Documentation

Mathlib.Data.Quot

Quotient types #

This module extends the core library's treatment of quotient types (Init.Core).

Tags #

quotient

theorem Setoid.ext {α : Sort u_1} {s : Setoid α} {t : Setoid α} :
(∀ (a b : α), Setoid.r a b Setoid.r a b) → s = t
theorem Quot.induction_on {α : Sort u} {r : ααProp} {β : Quot rProp} (q : Quot r) (h : (a : α) → β (Quot.mk r a)) :
β q
instance Quot.instInhabitedQuot {α : Sort u_1} (r : ααProp) [inst : Inhabited α] :
Equations
instance Quot.Subsingleton {α : Sort u_1} {ra : ααProp} [inst : Subsingleton α] :
Equations
def Quot.hrecOn₂ {α : Sort u_1} {β : Sort u_2} {ra : ααProp} {rb : ββProp} {φ : Quot raQuot rbSort u_3} (qa : Quot ra) (qb : Quot rb) (f : (a : α) → (b : β) → φ (Quot.mk ra a) (Quot.mk rb b)) (ca : ∀ {b : β} {a₁ a₂ : α}, ra a₁ a₂HEq (f a₁ b) (f a₂ b)) (cb : ∀ {a : α} {b₁ b₂ : β}, rb b₁ b₂HEq (f a b₁) (f a b₂)) :
φ qa qb

Recursion on two Quotient arguments a and b, result type depends on ⟦a⟧ and ⟦b⟧.

Equations
  • One or more equations did not get rendered due to their size.
def Quot.map {α : Sort u_1} {β : Sort u_2} {ra : ααProp} {rb : ββProp} (f : αβ) (h : (ra rb) f f) :
Quot raQuot rb

Map a function f : α → β→ β such that ra x y implies rb (f x) (f y) to a map Quot ra → Quot rb→ Quot rb.

Equations
def Quot.mapRight {α : Sort u_1} {ra : ααProp} {ra' : ααProp} (h : (a₁ a₂ : α) → ra a₁ a₂ra' a₁ a₂) :
Quot raQuot ra'

If ra is a subrelation of ra', then we have a natural map Quot ra → Quot ra'→ Quot ra'.

Equations
def Quot.factor {α : Type u_1} (r : ααProp) (s : ααProp) (h : (x y : α) → r x ys x y) :
Quot rQuot s

Weaken the relation of a quotient. This is the same as Quot.map id.

Equations
theorem Quot.factor_mk_eq {α : Type u_1} (r : ααProp) (s : ααProp) (h : (x y : α) → r x ys x y) :
theorem Quot.lift_mk {α : Sort u_2} {γ : Sort u_1} {r : ααProp} (f : αγ) (h : ∀ (a₁ a₂ : α), r a₁ a₂f a₁ = f a₂) (a : α) :
Quot.lift f h (Quot.mk r a) = f a
theorem Quot.liftOn_mk {α : Sort u_2} {γ : Sort u_1} {r : ααProp} (a : α) (f : αγ) (h : ∀ (a₁ a₂ : α), r a₁ a₂f a₁ = f a₂) :
Quot.liftOn (Quot.mk r a) f h = f a
@[simp]
theorem Quot.surjective_lift {α : Sort u_2} {γ : Sort u_1} {r : ααProp} {f : αγ} (h : ∀ (a₁ a₂ : α), r a₁ a₂f a₁ = f a₂) :
def Quot.lift₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {r : ααProp} {s : ββProp} (f : αβγ) (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂f a b₁ = f a b₂) (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂f a₁ b = f a₂ b) (q₁ : Quot r) (q₂ : Quot s) :
γ

Descends a function f : α → β → γ→ β → γ→ γ to quotients of α and β.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem Quot.lift₂_mk {α : Sort u_2} {β : Sort u_3} {γ : Sort u_1} {r : ααProp} {s : ββProp} (f : αβγ) (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂f a b₁ = f a b₂) (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂f a₁ b = f a₂ b) (a : α) (b : β) :
Quot.lift₂ f hr hs (Quot.mk r a) (Quot.mk s b) = f a b
def Quot.liftOn₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {r : ααProp} {s : ββProp} (p : Quot r) (q : Quot s) (f : αβγ) (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂f a b₁ = f a b₂) (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂f a₁ b = f a₂ b) :
γ

Descends a function f : α → β → γ→ β → γ→ γ to quotients of α and β and applies it.

