Documentation

Init.Data.Option.Lemmas

theorem Option.mem_iff {α : Type u_1} {a : α} {b : Option α} :
a b b = some a
theorem Option.mem_some {α : Type u_1} {a b : α} :
a some b b = a
theorem Option.mem_some_self {α : Type u_1} (a : α) :
a some a
theorem Option.some_ne_none {α : Type u_1} (x : α) :
some x none
theorem Option.forall {α : Type u_1} {p : Option αProp} :
(∀ (x : Option α), p x) p none ∀ (x : α), p (some x)
theorem Option.exists {α : Type u_1} {p : Option αProp} :
(∃ (x : Option α), p x) p none ∃ (x : α), p (some x)
theorem Option.get_mem {α : Type u_1} {o : Option α} (h : o.isSome = true) :
o.get h o
theorem Option.get_of_mem {α : Type u_1} {a : α} {o : Option α} (h : o.isSome = true) :
a oo.get h = a
theorem Option.not_mem_none {α : Type u_1} (a : α) :
¬a none
theorem Option.getD_of_ne_none {α : Type u_1} {x : Option α} (hx : x none) (y : α) :
some (x.getD y) = x
theorem Option.getD_eq_iff {α : Type u_1} {o : Option α} {a b : α} :
o.getD a = b o = some b o = none a = b
@[simp]
theorem Option.get!_none {α : Type u_1} [Inhabited α] :
none.get! = default
@[simp]
theorem Option.get!_some {α : Type u_1} [Inhabited α] {a : α} :
(some a).get! = a
theorem Option.get_eq_get! {α : Type u_1} [Inhabited α] (o : Option α) {h : o.isSome = true} :
o.get h = o.get!
theorem Option.get_eq_getD {α : Type u_1} {fallback : α} (o : Option α) {h : o.isSome = true} :
o.get h = o.getD fallback
theorem Option.some_get! {α : Type u_1} [Inhabited α] (o : Option α) :
o.isSome = truesome o.get! = o
theorem Option.get!_eq_getD {α : Type u_1} [Inhabited α] (o : Option α) :
o.get! = o.getD default
@[reducible, inline, deprecated Option.get!_eq_getD]
abbrev Option.get!_eq_getD_default {α : Type u_1} [Inhabited α] (o : Option α) :
o.get! = o.getD default
Equations
Instances For
    theorem Option.mem_unique {α : Type u_1} {o : Option α} {a b : α} (ha : a o) (hb : b o) :
    a = b
    theorem Option.ext {α : Type u_1} {o₁ o₂ : Option α} :
    (∀ (a : α), a o₁ a o₂)o₁ = o₂
    theorem Option.eq_none_iff_forall_not_mem {α✝ : Type u_1} {o : Option α✝} :
    o = none ∀ (a : α✝), ¬a o
    theorem Option.isSome_iff_exists {α✝ : Type u_1} {x : Option α✝} :
    x.isSome = true ∃ (a : α✝), x = some a
    theorem Option.isSome_eq_isSome {α✝ : Type u_1} {x : Option α✝} {α✝¹ : Type u_2} {y : Option α✝¹} :
    x.isSome = y.isSome (x = none y = none)
    @[simp]
    theorem Option.not_isSome {α✝ : Type u_1} {a : Option α✝} :
    a.isSome = false a.isNone = true
    theorem Option.eq_some_iff_get_eq {α✝ : Type u_1} {o : Option α✝} {a : α✝} :
    o = some a ∃ (h : o.isSome = true), o.get h = a
    theorem Option.eq_some_of_isSome {α : Type u_1} {o : Option α} (h : o.isSome = true) :
    o = some (o.get h)
    theorem Option.isSome_iff_ne_none {α✝ : Type u_1} {o : Option α✝} :
    o.isSome = true o none
    theorem Option.not_isSome_iff_eq_none {α✝ : Type u_1} {o : Option α✝} :
    ¬o.isSome = true o = none
    theorem Option.ne_none_iff_isSome {α✝ : Type u_1} {o : Option α✝} :
    o none o.isSome = true
    theorem Option.ne_none_iff_exists {α✝ : Type u_1} {o : Option α✝} :
