Documentation

Mathlib.Control.Functor

Functors #

This module provides additional lemmas, definitions, and instances for Functors.

Main definitions #

Tags #

functor, applicative

theorem Functor.map_id {F : Type u → Type v} {α : Type u} [Functor F] [LawfulFunctor F] :
(fun (x : F α) => id <$> x) = id
theorem Functor.map_comp_map {F : Type u → Type v} {α : Type u} {β : Type u} {γ : Type u} [Functor F] [LawfulFunctor F] (f : αβ) (g : βγ) :
((fun (x : F β) => g <$> x) fun (x : F α) => f <$> x) = fun (x : F α) => (g f) <$> x
theorem Functor.ext {F : Type u_1 → Type u_2} {F1 : Functor F} {F2 : Functor F} [LawfulFunctor F] [LawfulFunctor F] :
(∀ (α β : Type u_1) (f : αβ) (x : F α), f <$> x = f <$> x)F1 = F2
def id.mk {α : Sort u} :
αid α

Introduce id as a quasi-functor. (Note that where a lawful Monad or Applicative or Functor is needed, Id is the correct definition).

Equations
  • id.mk = id
Instances For
    def Functor.Const (α : Type u_1) (_β : Type u_2) :
    Type u_1

    Const α is the constant functor, mapping every type to α. When α has a monoid structure, Const α has an Applicative instance. (If α has an additive monoid structure, see Functor.AddConst.)

    Equations
    Instances For
      @[match_pattern]
      def Functor.Const.mk {α : Type u_1} {β : Type u_2} (x : α) :

      Const.mk is the canonical map α → Const α β (the identity), and it can be used as a pattern to extract this value.

      Equations
      Instances For
        def Functor.Const.mk' {α : Type u_1} (x : α) :

        Const.mk' is Const.mk but specialized to map α to Const α PUnit, where PUnit is the terminal object in Type*.

        Equations
        Instances For
          def Functor.Const.run {α : Type u_1} {β : Type u_2} (x : Functor.Const α β) :
          α

          Extract the element of α from the Const functor.

          Equations
          • x.run = x
          Instances For
            theorem Functor.Const.ext {α : Type u_1} {β : Type u_2} {x : Functor.Const α β} {y : Functor.Const α β} (h : x.run = y.run) :
            x = y
            def Functor.Const.map {γ : Type u_1} {α : Type u_2} {β : Type u_3} (_f : αβ) (x : Functor.Const γ β) :

            The map operation of the Const γ functor.

            Equations
            Instances For
              Equations
              Equations
              • =
              instance Functor.Const.instInhabited {α : Type u_1} {β : Type u_2} [Inhabited α] :
              Equations
              • Functor.Const.instInhabited = { default := default }
              def Functor.AddConst (α : Type u_1) (_β : Type u_2) :
              Type u_1

              AddConst α is a synonym for constant functor Const α, mapping every type to α. When α has an additive monoid structure, AddConst α has an Applicative instance. (If α has a multiplicative monoid structure, see Functor.Const.)

              Equations
              Instances For
                @[match_pattern]
                def Functor.AddConst.mk {α : Type u_1} {β : Type u_2} (x : α) :

                AddConst.mk is the canonical map α → AddConst α β, which is the identity, where AddConst α β = Const α β. It can be used as a pattern to extract this value.

                Equations
                Instances For
                  def Functor.AddConst.run {α : Type u_1} {β : Type u_2} :
                  Functor.AddConst α βα

                  Extract the element of α from the constant functor.

                  Equations
                  • Functor.AddConst.run = id
                  Instances For
                    Equations
                    • Functor.AddConst.functor = Functor.Const.functor
                    instance Functor.instInhabitedAddConst {α : Type u_1} {β : Type u_2} [Inhabited α] :
                    Equations
                    • Functor.instInhabitedAddConst = { default := default }
                    def Functor.Comp (F : Type u → Type w) (G : Type v → Type u) (α : Type v) :

                    Functor.Comp is a wrapper around Function.Comp for types. It prevents Lean's type class resolution mechanism from trying a Functor (Comp F id) when Functor F would do.

                    Equations
                    Instances For
                      @[match_pattern]
                      def Functor.Comp.mk {F : Type u → Type w} {G : Type v → Type u} {α : Type v} (x : F (G α)) :

                      Construct a term of Comp F G α from a term of F (G α), which is the same type. Can be used as a pattern to extract a term of F (G α).

                      Equations
                      Instances For
                        def Functor.Comp.run {F : Type u → Type w} {G : Type v → Type u} {α : Type v} (x : Functor.Comp F G α) :
                        F (G α)

                        Extract a term of F (G α) from a term of Comp F G α, which is the same type.

                        Equations
                        • x.run = x
                        Instances For
                          theorem Functor.Comp.ext {F : Type u → Type w} {G : Type v → Type u} {α : Type v} {x : Functor.Comp F G α} {y : Functor.Comp F G α} :
                          x.run = y.runx = y
                          instance Functor.Comp.instInhabited {F : Type u → Type w} {G : Type v → Type u} {α : Type v} [Inhabited (F (G α))] :
                          Equations
                          • Functor.Comp.instInhabited = { default := default }
                          def Functor.Comp.map {F : Type u → Type w} {G : Type v → Type u} [Functor F] [Functor G] {α : Type v} {β : Type v} (h : αβ) :
                          Functor.Comp F G αFunctor.Comp F G β

                          The map operation for the composition Comp F G of functors F and G.

