@[simp]

theorem monadLift_self {α : Type u} {m : Type u → Type v} (x : m α) : monadLift x = x

class LawfulFunctor (f : Type u → Type v) [Functor f] : Prop

The Functor typeclass only contains the operations of a functor. LawfulFunctor further asserts that these operations satisfy the laws of a functor, including the preservation of the identity and composition laws:

id <$> x = x
(h ∘ g) <$> x = h <$> g <$> x

• map_const {α β : Type u} : Functor.mapConst = Functor.map ∘ Function.const β
• id_map {α : Type u} (x : f α) : id <$> x = x
• comp_map {α β γ : Type u} (g : α → β) (h : β → γ) (x : f α) : (h ∘ g) <$> x = h <$> g <$> x

@[simp]

theorem id_map' {m : Type u_1 → Type u_2} {α : Type u_1} [Functor m] [LawfulFunctor m] (x : m α) : (fun (a : α) => a) <$> x = x

@[simp]

theorem Functor.map_map {f : Type u_1 → Type u_2} {α β γ : Type u_1} [Functor f] [LawfulFunctor f] (m : α → β) (g : β → γ) (x : f α) : g <$> m <$> x = (fun (a : α) => g (m a)) <$> x

class LawfulApplicative (f : Type u → Type v) [Applicative f] extends LawfulFunctor f : Prop

The Applicative typeclass only contains the operations of an applicative functor. LawfulApplicative further asserts that these operations satisfy the laws of an applicative functor:

pure id <*> v = v
pure (·∘·) <*> u <*> v <*> w = u <*> (v <*> w)
pure f <*> pure x = pure (f x)
u <*> pure y = pure (· y) <*> u

• map_const {α β : Type u} : Functor.mapConst = Functor.map ∘ Function.const β
• id_map {α : Type u} (x : f α) : id <$> x = x
• comp_map {α β γ : Type u} (g : α → β) (h : β → γ) (x : f α) : (h ∘ g) <$> x = h <$> g <$> x
• seqLeft_eq {α β : Type u} (x : f α) (y : f β) : x <* y = Function.const β <$> x <*> y
• seqRight_eq {α β : Type u} (x : f α) (y : f β) : x *> y = Function.const α id <$> x <*> y
• pure_seq {α β : Type u} (g : α → β) (x : f α) : pure g <*> x = g <$> x
• map_pure {α β : Type u} (g : α → β) (x : α) : g <$> pure x = pure (g x)
• seq_pure {α β : Type u} (g : f (α → β)) (x : α) : g <*> pure x = (fun (h : α → β) => h x) <$> g
• seq_assoc {α β γ : Type u} (x : f α) (g : f (α → β)) (h : f (β → γ)) : h <*> (g <*> x) = Function.comp <$> h <*> g <*> x

@[simp]

theorem pure_id_seq {f : Type u_1 → Type u_2} {α : Type u_1} [Applicative f] [LawfulApplicative f] (x : f α) : pure id <*> x = x

class LawfulMonad (m : Type u → Type v) [Monad m] extends LawfulApplicative m : Prop

The Monad typeclass only contains the operations of a monad. LawfulMonad further asserts that these operations satisfy the laws of a monad, including associativity and identity laws for bind:

pure x >>= f = f x
x >>= pure = x
x >>= f >>= g = x >>= (fun x => f x >>= g)

LawfulMonad.mk' is an alternative constructor containing useful defaults for many fields.

• map_const {α β : Type u} : Functor.mapConst = Functor.map ∘ Function.const β
• id_map {α : Type u} (x : m α) : id <$> x = x
• comp_map {α β γ : Type u} (g : α → β) (h : β → γ) (x : m α) : (h ∘ g) <$> x = h <$> g <$> x
• seqLeft_eq {α β : Type u} (x : m α) (y : m β) : x <* y = Function.const β <$> x <*> y
• seqRight_eq {α β : Type u} (x : m α) (y : m β) : x *> y = Function.const α id <$> x <*> y
• pure_seq {α β : Type u} (g : α → β) (x : m α) : pure g <*> x = g <$> x
• map_pure {α β : Type u} (g : α → β) (x : α) : g <$> pure x = pure (g x)
• seq_pure {α β : Type u} (g : m (α → β)) (x : α) : g <*> pure x = (fun (h : α → β) => h x) <$> g
• seq_assoc {α β γ : Type u} (x : m α) (g : m (α → β)) (h : m (β → γ)) : h <*> (g <*> x) = Function.comp <$> h <*> g <*> x
• bind_pure_comp {α β : Type u} (f : α → β) (x : m α) : (do let a ← x pure (f a)) = f <$> x
• bind_map {α β : Type u} (f : m (α → β)) (x : m α) : (do let x_1 ← f x_1 <$> x) = f <*> x
• pure_bind {α β : Type u} (x : α) (f : α → m β) : pure x >>= f = f x
• bind_assoc {α β γ : Type u} (x : m α) (f : α → m β) (g : β → m γ) : x >>= f >>= g = x >>= fun (x : α) => f x >>= g

@[simp]

theorem bind_pure {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] [LawfulMonad m] (x : m α) : x >>= pure = x

theorem map_eq_pure_bind {m : Type u_1 → Type u_2} {α β : Type u_1} [Monad m] [LawfulMonad m] (f : α → β) (x : m α) : f <$> x = do let a ← x pure (f a)