Equations
@[simp]
theorem Quot.liftOn₂_mk {α : Sort u_2} {β : Sort u_3} {γ : Sort u_1} {r : ααProp} {s : ββProp} (a : α) (b : β) (f : αβγ) (hr : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂f a b₁ = f a b₂) (hs : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂f a₁ b = f a₂ b) :
Quot.liftOn₂ (Quot.mk r a) (Quot.mk s b) f hr hs = f a b
def Quot.map₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {r : ααProp} {s : ββProp} {t : γγProp} (f : αβγ) (hr : (a : α) → (b₁ b₂ : β) → s b₁ b₂t (f a b₁) (f a b₂)) (hs : (a₁ a₂ : α) → (b : β) → r a₁ a₂t (f a₁ b) (f a₂ b)) (q₁ : Quot r) (q₂ : Quot s) :

Descends a function f : α → β → γ→ β → γ→ γ to quotients of α and β wih values in a quotient of γ.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem Quot.map₂_mk {α : Sort u_2} {β : Sort u_3} {γ : Sort u_1} {r : ααProp} {s : ββProp} {t : γγProp} (f : αβγ) (hr : (a : α) → (b₁ b₂ : β) → s b₁ b₂t (f a b₁) (f a b₂)) (hs : (a₁ a₂ : α) → (b : β) → r a₁ a₂t (f a₁ b) (f a₂ b)) (a : α) (b : β) :
Quot.map₂ f hr hs (Quot.mk r a) (Quot.mk s b) = Quot.mk t (f a b)
def Quot.recOnSubsingleton₂ {α : Sort u_1} {β : Sort u_2} {r : ααProp} {s : ββProp} {φ : Quot rQuot sSort u_3} [h : ∀ (a : α) (b : β), Subsingleton (φ (Quot.mk r a) (Quot.mk s b))] (q₁ : Quot r) (q₂ : Quot s) (f : (a : α) → (b : β) → φ (Quot.mk r a) (Quot.mk s b)) :
φ q₁ q₂

A binary version of Quot.recOnSubsingleton.

Equations
theorem Quot.induction_on₂ {α : Sort u_1} {β : Sort u_2} {r : ααProp} {s : ββProp} {δ : Quot rQuot sProp} (q₁ : Quot r) (q₂ : Quot s) (h : (a : α) → (b : β) → δ (Quot.mk r a) (Quot.mk s b)) :
δ q₁ q₂
theorem Quot.induction_on₃ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {r : ααProp} {s : ββProp} {t : γγProp} {δ : Quot rQuot sQuot tProp} (q₁ : Quot r) (q₂ : Quot s) (q₃ : Quot t) (h : (a : α) → (b : β) → (c : γ) → δ (Quot.mk r a) (Quot.mk s b) (Quot.mk t c)) :
δ q₁ q₂ q₃
instance Quot.lift.decidablePred {α : Sort u_1} (r : ααProp) (f : αProp) (h : ∀ (a b : α), r a bf a = f b) [hf : DecidablePred f] :
Equations
instance Quot.lift₂.decidablePred {α : Sort u_1} {β : Sort u_2} (r : ααProp) (s : ββProp) (f : αβProp) (ha : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂f a b₁ = f a b₂) (hb : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂f a₁ b = f a₂ b) [hf : (a : α) → DecidablePred (f a)] (q₁ : Quot r) :

Note that this provides DecidableRel (Quot.Lift₂ f ha hb) when α = β.

