allows this to be used when α is empty too. This hypothesis is absent on analogous definitions on stronger equivs like linear_equiv.of_left_inverse and ring_equiv.of_left_inverse as their typeclass assumptions are already sufficient to ensure non-emptiness.

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theorem equiv.of_left_inverse_apply_coe {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : nonempty α → β → α) (hf : ∀ (h : nonempty α), function.left_inverse (f_inv h) f) (a : α) :
↑(⇑(equiv.of_left_inverse f f_inv hf) a) = f a
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def equiv.of_left_inverse' {α : Sort u_1} {β : Type u_2} (f : α → β) (f_inv : β → α) (hf : function.left_inverse f_inv f) :
α ≃ ↥(set.range f)

If f : α → β has a left-inverse, then α is computably equivalent to the range of f.

Note that if α is empty, no such f_inv exists and so this definition can't be used, unlike the stronger but less convenient of_left_inverse.

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theorem equiv.of_injective_apply {α : Type u_1} {β : Type u_2} (f : α → β) (hf : function.injective f) (a : α) :
⇑(equiv.of_injective f hf) a = ⟨f a, _⟩
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def equiv.of_injective {α : Type u_1} {β : Type u_2} (f : α → β) (hf : function.injective f) :
α ≃ ↥(set.range f)

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

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theorem equiv.apply_of_injective_symm {α : Type u_1} {β : Type u_2} (f : α → β) (hf : function.injective f) (b : ↥(set.range f)) :
f (⇑((equiv.of_injective f hf).symm) b) = ↑b
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theorem equiv.self_comp_of_injective_symm {α : Type u_1} {β : Type u_2} (f : α → β) (hf : function.injective f) :
f ∘ ⇑((equiv.of_injective f hf).symm) = coe
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theorem equiv.of_left_inverse_eq_of_injective {α : Type u_1} {β : Type u_2} (f : α → β) (f_inv : nonempty α → β → α) (hf : ∀ (h : nonempty α), function.left_inverse (f_inv h) f) :
equiv.of_left_inverse f f_inv hf = equiv.of_injective f _
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theorem equiv.of_left_inverse'_eq_of_injective {α : Type u_1} {β : Type u_2} (f : α → β) (f_inv : β → α) (hf : function.left_inverse f_inv f) :
equiv.of_left_inverse' f f_inv hf = equiv.of_injective f _
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def equiv.of_bijective {α : Type 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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