    o none ∃ (x : α✝), some x = o
    theorem Option.ne_none_iff_exists' {α✝ : Type u_1} {o : Option α✝} :
    o none ∃ (x : α✝), o = some x
    theorem Option.bex_ne_none {α : Type u_1} {p : Option αProp} :
    (∃ (x : Option α), ∃ (x_1 : x none), p x) ∃ (x : α), p (some x)
    theorem Option.ball_ne_none {α : Type u_1} {p : Option αProp} :
    (∀ (x : Option α), x nonep x) ∀ (x : α), p (some x)
    @[simp]
    theorem Option.pure_def {α : Type u_1} :
    pure = some
    @[simp]
    theorem Option.bind_eq_bind {α β : Type u_1} :
    bind = Option.bind
    @[simp]
    theorem Option.bind_some {α : Type u_1} (x : Option α) :
    x.bind some = x
    @[simp]
    theorem Option.bind_none {α : Type u_1} {β : Type u_2} (x : Option α) :
    (x.bind fun (x : α) => none) = none
    theorem Option.bind_eq_some {α✝ : Type u_1} {b : α✝} {α✝¹ : Type u_2} {x : Option α✝¹} {f : α✝¹Option α✝} :
    x.bind f = some b ∃ (a : α✝¹), x = some a f a = some b
    @[simp]
    theorem Option.bind_eq_none {α : Type u_1} {β : Type u_2} {o : Option α} {f : αOption β} :
    o.bind f = none ∀ (a : α), o = some af a = none
    theorem Option.bind_eq_none' {α : Type u_1} {β : Type u_2} {o : Option α} {f : αOption β} :
    o.bind f = none ∀ (b : β) (a : α), a o¬b f a
    theorem Option.mem_bind_iff {α : Type u_1} {β : Type u_2} {b : β} {o : Option α} {f : αOption β} :
    b o.bind f ∃ (a : α), a o b f a
    theorem Option.bind_comm {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : αβOption γ} (a : Option α) (b : Option β) :
    (a.bind fun (x : α) => b.bind (f x)) = b.bind fun (y : β) => a.bind fun (x : α) => f x y
    theorem Option.bind_assoc {α : Type u_1} {β : Type u_2} {γ : Type u_3} (x : Option α) (f : αOption β) (g : βOption γ) :
    (x.bind f).bind g = x.bind fun (y : α) => (f y).bind g
    theorem Option.join_eq_some {α✝ : Type u_1} {a : α✝} {x : Option (Option α✝)} :
    x.join = some a x = some (some a)
    theorem Option.join_ne_none {α✝ : Type u_1} {x : Option (Option α✝)} :
    x.join none ∃ (z : α✝), x = some (some z)
    theorem Option.join_ne_none' {α✝ : Type u_1} {x : Option (Option α✝)} :
    ¬x.join = none ∃ (z : α✝), x = some (some z)
    theorem Option.join_eq_none {α✝ : Type u_1} {o : Option (Option α✝)} :
    o.join = none o = none o = some none
    theorem Option.bind_id_eq_join {α : Type u_1} {x : Option (Option α)} :
    x.bind id = x.join
    @[simp]
    theorem Option.map_eq_map {α✝ α✝¹ : Type u_1} {f : α✝α✝¹} :
    theorem Option.map_none {α✝ α✝¹ : Type u_1} {f : α✝α✝¹} :
    f <$> none = none
    theorem Option.map_some {α✝ α✝¹ : Type u_1} {f : α✝α✝¹} {a : α✝} :
    f <$> some a = some (f a)
    @[simp]
    theorem Option.map_eq_some' {α✝ : Type u_1} {b : α✝} {α✝¹ : Type u_2} {x : Option α✝¹} {f : α✝¹α✝} :
    Option.map f x = some b ∃ (a : α✝¹), x = some a f a = b
    theorem Option.map_eq_some {α✝ α✝¹ : Type u_1} {f : α✝α✝¹} {x : Option α✝} {b : α✝¹} :
    f <$> x = some b ∃ (a : α✝), x = some a f a = b
    @[simp]
    theorem Option.map_eq_none' {α✝ : Type u_1} {x : Option α✝} {α✝¹ : Type u_2} {f : α✝α✝¹} :
