Extra definitions on `Option` #

This file defines more operations involving Option α. Lemmas about them are located in other files under Mathlib.Data.Option. Other basic operations on Option are defined in the core library.

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inductive Option.rel {α : Type u_1} {β : Type u_2} (r : α → β → Prop) :

Option α → Option β → Prop

some: ∀ {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {a : α} {b : β}, r a b → Option.rel r (some a) (some b)
If a ~ b, then some a ~ some b
none: ∀ {α : Type u_1} {β : Type u_2} {r : α → β → Prop}, Option.rel r none none
none ~ none

Lifts a relation α → β → Prop to a relation Option α → Option β → Prop by just adding none ~ none.

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def Option.traverse {F : Type u → Type v} [Applicative F] {α : Type u_1} {β : Type u} (f : α → F β) :

Option α → F (Option β)

Traverse an object of Option α with a function f : α → F β for an applicative F.

Equations

Option.traverse f x = match x with | none => pure none | some x => some <$> f x

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def Option.maybe {m : Type u → Type v} [Monad m] {α : Type u} :

Option (m α) → m (Option α)

If you maybe have a monadic computation in a [Monad m] which produces a term of type α, then there is a naturally associated way to always perform a computation in m which maybe produces a result.

Equations

Option.maybe x = match x with | none => pure none | some fn => some <$> fn

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@[deprecated Option.getDM]

def Option.getDM' {m : Type u_1 → Type u_2} {α : Type u_1} [Monad m] (x : m (Option α)) (y : m α) :

m α

Equations

Option.getDM' x y = do let __do_lift ← x Option.getDM __do_lift y

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def Option.elim' {α : Type u_1} {β : Type u_2} (b : β) (f : α → β) :

Option α → β

An elimination principle for Option. It is a nondependent version of Option.rec.

Equations

Option.elim' b f x = match x with | some a => f a | none => b

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@[simp]

theorem Option.elim'_none {α : Type u_1} {β : Type u_2} (b : β) (f : α → β) :

Option.elim' b f none = b

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@[simp]

theorem Option.elim'_some {α : Type u_1} {β : Type u_2} {a : α} (b : β) (f : α → β) :

Option.elim' b f (some a) = f a

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theorem Option.elim'_eq_elim {α : Type u_3} {β : Type u_4} (b : β) (f : α → β) (a : Option α) :

Option.elim' b f a = Option.elim a b f

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theorem Option.mem_some_iff {α : Type u_3} {a : α} {b : α} :

a ∈ some b ↔ b = a

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@[inline]

def Option.decidableEqNone {α : Type u_1} {o : Option α} :

Decidable (o = none)

o = none is decidable even if the wrapped type does not have decidable equality. This is not an instance because it is not definitionally equal to Option.decidableEq. Try to use o.isNone or o.isSome instead.

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