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_3} {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_4} {r : ααProp} {β : Quot rProp} (q : Quot r) (h : ∀ (a : α), β (Quot.mk r a)) :
β q
instance Quot.instInhabited_mathlib {α : Sort u_1} (r : ααProp) [Inhabited α] :
Equations
instance Quot.Subsingleton {α : Sort u_1} {ra : ααProp} [Subsingleton α] :
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  • =
@[deprecated Quot.recOn]
def Quot.recOn' {α : Sort u} {r : ααProp} {motive : Quot rSort v} (q : Quot r) (f : (a : α) → motive (Quot.mk r a)) (h : ∀ (a b : α) (p : r a b), f a = f b) :
motive q

Alias of Quot.recOn.


Dependent recursion principle for Quot. This constructor can be tricky to use, so you should consider the simpler versions if they apply:

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    @[deprecated Quot.recOnSubsingleton]
    def Quot.recOnSubsingleton' {α : Sort u} {r : ααProp} {motive : Quot rSort v} [h : ∀ (a : α), Subsingleton (motive (Quot.mk r a))] (q : Quot r) (f : (a : α) → motive (Quot.mk r a)) :
    motive q

    Alias of Quot.recOnSubsingleton.


    Dependent induction principle for a quotient, when the target type is a Subsingleton. In this case the quotient's side condition is trivial so any function can be lifted.

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      instance Quot.instUnique {α : Sort u_1} {ra : ααProp} [Unique 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
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        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 such that ra x y implies rb (f x) (f y) to a map Quot ra → Quot rb.

        Equations
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          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'.

          Equations
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            def Quot.factor {α : Type u_4} (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
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              theorem Quot.factor_mk_eq {α : Type u_4} (r : ααProp) (s : ααProp) (h : ∀ (x y : α), r x ys x y) :
              theorem Quot.lift_mk {α : Sort u_1} {γ : Sort u_4} {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_1} {γ : Sort u_4} {r : ααProp} (a : α) (f : αγ) (h : ∀ (a₁ a₂ : α), r a₁ a₂f a₁ = f a₂) :
              (Quot.mk r a).liftOn f h = f a
              @[simp]
              theorem Quot.surjective_lift {α : Sort u_1} {γ : Sort u_4} {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_4} {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 to quotients of α and β.

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                @[simp]
                theorem Quot.lift₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {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_4} {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 to quotients of α and β and applies it.

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                  @[simp]
                  theorem Quot.liftOn₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {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.mk r a).liftOn₂ (Quot.mk s b) f hr hs = f a b
                  def Quot.map₂ {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {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 to quotients of α and β with values in a quotient of γ.

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                    @[simp]
                    theorem Quot.map₂_mk {α : Sort u_1} {β : Sort u_2} {γ : Sort u_4} {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_5} [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.

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                      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_4} {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.instDecidableLiftOnOfDecidablePred_mathlib {α : Sort u_1} (r : ααProp) (q : Quot r) (f : αProp) (h : ∀ (a b : α), r a bf a = f b) [DecidablePred f] :
                      Decidable (q.liftOn f h)
                      Equations
                      instance Quot.instDecidableLiftOn₂OfDecidablePred {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) [(a : α) → DecidablePred (f a)] :
                      Decidable (q₁.liftOn₂ q₂ f ha hb)
                      Equations

                      The canonical quotient map into a Quotient.

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                      • One or more equations did not get rendered due to their size.
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                        Pretty printer defined by notation3 command.

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                        • One or more equations did not get rendered due to their size.
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                          instance Quotient.instInhabitedQuotient {α : Sort u_1} (s : Setoid α) [Inhabited α] :
                          Equations
                          Equations
                          • =
                          instance Quotient.instIsEquivEquiv {α : Type u_4} [Setoid α] :
                          IsEquiv α fun (x1 x2 : α) => x1 x2
                          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 a 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⟧.