    Option.map f x = none x = none
    theorem Option.isSome_map {α α✝ : Type u_1} {f : αα✝} {x : Option α} :
    (f <$> x).isSome = x.isSome
    @[simp]
    theorem Option.isSome_map' {α : Type u_1} {α✝ : Type u_2} {f : αα✝} {x : Option α} :
    (Option.map f x).isSome = x.isSome
    @[simp]
    theorem Option.isNone_map' {α : Type u_1} {α✝ : Type u_2} {f : αα✝} {x : Option α} :
    (Option.map f x).isNone = x.isNone
    theorem Option.map_eq_none {α✝ α✝¹ : Type u_1} {f : α✝α✝¹} {x : Option α✝} :
    f <$> x = none x = none
    theorem Option.map_eq_bind {α : Type u_1} {α✝ : Type u_2} {f : αα✝} {x : Option α} :
    Option.map f x = x.bind (some f)
    theorem Option.map_congr {α : Type u_1} {α✝ : Type u_2} {f g : αα✝} {x : Option α} (h : ∀ (a : α), a xf a = g a) :
    @[simp]
    theorem Option.map_id_fun {α : Type u} :
    theorem Option.map_id' {α : Type u_1} {x : Option α} :
    Option.map (fun (a : α) => a) x = x
    @[simp]
    theorem Option.map_id_fun' {α : Type u} :
    (Option.map fun (a : α) => a) = id
    theorem Option.get_map {α : Type u_1} {β : Type u_2} {f : αβ} {o : Option α} {h : (Option.map f o).isSome = true} :
    (Option.map f o).get h = f (o.get )
    @[simp]
    theorem Option.map_map {β : Type u_1} {γ : Type u_2} {α : Type u_3} (h : βγ) (g : αβ) (x : Option α) :
    theorem Option.comp_map {β : Type u_1} {γ : Type u_2} {α : Type u_3} (h : βγ) (g : αβ) (x : Option α) :
    @[simp]
    theorem Option.map_comp_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : αβ) (g : βγ) :
    theorem Option.mem_map_of_mem {α : Type u_1} {β : Type u_2} {x : Option α} {a : α} (g : αβ) (h : a x) :
    g a Option.map g x
    @[simp]
    theorem Option.map_if {α : Type u_1} {β : Type u_2} {c : Prop} {a : α} {f : αβ} [Decidable c] :
    Option.map f (if c then some a else none) = if c then some (f a) else none
    @[simp]
    theorem Option.map_dif {α : Type u_1} {β : Type u_2} {c : Prop} {f : αβ} [Decidable c] {a : cα} :
    Option.map f (if h : c then some (a h) else none) = if h : c then some (f (a h)) else none
    @[simp]
    theorem Option.filter_none {α : Type u_1} (p : αBool) :
    Option.filter p none = none
    theorem Option.filter_some {α✝ : Type u_1} {p : α✝Bool} {a : α✝} :
    Option.filter p (some a) = if p a = true then some a else none
    theorem Option.isSome_filter_of_isSome {α : Type u_1} (p : αBool) (o : Option α) (h : (Option.filter p o).isSome = true) :
    o.isSome = true
    @[simp]
    theorem Option.filter_eq_none {α : Type u_1} {o : Option α} {p : αBool} :
    Option.filter p o = none o = none ∀ (a : α), a o¬p a = true
    @[simp]
    theorem Option.filter_eq_some {α : Type u_1} {a : α} {o : Option α} {p : αBool} :
    theorem Option.mem_filter_iff {α : Type u_1} {p : αBool} {a : α} {o : Option α} :
    a Option.filter p o a o p a = true
    @[simp]
    theorem Option.all_guard {α : Type u_1} {q : αBool} (p : αProp) [DecidablePred p] (a : α) :
    Option.all q (Option.guard p a) = (!decide (p a) || q a)
    @[simp]
    theorem Option.any_guard {α : Type u_1} {q : αBool} (p : αProp) [DecidablePred p] (a : α) :