Equations
instance Quot.instDecidableLiftOnProp {α : Sort u_1} (r : ααProp) (q : Quot r) (f : αProp) (h : ∀ (a b : α), r a bf a = f b) [inst : DecidablePred f] :
Equations
instance Quot.instDecidableLiftOn₂Prop {α : Sort u_1} {β : Sort u_2} (r : ααProp) (s : ββProp) (q₁ : Quot r) (q₂ : Quot s) (f : αβProp) (ha : ∀ (a : α) (b₁ b₂ : β), s b₁ b₂f a b₁ = f a b₂) (hb : ∀ (a₁ a₂ : α) (b : β), r a₁ a₂f a₁ b = f a₂ b) [inst : (a : α) → DecidablePred (f a)] :
Decidable (Quot.liftOn₂ q₁ q₂ f ha hb)
Equations

The canonical quotient map into a Quotient.

Equations
  • One or more equations did not get rendered due to their size.
instance Quotient.instInhabitedQuotient {α : Sort u_1} (s : Setoid α) [inst : Inhabited α] :
Equations
def Quotient.hrecOn₂ {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] {φ : Quotient saQuotient sbSort u_3} (qa : Quotient sa) (qb : Quotient sb) (f : (a : α) → (b : β) → φ (Quotient.mk sa a) (Quotient.mk sb b)) (c : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ a₂b₁ b₂HEq (f a₁ b₁) (f a₂ b₂)) :
φ qa qb

Induction on two Quotient arguments a and b, result type depends on ⟦a⟧ and ⟦b⟧.

Equations
def Quotient.map {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] (f : αβ) (h : ((fun x x_1 => x x_1) fun x x_1 => x x_1) f f) :
Quotient saQuotient sb

Map a function f : α → β→ β that sends equivalent elements to equivalent elements to a function Quotient sa → Quotient sb→ Quotient sb. Useful to define unary operations on quotients.

Equations
@[simp]
theorem Quotient.map_mk {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] (f : αβ) (h : ((fun x x_1 => x x_1) fun x x_1 => x x_1) f f) (x : α) :
def Quotient.map₂ {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] {γ : Sort u_3} [sc : Setoid γ] (f : αβγ) (h : ((fun x x_1 => x x_1) (fun x x_1 => x x_1) fun x x_1 => x x_1) f f) :
Quotient saQuotient sbQuotient sc

Map a function f : α → β → γ→ β → γ→ γ that sends equivalent elements to equivalent elements to a function f : Quotient sa → Quotient sb → Quotient sc→ Quotient sb → Quotient sc→ Quotient sc. Useful to define binary operations on quotients.

Equations
  • One or more equations did not get rendered due to their size.
@[simp]
theorem Quotient.map₂_mk {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] {γ : Sort u_3} [sc : Setoid γ] (f : αβγ) (h : ((fun x x_1 => x x_1) (fun x x_1 => x x_1) fun x x_1 => x x_1) f f) (x : α) (y : β) :
Quotient.map₂ f h (Quotient.mk sa x) (Quotient.mk sb y) = Quotient.mk sc (f x y)
instance Quotient.lift.decidablePred {α : Sort u_1} [sa : Setoid α] (f : αProp) (h : ∀ (a b : α), a bf a = f b) [inst : DecidablePred f] :
Equations
instance Quotient.lift₂.decidablePred {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] (f : αβProp) (h : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ a₂b₁ b₂f a₁ b₁ = f a₂ b₂) [hf : (a : α) → DecidablePred (f a)] (q₁ : Quotient sa) :

Note that this provides DecidableRel (Quotient.lift₂ f h) when α = β.