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                            def Quotient.map {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] (f : αβ) (h : ((fun (x1 x2 : α) => x1 x2) fun (x1 x2 : β) => x1 x2) f f) :
                            Quotient saQuotient sb

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

                            Equations
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                              @[simp]
                              theorem Quotient.map_mk {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] (f : αβ) (h : ((fun (x1 x2 : α) => x1 x2) fun (x1 x2 : β) => x1 x2) f f) (x : α) :
                              Quotient.map f h x = f x
                              def Quotient.map₂ {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] {γ : Sort u_4} [sc : Setoid γ] (f : αβγ) (h : ((fun (x1 x2 : α) => x1 x2) (fun (x1 x2 : β) => x1 x2) fun (x1 x2 : γ) => x1 x2) f f) :
                              Quotient saQuotient sbQuotient sc

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

                              Equations
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                                @[simp]
                                theorem Quotient.map₂_mk {α : Sort u_1} {β : Sort u_2} [sa : Setoid α] [sb : Setoid β] {γ : Sort u_4} [sc : Setoid γ] (f : αβγ) (h : ((fun (x1 x2 : α) => x1 x2) (fun (x1 x2 : β) => x1 x2) fun (x1 x2 : γ) => x1 x2) f f) (x : α) (y : β) :
                                Quotient.map₂ f h x y = 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) [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.instDecidableLiftOnOfDecidablePred_mathlib {α : Sort u_1} [sa : Setoid α] (q : Quotient sa) (f : αProp) (h : ∀ (a b : α), a bf a = f b) [DecidablePred f] :
                                Decidable (q.liftOn f h)
                                Equations
                                instance Quotient.instDecidableLiftOn₂OfDecidablePred_mathlib {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) [(a : α) → DecidablePred (f a)] :
                                Decidable (q₁.liftOn₂ q₂ f h)
                                Equations
                                theorem Quot.eq {α : Type u_3} {r : ααProp} {x : α} {y : α} :
                                @[simp]
                                theorem Quotient.eq {α : Sort u_1} [r : Setoid α] {x : α} {y : α} :
                                x = y x y
                                theorem Quotient.forall {α : Sort u_3} {s : Setoid α} {p : Quotient sProp} :
                                (∀ (a : Quotient s), p a) ∀ (a : α), p a
                                theorem Quotient.exists {α : Sort u_3} {s : Setoid α} {p : Quotient sProp} :
                                (∃ (a : Quotient s), p a) ∃ (a : α), p 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) :
                                Quotient.lift f h x = f x
                                @[simp]
                                theorem Quotient.lift_comp_mk {α : Sort u_1} {β : Sort u_2} [Setoid α] (f : αβ) (h : ∀ (a b : α), a bf a = f b) :
                                @[simp]
                                theorem Quotient.lift₂_mk {α : Sort u_3} {β : Sort u_4} {γ : Sort u_5} [Setoid α] [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 a 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) :
                                x.liftOn f h = f x
                                @[simp]
                                theorem Quotient.liftOn₂_mk {α : Sort u_3} {β : Sort u_4} [Setoid α] (f : ααβ) (h : ∀ (a₁ a₂ b₁ b₂ : α), a₁ b₁a₂ b₂f a₁ a₂ = f b₁ b₂) (x : α) (y : α) :
                                x.liftOn₂ 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.

                                theorem surjective_quotient_mk' (α : Sort u_3) [s : Setoid α] :
                                Function.Surjective Quotient.mk'

                                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
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                                  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.

