Traversing collections #

This file proves basic properties of traversable and applicative functors and defines PureTransformation F, the natural applicative transformation from the identity functor to F.

References #

Inspired by [The Essence of the Iterator Pattern][gibbons2009].

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def Traversable.PureTransformation (F : Type u → Type u) [inst : Applicative F] [inst : LawfulApplicative F] :

ApplicativeTransformation Id F

The natural applicative transformation from the identity functor to F, defined by pure : Π {α}, α → F α→ F α.

Equations

One or more equations did not get rendered due to their size.

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@[simp]

theorem Traversable.pureTransformation_apply (F : Type u → Type u) [inst : Applicative F] [inst : LawfulApplicative F] {α : Type u} (x : id α) :

(fun {α} => ApplicativeTransformation.app (Traversable.PureTransformation F) α) α x = pure x

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theorem Traversable.map_eq_traverse_id {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {β : Type u} {γ : Type u} (f : β → γ) :

Functor.map f = traverse (pure ∘ f)

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theorem Traversable.map_traverse {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] {α : Type u} {β : Type u} {γ : Type u} (g : α → F β) (f : β → γ) (x : t α) :

Functor.map f <$> traverse g x = traverse (Functor.map f ∘ g) x

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theorem Traversable.traverse_map {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] {α : Type u} {β : Type u} {γ : Type u} (f : β → F γ) (g : α → β) (x : t α) :

traverse f (g <$> x) = traverse (f ∘ g) x

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theorem Traversable.pure_traverse {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] {α : Type u} (x : t α) :

traverse pure x = pure x

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theorem Traversable.id_sequence {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {α : Type u} (x : t α) :

sequence (pure <$> x) = pure x

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theorem Traversable.comp_sequence {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} {G : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] [inst : Applicative G] [inst : LawfulApplicative G] {α : Type u} (x : t (F (G α))) :

sequence (Functor.Comp.mk <$> x) = Functor.Comp.mk (sequence <$> sequence x)

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theorem Traversable.naturality' {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} {G : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] [inst : Applicative G] [inst : LawfulApplicative G] {α : Type u} (η : ApplicativeTransformation F G) (x : t (F α)) :

(fun {α} => ApplicativeTransformation.app η α) (t α) (sequence x) = sequence ((fun {α} => ApplicativeTransformation.app η α) α <$> x)

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theorem Traversable.traverse_id {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {α : Type u} :

traverse pure = pure

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theorem Traversable.traverse_comp {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} {G : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] [inst : Applicative G] [inst : LawfulApplicative G] {α : Type u} {β : Type u} {γ : Type u} (g : α → F β) (h : β → G γ) :

traverse (Functor.Comp.mk ∘ Functor.map h ∘ g) = Functor.Comp.mk ∘ Functor.map (traverse h) ∘ traverse g

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theorem Traversable.traverse_eq_map_id' {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {β : Type u} {γ : Type u} (f : β → γ) :

traverse (pure ∘ f) = pure ∘ Functor.map f

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theorem Traversable.traverse_map' {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {G : Type u → Type u} [inst : Applicative G] [inst : LawfulApplicative G] {α : Type u} {β : Type u} {γ : Type u} (g : α → β) (h : β → G γ) :

traverse (h ∘ g) = traverse h ∘ Functor.map g

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theorem Traversable.map_traverse' {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {G : Type u → Type u} [inst : Applicative G] [inst : LawfulApplicative G] {α : Type u} {β : Type u} {γ : Type u} (g : α → G β) (h : β → γ) :

traverse (Functor.map h ∘ g) = Functor.map (Functor.map h) ∘ traverse g

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theorem Traversable.naturality_pf {t : Type u → Type u} [inst : Traversable t] [inst : IsLawfulTraversable t] {F : Type u → Type u} {G : Type u → Type u} [inst : Applicative F] [inst : LawfulApplicative F] [inst : Applicative G] [inst : LawfulApplicative G] {α : Type u} {β : Type u} (η : ApplicativeTransformation F G) (f : α → F β) :

traverse ((fun {α} => ApplicativeTransformation.app η α) β ∘ f) = (fun {α} => ApplicativeTransformation.app η α) (t β) ∘ traverse f