                          Equations
                          Instances For
                            instance Functor.Comp.functor {F : Type u → Type w} {G : Type v → Type u} [Functor F] [Functor G] :
                            Equations
                            theorem Functor.Comp.map_mk {F : Type u → Type w} {G : Type v → Type u} [Functor F] [Functor G] {α : Type v} {β : Type v} (h : αβ) (x : F (G α)) :
                            h <$> Functor.Comp.mk x = Functor.Comp.mk ((fun (x : G α) => h <$> x) <$> x)
                            @[simp]
                            theorem Functor.Comp.run_map {F : Type u → Type w} {G : Type v → Type u} [Functor F] [Functor G] {α : Type v} {β : Type v} (h : αβ) (x : Functor.Comp F G α) :
                            (h <$> x).run = (fun (x : G α) => h <$> x) <$> x.run
                            theorem Functor.Comp.id_map {F : Type u → Type w} {G : Type v → Type u} [Functor F] [Functor G] [LawfulFunctor F] [LawfulFunctor G] {α : Type v} (x : Functor.Comp F G α) :
                            theorem Functor.Comp.comp_map {F : Type u → Type w} {G : Type v → Type u} [Functor F] [Functor G] [LawfulFunctor F] [LawfulFunctor G] {α : Type v} {β : Type v} {γ : Type v} (g' : αβ) (h : βγ) (x : Functor.Comp F G α) :
                            Equations
                            • =
                            theorem Functor.Comp.functor_comp_id {F : Type u_1 → Type u_2} [AF : Functor F] [LawfulFunctor F] :
                            Functor.Comp.functor = AF
                            theorem Functor.Comp.functor_id_comp {F : Type u_1 → Type u_2} [AF : Functor F] [LawfulFunctor F] :
                            Functor.Comp.functor = AF
                            def Functor.Comp.seq {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] {α : Type v} {β : Type v} :
                            Functor.Comp F G (αβ)(UnitFunctor.Comp F G α)Functor.Comp F G β

                            The <*> operation for the composition of applicative functors.

                            Equations
                            Instances For
                              instance Functor.Comp.instPure {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] :
                              Equations
                              instance Functor.Comp.instSeq {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] :
                              Equations
                              @[simp]
                              theorem Functor.Comp.run_pure {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] {α : Type v} (x : α) :
                              (pure x).run = pure (pure x)
                              @[simp]
                              theorem Functor.Comp.run_seq {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] {α : Type v} {β : Type v} (f : Functor.Comp F G (αβ)) (x : Functor.Comp F G α) :
                              (Seq.seq f fun (x_1 : Unit) => x).run = Seq.seq ((fun (x1 : G (αβ)) (x2 : G α) => Seq.seq x1 fun (x : Unit) => x2) <$> f.run) fun (x_1 : Unit) => x.run
                              Equations
                              • Functor.Comp.instApplicativeComp = Applicative.mk
                              def Functor.Liftp {F : Type u → Type u} [Functor F] {α : Type u} (p : αProp) (x : F α) :

                              If we consider x : F α to, in some sense, contain values of type α, predicate Liftp p x holds iff every value contained by x satisfies p.

                              Equations
                              Instances For
                                def Functor.Liftr {F : Type u → Type u} [Functor F] {α : Type u} (r : ααProp) (x : F α) (y : F α) :

                                If we consider x : F α to, in some sense, contain values of type α, then Liftr r x y relates x and y iff (1) x and y have the same shape and (2) we can pair values a from x and b from y so that r a b holds.

                                Equations
                                • One or more equations did not get rendered due to their size.
                                Instances For
                                  def Functor.supp {F : Type u → Type u} [Functor F] {α : Type u} (x : F α) :
                                  Set α

                                  If we consider x : F α to, in some sense, contain values of type α, then supp x is the set of values of type α that x contains.

                                  Equations
                                  Instances For
                                    theorem Functor.of_mem_supp {F : Type u → Type u} [Functor F] {α : Type u} {x : F α} {p : αProp} (h : Functor.Liftp p x) (y : α) :
                                    y Functor.supp xp y
                                    @[reducible, inline]
                                    abbrev Functor.mapConstRev {f : Type u → Type v} [Functor f] {α : Type u} {β : Type u} :
                                    f βαf α

                                    If f is a functor, if fb : f β and a : α, then mapConstRev fb a is the result of applying f.map to the constant function β → α sending everything to a, and then evaluating at fb. In other words it's const a <$> fb.

                                    Equations
                                    Instances For

                                      If f is a functor, if fb : f β and a : α, then mapConstRev fb a is the result of applying f.map to the constant function β → α sending everything to a, and then evaluating at fb. In other words it's const a <$> fb.

                                      Equations
                                      Instances For