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} {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) [Inhabited α] :
Equations
instance Quot.Subsingleton {α : Sort u_1} {ra : ααProp} [Subsingleton α] :
Equations
  • =
instance Quot.instUniqueQuot {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
Instances For
    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
    Instances For
      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
      Instances For
        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.liftOn (Quot.mk r a) 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.

          Equations
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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.

            Equations
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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.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_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.

              Equations
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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.

                Equations
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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.instDecidableLiftOnProp {α : Sort u_1} (r : ααProp) (q : Quot r) (f : αProp) (h : ∀ (a b : α), r a bf a = f b) [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) [(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.
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                    Pretty printer defined by notation3 command.

                    Equations
                    • 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.instIsEquivEquivInstHasEquiv {α : Type u_4} [Setoid α] :
                      IsEquiv α fun (x x_1 : α) => x x_1
                      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⟧.

                      Equations
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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 (x x_1 : α) => x x_1) fun (x x_1 : β) => x x_1) 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 (x x_1 : α) => x x_1) fun (x x_1 : β) => x x_1) 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 (x x_1 : α) => x x_1) (fun (x x_1 : β) => x x_1) fun (x x_1 : γ) => x x_1) 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 (x x_1 : α) => x x_1) (fun (x x_1 : β) => x x_1) fun (x x_1 : γ) => x x_1) 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.instDecidableLiftOnProp {α : Sort u_1} [sa : Setoid α] (q : Quotient sa) (f : αProp) (h : ∀ (a b : α), a bf a = f b) [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) [(a : α) → DecidablePred (f a)] :
                            Equations
                            theorem Quot.eq {α : Type u_3} {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} :
                            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) :
                            Quotient.liftOn x 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) :
                            Quotient.liftOn₂ x y f h = f x y
                            theorem surjective_quot_mk {α : Sort u_1} (r : ααProp) :

                            Quot.mk r 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.

                              Equations
                              Instances For
                                @[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
                                Instances For
                                  @[simp]
                                  theorem Quotient.out_eq {α : Sort u_1} [s : Setoid α] (q : Quotient s) :
                                  Quotient.out q = q
                                  theorem Quotient.mk_out {α : Sort u_1} [Setoid α] (a : α) :
                                  Quotient.out a a
                                  theorem Quotient.mk_eq_iff_out {α : Sort u_1} [s : Setoid α] {x : α} {y : Quotient s} :
                                  x = y x Quotient.out y
                                  theorem Quotient.eq_mk_iff_out {α : Sort u_1} [s : Setoid α] {x : Quotient s} {y : α} :
                                  x = y Quotient.out x y
                                  @[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_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
                                  Instances For
                                    @[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.

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

                                        Constructor for Trunc α

                                        Equations
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                                          instance Trunc.instInhabitedTrunc {α : Sort u_1} [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
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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
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                                              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
                                              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.

                                                Equations
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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.

                                                  Equations
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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.

                                                    Equations
                                                    Instances For
                                                      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.

                                                      Equations
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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).

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                                                        • Trunc.out = Quot.out
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                                                          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.

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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.

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                                                              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) :
                                                              @[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) :
                                                              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.

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                                                                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_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.

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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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                                                                      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) :
                                                                      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⟧.

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                                                                        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) :
                                                                        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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                                                                          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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                                                                            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} :
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                                                                            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.

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                                                                              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] :
                                                                              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) :
                                                                              Quotient.liftOn' x 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) :
                                                                              Quotient.liftOn₂' a 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'Prop {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] :
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                                                                              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) [(a : α) → DecidablePred (f a)] :
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                                                                              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