                                  Instances For
                                    @[simp]
                                    theorem Quot.out_eq {α : Sort u_1} {r : ααProp} (q : Quot r) :
                                    Quot.mk r q.out = q
                                    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
                                    Instances For
                                      @[simp]
                                      theorem Quotient.out_eq {α : Sort u_1} [s : Setoid α] (q : Quotient s) :
                                      q.out = q
                                      theorem Quotient.mk_out {α : Sort u_1} [Setoid α] (a : α) :
                                      a.out a
                                      theorem Quotient.mk_eq_iff_out {α : Sort u_1} [s : Setoid α] {x : α} {y : Quotient s} :
                                      x = y x y.out
                                      theorem Quotient.eq_mk_iff_out {α : Sort u_1} [s : Setoid α] {x : Quotient s} {y : α} :
                                      x = y x.out y
                                      @[simp]
                                      theorem Quotient.out_equiv_out {α : Sort u_1} {s : Setoid α} {x : Quotient s} {y : Quotient s} :
                                      x.out y.out x = y
                                      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} :
                                      x.out = y.out x = y
                                      instance piSetoid {ι : Sort u_3} {α : ιSort u_4} [(i : ι) → Setoid (α i)] :
                                      Setoid ((i : ι) → α i)
                                      Equations
                                      • piSetoid = { r := fun (a b : (i : ι) → α i) => ∀ (i : ι), a i b i, iseqv := }
                                      noncomputable def Quotient.choice {ι : Type u_3} {α : ιType u_4} [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
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                                        @[simp]
                                        theorem Quotient.choice_eq {ι : Type u_3} {α : ιType u_4} [(i : ι) → Setoid (α i)] (f : (i : ι) → α i) :
                                        (Quotient.choice fun (i : ι) => f i) = f
                                        theorem Quotient.induction_on_pi {ι : Type u_3} {α : ιSort u_4} [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 : ι) => a i) :
                                        p f

                                        Truncation #

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

                                        Always-true relation as a Setoid.

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

                                        Equations
                                        • trueSetoid = { r := fun (x x : α) => True, iseqv := }
                                        Instances For
                                          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.

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

                                            Constructor for Trunc α

                                            Equations
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                                              instance Trunc.instInhabited {α : Sort u_1} [Inhabited α] :
                                              Equations
                                              • Trunc.instInhabited = { 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
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                                                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_1} {β : Sort u_2} (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
                                                Instances For
                                                  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 : α), 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
                                                  Equations
                                                  • =
                                                  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
                                                  • q.bind f = q.liftOn f
                                                  Instances For
                                                    def Trunc.map {α : Sort u_1} {β : Sort u_2} (f : αβ) (q : Trunc α) :

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

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                                                      def Trunc.rec {α : Sort u_1} {C : Trunc αSort u_3} (f : (a : α) → C (Trunc.mk a)) (h : ∀ (a b : α), f a = f b) (q : Trunc α) :
                                                      C q

                                                      Recursion/induction principle for Trunc.

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                                                        def Trunc.recOn {α : Sort u_1} {C : Trunc αSort u_3} (q : Trunc α) (f : (a : α) → C (Trunc.mk a)) (h : ∀ (a b : α), f a = f b) :
                                                        C q

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

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                                                          def Trunc.recOnSubsingleton {α : Sort u_1} {C : Trunc αSort u_3} [∀ (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.

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                                                            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
                                                            Instances For
                                                              @[simp]
                                                              theorem Trunc.out_eq {α : Sort u_1} (q : Trunc α) :
                                                              Trunc.mk q.out = q
                                                              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.

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                                                                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_4} {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
                                                                • q.liftOn' f h = q.liftOn f h
                                                                Instances For
                                                                  @[simp]
                                                                  theorem Quotient.liftOn'_mk'' {α : Sort u_1} {φ : Sort u_4} {s₁ : Setoid α} (f : αφ) (h : ∀ (a b : α), Setoid.r a bf a = f b) (x : α) :
                                                                  (Quotient.mk'' x).liftOn' f h = f x
                                                                  @[simp]
                                                                  theorem Quotient.surjective_liftOn' {α : Sort u_1} {φ : Sort u_4} {s₁ : Setoid α} {f : αφ} (h : ∀ (a b : α), Setoid.r a bf a = f b) :
                                                                  (Function.Surjective fun (x : Quotient s₁) => x.liftOn' f h) Function.Surjective f
                                                                  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
                                                                  • q₁.liftOn₂' q₂ f h = q₁.liftOn₂ q₂ f h
                                                                  Instances For
                                                                    @[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) :
                                                                    (Quotient.mk'' a).liftOn₂' (Quotient.mk'' b) f h = f a b
                                                                    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_5} [∀ (a : α), Subsingleton (φ 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
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                                                                      def Quotient.recOnSubsingleton₂' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {φ : Quotient s₁Quotient s₂Sort u_5} [∀ (a : α) (b : β), Subsingleton (φ a 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.