Equations
instance Quotient.instDecidableLiftOnProp {α : Sort u_1} [sa : Setoid α] (q : Quotient sa) (f : αProp) (h : ∀ (a b : α), a bf a = f b) [inst : DecidablePred f] :
Equations
instance Quotient.instDecidableLiftOn₂Prop {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] (q₁ : Quotient sa) (q₂ : Quotient sb) (f : αβProp) (h : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ a₂b₁ b₂f a₁ b₁ = f a₂ b₂) [inst : (a : α) → DecidablePred (f a)] :
Equations
theorem Quot.eq {α : Type u_1} {r : ααProp} {x : α} {y : α} :
Quot.mk r x = Quot.mk r y EqvGen r x y
@[simp]
theorem Quotient.eq {α : Sort u_1} [r : Setoid α] {x : α} {y : α} :
theorem forall_quotient_iff {α : Type u_1} [r : Setoid α] {p : Quotient rProp} :
((a : Quotient r) → p a) (a : α) → p (Quotient.mk r a)
@[simp]
theorem Quotient.lift_mk {α : Sort u_1} {β : Sort u_2} [s : Setoid α] (f : αβ) (h : ∀ (a b : α), a bf a = f b) (x : α) :
@[simp]
theorem Quotient.lift_comp_mk {α : Sort u_1} {β : Sort u_2} [inst : Setoid α] (f : αβ) (h : ∀ (a b : α), a bf a = f b) :
@[simp]
theorem Quotient.lift₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} [inst : Setoid α] [inst : Setoid β] (f : αβγ) (h : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), a₁ b₁a₂ b₂f a₁ a₂ = f b₁ b₂) (a : α) (b : β) :
Quotient.lift₂ f h (Quotient.mk inst a) (Quotient.mk inst b) = f a b
theorem Quotient.liftOn_mk {α : Sort u_1} {β : Sort u_2} [s : Setoid α] (f : αβ) (h : ∀ (a b : α), a bf a = f b) (x : α) :
@[simp]
theorem Quotient.liftOn₂_mk {α : Sort u_1} {β : Sort u_2} [inst : Setoid α] (f : ααβ) (h : ∀ (a₁ a₂ b₁ b₂ : α), a₁ b₁a₂ b₂f a₁ a₂ = f b₁ b₂) (x : α) (y : α) :
Quotient.liftOn₂ (Quotient.mk inst x) (Quotient.mk inst y) f h = f x y
theorem surjective_quot_mk {α : Sort u_1} (r : ααProp) :

Quot.mk r is a surjective function.

Quotient.mk is a surjective function.

noncomputable def Quot.out {α : Sort u_1} {r : ααProp} (q : Quot r) :
α

Choose an element of the equivalence class using the axiom of choice. Sound but noncomputable.

Equations
unsafe def Quot.unquot {α : Sort u_1} {r : ααProp} :
Quot rα

Unwrap the VM representation of a quotient to obtain an element of the equivalence class. Computable but unsound.

Equations
@[simp]
theorem Quot.out_eq {α : Sort u_1} {r : ααProp} (q : Quot r) :
noncomputable def Quotient.out {α : Sort u_1} [s : Setoid α] :
Quotient sα

Choose an element of the equivalence class using the axiom of choice. Sound but noncomputable.

Equations
  • Quotient.out = Quot.out
@[simp]
theorem Quotient.out_eq {α : Sort u_1} [s : Setoid α] (q : Quotient s) :
theorem Quotient.mk_out {α : Sort u_1} [inst : Setoid α] (a : α) :
theorem Quotient.mk_eq_iff_out {α : Sort u_1} [s : Setoid α] {x : α} {y : Quotient s} :
theorem Quotient.eq_mk_iff_out {α : Sort u_1} [s : Setoid α] {x : Quotient s} {y : α} :
@[simp]
theorem Quotient.out_equiv_out {α : Sort u_1} {s : Setoid α} {x : Quotient s} {y : Quotient s} :
theorem Quotient.out_injective {α : Sort u_1} {s : Setoid α} :
Function.Injective Quotient.out
@[simp]
theorem Quotient.out_inj {α : Sort u_1} {s : Setoid α} {x : Quotient s} {y : Quotient s} :
instance piSetoid {ι : Sort u_1} {α : ιSort u_2} [inst : (i : ι) → Setoid (α i)] :
Setoid ((i : ι) → α i)
Equations
  • piSetoid = { r := fun a b => ∀ (i : ι), a i b i, iseqv := (_ : Equivalence fun a b => ∀ (i : ι), a i b i) }
noncomputable def Quotient.choice {ι : Type u_1} {α : ιType u_2} [S : (i : ι) → Setoid (α i)] (f : (i : ι) → Quotient (S i)) :
Quotient inferInstance

Given a function f : Π i, Quotient (S i), returns the class of functions Π i, α i sending each i to an element of the class f i.