Use simp [← bind_pure_comp] rather than simp [map_eq_pure_bind], as bind_pure_comp is in the default simp set, so also using map_eq_pure_bind would cause a loop.

theorem seq_eq_bind_map {m : Type u → Type u_1} {α β : Type u} [Monad m] [LawfulMonad m] (f : m (α → β)) (x : m α) : f <*> x = do let x_1 ← f x_1 <$> x

theorem bind_congr {m : Type u_1 → Type u_2} {α β : Type u_1} [Bind m] {x : m α} {f g : α → m β} (h : ∀ (a : α), f a = g a) : x >>= f = x >>= g

theorem bind_pure_unit {m : Type u_1 → Type u_2} [Monad m] [LawfulMonad m] {x : m PUnit} : (do x pure PUnit.unit) = x

theorem map_congr {m : Type u_1 → Type u_2} {α β : Type u_1} [Functor m] {x : m α} {f g : α → β} (h : ∀ (a : α), f a = g a) : f <$> x = g <$> x

theorem seq_eq_bind {m : Type u → Type u_1} {α β : Type u} [Monad m] [LawfulMonad m] (mf : m (α → β)) (x : m α) : mf <*> x = do let f ← mf f <$> x

theorem seqRight_eq_bind {m : Type u_1 → Type u_2} {α β : Type u_1} [Monad m] [LawfulMonad m] (x : m α) (y : m β) : x *> y = do let _ ← x y

theorem seqLeft_eq_bind {m : Type u_1 → Type u_2} {α β : Type u_1} [Monad m] [LawfulMonad m] (x : m α) (y : m β) : x <* y = do let a ← x let _ ← y pure a

@[simp]

theorem map_bind {m : Type u_1 → Type u_2} {β γ α : Type u_1} [Monad m] [LawfulMonad m] (f : β → γ) (x : m α) (g : α → m β) : f <$> (x >>= g) = do let a ← x f <$> g a

@[simp]

theorem bind_map_left {m : Type u_1 → Type u_2} {α β γ : Type u_1} [Monad m] [LawfulMonad m] (f : α → β) (x : m α) (g : β → m γ) : (do let b ← f <$> x g b) = do let a ← x g (f a)

theorem Functor.map_unit {m : Type u_1 → Type u_2} [Monad m] [LawfulMonad m] {a : m PUnit} : (fun (x : PUnit) => PUnit.unit) <$> a = a

@[simp]

theorem LawfulMonad.map_pure' {m : Type u_1 → Type u_2} {α β : Type u_1} {f : α → β} [Monad m] [LawfulMonad m] {a : α} : f <$> pure a = pure (f a)

This is just a duplicate of LawfulApplicative.map_pure, but sometimes applies when that doesn't.

It is named with a prime to avoid conflict with the inherited field LawfulMonad.map_pure.

@[simp]

theorem LawfulMonad.map_map {α α✝ a✝ : Type u_1} {g : α✝ → a✝} {f : α → α✝} {m : Type u_1 → Type u_2} [Monad m] [LawfulMonad m] {x : m α} : g <$> f <$> x = (fun (a : α) => g (f a)) <$> x

This is just a duplicate of Functor.map_map, but sometimes applies when that doesn't.

theorem LawfulMonad.mk' (m : Type u → Type v) [Monad m] (id_map : ∀ {α : Type u} (x : m α), id <$> x = x) (pure_bind : ∀ {α β : Type u} (x : α) (f : α → m β), pure x >>= f = f x) (bind_assoc : ∀ {α β γ : Type u} (x : m α) (f : α → m β) (g : β → m γ), x >>= f >>= g = x >>= fun (x : α) => f x >>= g) (map_const : ∀ {α β : Type u} (x : α) (y : m β), Functor.mapConst x y = Function.const β x <$> y := by intros; rfl) (seqLeft_eq : ∀ {α β : Type u} (x : m α) (y : m β), x <* y = do let a ← x let _ ← y pure a := by intros; rfl) (seqRight_eq : ∀ {α β : Type u} (x : m α) (y : m β), x *> y = do let _ ← x y := by intros; rfl) (bind_pure_comp : ∀ {α β : Type u} (f : α → β) (x : m α), (do let y ← x pure (f y)) = f <$> x := by intros; rfl) (bind_map : ∀ {α β : Type u} (f : m (α → β)) (x : m α), (do let x_1 ← f x_1 <$> x) = f <*> x := by intros; rfl) : LawfulMonad m

An alternative constructor for LawfulMonad which has more defaultable fields in the common case.

Id

@[simp]

theorem Id.map_eq {α β : Type u_1} (x : Id α) (f : α → β) : f <$> x = f x

@[simp]

theorem Id.bind_eq {α β : Type u_1} (x : Id α) (f : α → id β) : x >>= f = f x

@[simp]

theorem Id.pure_eq {α : Type u_1} (a : α) : pure a = a

instance Id.instLawfulMonad : LawfulMonad Id

Option

instance instLawfulMonadOption : LawfulMonad Option

instance instLawfulApplicativeOption : LawfulApplicative Option

instance instLawfulFunctorOption : LawfulFunctor Option