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                                                                        def Quotient.hrecOn' {α : Sort u_1} {s₁ : Setoid α} {φ : Quotient s₁Sort u_5} (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⟧.

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                                                                          @[simp]
                                                                          theorem Quotient.hrecOn'_mk'' {α : Sort u_1} {s₁ : Setoid α} {φ : Quotient s₁Sort u_5} (f : (a : α) → φ (Quotient.mk'' a)) (c : ∀ (a₁ a₂ : α), a₁ a₂HEq (f a₁) (f a₂)) (x : α) :
                                                                          (Quotient.mk'' x).hrecOn' f c = f x
                                                                          def Quotient.hrecOn₂' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {φ : Quotient s₁Quotient s₂Sort u_5} (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
                                                                          • qa.hrecOn₂' qb f c = qa.hrecOn₂ qb f c
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                                                                            @[simp]
                                                                            theorem Quotient.hrecOn₂'_mk'' {α : Sort u_1} {β : Sort u_2} {s₁ : Setoid α} {s₂ : Setoid β} {φ : Quotient s₁Quotient s₂Sort u_5} (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.mk'' x).hrecOn₂' qb f c = qb.hrecOn' (f 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 that sends equivalent elements to equivalent elements to a function Quotient sa → Quotient sb. Useful to define unary operations on quotients.

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                                                                              @[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.

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                                                                              Instances For
                                                                                @[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) :
                                                                                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
                                                                                • a.out' = a.out
                                                                                Instances For
                                                                                  @[simp]
                                                                                  theorem Quotient.out_eq' {α : Sort u_1} {s₁ : Setoid α} (q : Quotient s₁) :
                                                                                  Quotient.mk'' q.out' = q
                                                                                  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] :
                                                                                  Quotient.mk'' = Quotient.mk s
                                                                                  @[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) :
                                                                                  x.liftOn' f h = f x
                                                                                  @[simp]
                                                                                  theorem Quotient.liftOn₂'_mk {α : Sort u_1} {β : Sort u_2} {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) :
                                                                                  a.liftOn₂' b f h = f a b
                                                                                  @[simp]
                                                                                  theorem Quotient.map'_mk {α : Sort u_1} {β : Sort u_2} [s : Setoid α] [t : Setoid β] (f : αβ) (h : (Setoid.r Setoid.r) f f) (x : α) :
                                                                                  Quotient.map' f h x = f x
                                                                                  instance Quotient.instDecidableLiftOn'OfDecidablePred {α : Sort u_1} {s₁ : Setoid α} (q : Quotient s₁) (f : αProp) (h : ∀ (a b : α), Setoid.r a bf a = f b) [DecidablePred f] :
                                                                                  Decidable (q.liftOn' f h)
                                                                                  Equations
                                                                                  instance Quotient.instDecidableLiftOn₂'OfDecidablePred {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) [(a : α) → DecidablePred (f a)] :
                                                                                  Decidable (q₁.liftOn₂' q₂ f h)
                                                                                  Equations
                                                                                  @[simp]
                                                                                  theorem Equivalence.quot_mk_eq_iff {α : Type u_3} {r : ααProp} (h : Equivalence r) (x : α) (y : α) :
                                                                                  Quot.mk r x = Quot.mk r y r x y