Equations
@[simp]
theorem Quotient.choice_eq {ι : Type u_1} {α : ιType u_2} [inst : (i : ι) → Setoid (α i)] (f : (i : ι) → α i) :
(Quotient.choice fun i => Quotient.mk (inst i) (f i)) = Quotient.mk inferInstance f
theorem Quotient.induction_on_pi {ι : Type u_1} {α : ιSort u_2} [s : (i : ι) → Setoid (α i)] {p : ((i : ι) → Quotient (s i)) → Prop} (f : (i : ι) → Quotient (s i)) (h : (a : (i : ι) → α i) → p fun i => Quotient.mk (s i) (a i)) :
p f

Truncation #

theorem true_equivalence {α : Sort u_1} :
Equivalence fun x x => True
def trueSetoid {α : Sort u_1} :

Always-true relation as a Setoid.

Note that in later files the preferred spelling is ⊤ : Setoid α⊤ : Setoid α.

Equations
def Trunc (α : Sort u) :

Trunc α is the quotient of α by the always-true relation. This is related to the propositional truncation in HoTT, and is similar in effect to Nonempty α, but unlike Nonempty α, Trunc α is data, so the VM representation is the same as α, and so this can be used to maintain computability.

Equations
def Trunc.mk {α : Sort u_1} (a : α) :

Constructor for Trunc α

Equations
instance Trunc.instInhabitedTrunc {α : Sort u_1} [inst : Inhabited α] :
Equations
  • Trunc.instInhabitedTrunc = { default := Trunc.mk default }
def Trunc.lift {α : Sort u_1} {β : Sort u_2} (f : αβ) (c : ∀ (a b : α), f a = f b) :
Trunc αβ

Any constant function lifts to a function out of the truncation

Equations
theorem Trunc.ind {α : Sort u_1} {β : Trunc αProp} :
((a : α) → β (Trunc.mk a)) → (q : Trunc α) → β q
theorem Trunc.lift_mk {α : Sort u_2} {β : Sort u_1} (f : αβ) (c : ∀ (a b : α), f a = f b) (a : α) :
Trunc.lift f c (Trunc.mk a) = f a
def Trunc.liftOn {α : Sort u_1} {β : Sort u_2} (q : Trunc α) (f : αβ) (c : ∀ (a b : α), f a = f b) :
β

Lift a constant function on q : Trunc α.

Equations
theorem Trunc.induction_on {α : Sort u_1} {β : Trunc αProp} (q : Trunc α) (h : (a : α) → β (Trunc.mk a)) :
β q
theorem Trunc.exists_rep {α : Sort u_1} (q : Trunc α) :
a, Trunc.mk a = q
theorem Trunc.induction_on₂ {α : Sort u_1} {β : Sort u_2} {C : Trunc αTrunc βProp} (q₁ : Trunc α) (q₂ : Trunc β) (h : (a : α) → (b : β) → C (Trunc.mk a) (Trunc.mk b)) :
C q₁ q₂
theorem Trunc.eq {α : Sort u_1} (a : Trunc α) (b : Trunc α) :
a = b
def Trunc.bind {α : Sort u_1} {β : Sort u_2} (q : Trunc α) (f : αTrunc β) :

The bind operator for the Trunc monad.

Equations
def Trunc.map {α : Sort u_1} {β : Sort u_2} (f : αβ) (q : Trunc α) :

A function f : α → β→ β defines a function map f : Trunc α → Trunc β→ Trunc β.

Equations
def Trunc.rec {α : Sort u_1} {C : Trunc αSort u_2} (f : (a : α) → C (Trunc.mk a)) (h : ∀ (a b : α), (_ : Trunc.mk a = Trunc.mk b)f a = f b) (q : Trunc α) :
C q

Recursion/induction principle for Trunc.

Equations
def Trunc.recOn {α : Sort u_1} {C : Trunc αSort u_2} (q : Trunc α) (f : (a : α) → C (Trunc.mk a)) (h : ∀ (a b : α), (_ : Trunc.mk a = Trunc.mk b)f a = f b) :
C q

A version of Trunc.rec taking q : Trunc α as the first argument.

