HepLean Documentation

Mathlib.Control.Applicative

applicative instances #

This file provides Applicative instances for concrete functors:

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theorem Applicative.map_seq_map {F : Type u → Type v} [Applicative F] [LawfulApplicative F] {α β γ σ : Type u} (f : α → β → γ) (g : σ → β) (x : F α) (y : F σ) :
f <$> x <*> g <$> y = ((fun (x : β → γ) => x ∘ g) ∘ f) <$> x <*> y
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theorem Applicative.pure_seq_eq_map' {F : Type u → Type v} [Applicative F] [LawfulApplicative F] {α β : Type u} (f : α → β) :
(fun (x : F α) => pure f <*> x) = fun (x : F α) => f <$> x
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theorem Applicative.ext {F : Type u → Type u_1} {A1 A2 : Applicative F} [LawfulApplicative F] [LawfulApplicative F] :
(∀ {α : Type u} (x : α), pure x = pure x) → (∀ {α β : Type u} (f : F (α → β)) (x : F α), f <*> x = f <*> x) → A1 = A2
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instance instCommApplicativeId :
CommApplicative Id
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theorem Functor.Comp.map_pure {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] {α β : Type v} (f : α → β) (x : α) :
f <$> pure x = pure (f x)
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theorem Functor.Comp.seq_pure {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] {α β : Type v} (f : Functor.Comp F G (α → β)) (x : α) :
f <*> pure x = (fun (g : α → β) => g x) <$> f
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theorem Functor.Comp.seq_assoc {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] {α β γ : Type v} (x : Functor.Comp F G α) (f : Functor.Comp F G (α → β)) (g : Functor.Comp F G (β → γ)) :
g <*> (f <*> x) = Function.comp <$> g <*> f <*> x
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theorem Functor.Comp.pure_seq_eq_map {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] {α β : Type v} (f : α → β) (x : Functor.Comp F G α) :
pure f <*> x = f <$> x
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instance Functor.Comp.instLawfulApplicativeComp {F : Type u → Type w} {G : Type v → Type u} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] :
LawfulApplicative (Functor.Comp F G)
Equations
  • ⋯ = ⋯
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theorem Functor.Comp.applicative_id_comp {F : Type u_1 → Type u_2} [AF : Applicative F] [LawfulApplicative F] :
Functor.Comp.instApplicativeComp = AF
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theorem Functor.Comp.applicative_comp_id {F : Type u_1 → Type u_2} [AF : Applicative F] [LawfulApplicative F] :
Functor.Comp.instApplicativeComp = AF
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instance Functor.Comp.instCommApplicative {f : Type u → Type w} {g : Type v → Type u} [Applicative f] [Applicative g] [CommApplicative f] [CommApplicative g] :
CommApplicative (Functor.Comp f g)
Equations
  • ⋯ = ⋯
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theorem Comp.seq_mk {α β : Type w} {f : Type u → Type v} {g : Type w → Type u} [Applicative f] [Applicative g] (h : f (g (α → β))) (x : f (g α)) :
Functor.Comp.mk h <*> Functor.Comp.mk x = Functor.Comp.mk ((fun (x1 : g (α → β)) (x2 : g α) => x1 <*> x2) <$> h <*> x)
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instance instApplicativeConstOfOneOfMul {α : Type u_1} [One α] [Mul α] :
Applicative (Functor.Const α)
Equations
  • instApplicativeConstOfOneOfMul = Applicative.mk
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instance instLawfulApplicativeConst {α : Type u_1} [Monoid α] :
LawfulApplicative (Functor.Const α)
Equations
  • ⋯ = ⋯
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instance instApplicativeAddConstOfZeroOfAdd {α : Type u_1} [Zero α] [Add α] :
Applicative (Functor.AddConst α)
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  • instApplicativeAddConstOfZeroOfAdd = Applicative.mk
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instance instLawfulApplicativeAddConst {α : Type u_1} [AddMonoid α] :
LawfulApplicative (Functor.AddConst α)
Equations
  • ⋯ = ⋯