Equations
def Trunc.recOnSubsingleton {α : Sort u_1} {C : Trunc αSort u_2} [inst : ∀ (a : α), Subsingleton (C (Trunc.mk a))] (q : Trunc α) (f : (a : α) → C (Trunc.mk a)) :
C q

A version of Trunc.recOn assuming the codomain is a Subsingleton.

Equations
noncomputable def Trunc.out {α : Sort u_1} :
Trunc αα

Noncomputably extract a representative of Trunc α (using the axiom of choice).

Equations
  • Trunc.out = Quot.out
@[simp]
theorem Trunc.out_eq {α : Sort u_1} (q : Trunc α) :
theorem Trunc.nonempty {α : Sort u_1} (q : Trunc α) :

Quotient with implicit Setoid #

Versions of quotient definitions and lemmas ending in ' use unification instead of typeclass inference for inferring the Setoid argument. This is useful when there are several different quotient relations on a type, for example quotient groups, rings and modules.

def Quotient.mk'' {α : Sort u_1} {s₁ : Setoid α} (a : α) :

A version of Quotient.mk taking {s : Setoid α} as an implicit argument instead of an instance argument.

Equations
theorem Quotient.surjective_Quotient_mk'' {α : Sort u_1} {s₁ : Setoid α} :
Function.Surjective Quotient.mk''

Quotient.mk'' is a surjective function.

def Quotient.liftOn' {α : Sort u_1} {φ : Sort u_2} {s₁ : Setoid α} (q : Quotient s₁) (f : αφ) (h : ∀ (a b : α), Setoid.r a bf a = f b) :
φ

A version of Quotient.liftOn taking {s : Setoid α} as an implicit argument instead of an instance argument.

Equations
@[simp]
theorem Quotient.liftOn'_mk'' {α : Sort u_1} {φ : Sort u_2} {s₁ : Setoid α} (f : αφ) (h : ∀ (a b : α), Setoid.r a bf a = f b) (x : α) :
@[simp]
theorem Quotient.surjective_liftOn' {α : Sort u_1} {φ : Sort u_2} {s₁ : Setoid α} {f : αφ} (h : ∀ (a b : α), Setoid.r a bf a = f b) :
def Quotient.liftOn₂' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : Setoid α} {s₂ : Setoid β} (q₁ : Quotient s₁) (q₂ : Quotient s₂) (f : αβγ) (h : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), Setoid.r a₁ b₁Setoid.r a₂ b₂f a₁ a₂ = f b₁ b₂) :
γ

A version of Quotient.liftOn₂ taking {s₁ : Setoid α} {s₂ : Setoid β} as implicit arguments instead of instance arguments.

Equations
@[simp]
theorem Quotient.liftOn₂'_mk'' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : Setoid α} {s₂ : Setoid β} (f : αβγ) (h : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), Setoid.r a₁ b₁Setoid.r a₂ b₂f a₁ a₂ = f b₁ b₂) (a : α) (b : β) :
theorem Quotient.ind' {α : Sort u_1} {s₁ : Setoid α} {p : Quotient s₁Prop} (h : (a : α) → p (Quotient.mk'' a)) (q : Quotient s₁) :
p q

A version of Quotient.ind taking {s : Setoid α} as an implicit argument instead of an instance argument.

theorem Quotient.ind₂' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {p : Quotient s₁Quotient s₂Prop} (h : (a₁ : α) → (a₂ : β) → p (Quotient.mk'' a₁) (Quotient.mk'' a₂)) (q₁ : Quotient s₁) (q₂ : Quotient s₂) :
p q₁ q₂

A version of Quotient.ind₂ taking {s₁ : Setoid α} {s₂ : Setoid β} as implicit arguments instead of instance arguments.

theorem Quotient.inductionOn' {α : Sort u_1} {s₁ : Setoid α} {p : Quotient s₁Prop} (q : Quotient s₁) (h : (a : α) → p (Quotient.mk'' a)) :
p q

A version of Quotient.inductionOn taking {s : Setoid α} as an implicit argument instead of an instance argument.

theorem Quotient.inductionOn₂' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {p : Quotient s₁Quotient s₂Prop} (q₁ : Quotient s₁) (q₂ : Quotient s₂) (h : (a₁ : α) → (a₂ : β) → p (Quotient.mk'' a₁) (Quotient.mk'' a₂)) :
p q₁ q₂

A version of Quotient.inductionOn₂ taking {s₁ : Setoid α} {s₂ : Setoid β} as implicit arguments instead of instance arguments.

theorem Quotient.inductionOn₃' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : Setoid α} {s₂ : Setoid β} {s₃ : Setoid γ} {p : Quotient s₁Quotient s₂Quotient s₃Prop} (q₁ : Quotient s₁) (q₂ : Quotient s₂) (q₃ : Quotient s₃) (h : (a₁ : α) → (a₂ : β) → (a₃ : γ) → p (Quotient.mk'' a₁) (Quotient.mk'' a₂) (Quotient.mk'' a₃)) :
p q₁ q₂ q₃

A version of Quotient.inductionOn₃ taking {s₁ : Setoid α} {s₂ : Setoid β} {s₃ : Setoid γ} as implicit arguments instead of instance arguments.

def Quotient.recOnSubsingleton' {α : Sort u_1} {s₁ : Setoid α} {φ : Quotient s₁Sort u_2} [inst : ∀ (a : α), Subsingleton (φ (Quotient.mk s₁ a))] (q : Quotient s₁) (f : (a : α) → φ (Quotient.mk'' a)) :
φ q

A version of Quotient.recOnSubsingleton taking {s₁ : Setoid α} as an implicit argument instead of an instance argument.

Equations
def Quotient.recOnSubsingleton₂' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {φ : Quotient s₁Quotient s₂Sort u_3} [inst : ∀ (a : α) (b : β), Subsingleton (φ (Quotient.mk s₁ a) (Quotient.mk s₂ b))] (q₁ : Quotient s₁) (q₂ : Quotient s₂) (f : (a₁ : α) → (a₂ : β) → φ (Quotient.mk'' a₁) (Quotient.mk'' a₂)) :
φ q₁ q₂

A version of Quotient.recOnSubsingleton₂ taking {s₁ : Setoid α} {s₂ : Setoid α} as implicit arguments instead of instance arguments.

Equations
def Quotient.hrecOn' {α : Sort u_1} {s₁ : Setoid α} {φ : Quotient s₁Sort u_2} (qa : Quotient s₁) (f : (a : α) → φ (Quotient.mk'' a)) (c : ∀ (a₁ a₂ : α), a₁ a₂HEq (f a₁) (f a₂)) :
φ qa

Recursion on a Quotient argument a, result type depends on ⟦a⟧.

Equations
@[simp]
theorem Quotient.hrecOn'_mk'' {α : Sort u_1} {s₁ : Setoid α} {φ : Quotient s₁Sort u_2} (f : (a : α) → φ (Quotient.mk'' a)) (c : ∀ (a₁ a₂ : α), a₁ a₂HEq (f a₁) (f a₂)) (x : α) :
def Quotient.hrecOn₂' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {φ : Quotient s₁Quotient s₂Sort u_3} (qa : Quotient s₁) (qb : Quotient s₂) (f : (a : α) → (b : β) → φ (Quotient.mk'' a) (Quotient.mk'' b)) (c : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ a₂b₁ b₂HEq (f a₁ b₁) (f a₂ b₂)) :
φ qa qb

Recursion on two Quotient arguments a and b, result type depends on ⟦a⟧ and ⟦b⟧.

Equations
@[simp]
theorem Quotient.hrecOn₂'_mk'' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {φ : Quotient s₁Quotient s₂Sort u_3} (f : (a : α) → (b : β) → φ (Quotient.mk'' a) (Quotient.mk'' b)) (c : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), a₁ a₂b₁ b₂HEq (f a₁ b₁) (f a₂ b₂)) (x : α) (qb : Quotient s₂) :
Quotient.hrecOn₂' (Quotient.mk'' x) qb f c = Quotient.hrecOn' qb (f x) fun x x_1 => c x x x x_1 (_ : x x)
def Quotient.map' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} (f : αβ) (h : (Setoid.r Setoid.r) f f) :
Quotient s₁Quotient s₂

Map a function f : α → β→ β that sends equivalent elements to equivalent elements to a function Quotient sa → Quotient sb→ Quotient sb. Useful to define unary operations on quotients.

Equations
@[simp]
theorem Quotient.map'_mk'' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} (f : αβ) (h : (Setoid.r Setoid.r) f f) (x : α) :
def Quotient.map₂' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : Setoid α} {s₂ : Setoid β} {s₃ : Setoid γ} (f : αβγ) (h : (Setoid.r Setoid.r Setoid.r) f f) :
Quotient s₁Quotient s₂Quotient s₃

A version of Quotient.map₂ using curly braces and unification.

Equations
@[simp]
theorem Quotient.map₂'_mk'' {α : Sort u_1} {β : Sort u_2} {γ : Sort u_3} {s₁ : Setoid α} {s₂ : Setoid β} {s₃ : Setoid γ} (f : αβγ) (h : (Setoid.r Setoid.r Setoid.r) f f) (x : α) :
Quotient.map₂' f h (Quotient.mk'' x) = Quotient.map' (f x) (h x x (_ : x x))
theorem Quotient.exact' {α : Sort u_1} {s₁ : Setoid α} {a : α} {b : α} :
theorem Quotient.sound' {α : Sort u_1} {s₁ : Setoid α} {a : α} {b : α} :
@[simp]
theorem Quotient.eq' {α : Sort u_1} [s₁ : Setoid α] {a : α} {b : α} :
@[simp]
theorem Quotient.eq'' {α : Sort u_1} {s₁ : Setoid α} {a : α} {b : α} :
noncomputable def Quotient.out' {α : Sort u_1} {s₁ : Setoid α} (a : Quotient s₁) :
α

A version of Quotient.out taking {s₁ : Setoid α} as an implicit argument instead of an instance argument.

Equations
@[simp]
theorem Quotient.out_eq' {α : Sort u_1} {s₁ : Setoid α} (q : Quotient s₁) :
theorem Quotient.mk_out' {α : Sort u_1} {s₁ : Setoid α} (a : α) :
theorem Quotient.mk''_eq_mk {α : Sort u_1} [s : Setoid α] (x : α) :
@[simp]
theorem Quotient.liftOn'_mk {α : Sort u_1} {β : Sort u_2} [s : Setoid α] (x : α) (f : αβ) (h : ∀ (a b : α), Setoid.r a bf a = f b) :
@[simp]
theorem Quotient.liftOn₂'_mk {α : Sort u_2} {β : Sort u_1} {γ : Sort u_3} [s : Setoid α] [t : Setoid β] (f : αβγ) (h : ∀ (a₁ : α) (a₂ : β) (b₁ : α) (b₂ : β), Setoid.r a₁ b₁Setoid.r a₂ b₂f a₁ a₂ = f b₁ b₂) (a : α) (b : β) :
@[simp]
theorem Quotient.map'_mk {α : Sort u_2} {β : Sort u_1} [s : Setoid α] [t : Setoid β] (f : αβ) (h : (Setoid.r Setoid.r) f f) (x : α) :
instance Quotient.instDecidableLiftOn'Prop {α : Sort u_1} {s₁ : Setoid α} (q : Quotient s₁) (f : αProp) (h : ∀ (a b : α), Setoid.r a bf a = f b) [inst : DecidablePred f] :
Equations
instance Quotient.instDecidableLiftOn₂'Prop {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} (q₁ : Quotient s₁) (q₂ : Quotient s₂) (f : αβProp) (h : ∀ (a₁ : α) (b₁ : β) (a₂ : α) (b₂ : β), Setoid.r a₁ a₂Setoid.r b₁ b₂f a₁ b₁ = f a₂ b₂) [inst : (a : α) → DecidablePred (f a)] :
Equations
  • One or more equations did not get rendered due to their size.