    Option.any q (Option.guard p a) = (decide (p a) && q a)
    theorem Option.bind_map_comm {α : Type u_1} {β : Type u_2} {x : Option (Option α)} {f : αβ} :
    x.bind (Option.map f) = (Option.map (Option.map f) x).bind id
    theorem Option.bind_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : αβ} {g : βOption γ} {x : Option α} :
    (Option.map f x).bind g = x.bind (g f)
    @[simp]
    theorem Option.map_bind {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : αOption β} {g : βγ} {x : Option α} :
    Option.map g (x.bind f) = x.bind (Option.map g f)
    theorem Option.join_map_eq_map_join {α : Type u_1} {β : Type u_2} {f : αβ} {x : Option (Option α)} :
    (Option.map (Option.map f) x).join = Option.map f x.join
    theorem Option.join_join {α : Type u_1} {x : Option (Option (Option α))} :
    x.join.join = (Option.map Option.join x).join
    theorem Option.mem_of_mem_join {α : Type u_1} {a : α} {x : Option (Option α)} (h : a x.join) :
    some a x
    @[simp]
    theorem Option.some_orElse {α : Type u_1} (a : α) (x : Option α) :
    (some a <|> x) = some a
    @[simp]
    theorem Option.none_orElse {α : Type u_1} (x : Option α) :
    (none <|> x) = x
    @[simp]
    theorem Option.orElse_none {α : Type u_1} (x : Option α) :
    (x <|> none) = x
    theorem Option.map_orElse {α : Type u_1} {α✝ : Type u_2} {f : αα✝} {x y : Option α} :
    Option.map f (x <|> y) = (Option.map f x <|> Option.map f y)
    @[simp]
    theorem Option.guard_eq_some {α✝ : Type u_1} {p : α✝Prop} {a b : α✝} [DecidablePred p] :
    Option.guard p a = some b a = b p a
    @[simp]
    theorem Option.guard_isSome {α✝ : Type u_1} {p : α✝Prop} {a : α✝} [DecidablePred p] :
    (Option.guard p a).isSome = true p a
    @[simp]
    theorem Option.guard_eq_none {α✝ : Type u_1} {p : α✝Prop} {a : α✝} [DecidablePred p] :
    Option.guard p a = none ¬p a
    @[simp]
    theorem Option.guard_pos {α✝ : Type u_1} {p : α✝Prop} {a : α✝} [DecidablePred p] (h : p a) :
    theorem Option.guard_congr {α : Type u_1} {f g : αProp} [DecidablePred f] [DecidablePred g] (h : ∀ (a : α), f a g a) :
    @[simp]
    theorem Option.guard_false {α : Type u_1} :
    (Option.guard fun (x : α) => False) = fun (x : α) => none
    @[simp]
    theorem Option.guard_true {α : Type u_1} :
    (Option.guard fun (x : α) => True) = some
    theorem Option.guard_comp {α : Type u_1} {β : Type u_2} {p : αProp} [DecidablePred p] {f : βα} :
    theorem Option.liftOrGet_eq_or_eq {α : Type u_1} {f : ααα} (h : ∀ (a b : α), f a b = a f a b = b) (o₁ o₂ : Option α) :
    Option.liftOrGet f o₁ o₂ = o₁ Option.liftOrGet f o₁ o₂ = o₂
    @[simp]
    theorem Option.liftOrGet_none_left {α : Type u_1} {f : ααα} {b : Option α} :
    Option.liftOrGet f none b = b
    @[simp]
    theorem Option.liftOrGet_none_right {α : Type u_1} {f : ααα} {a : Option α} :
    Option.liftOrGet f a none = a
    @[simp]
    theorem Option.liftOrGet_some_some {α : Type u_1} {f : ααα} {a b : α} :
    Option.liftOrGet f (some a) (some b) = some (f a b)
    @[simp]
    theorem Option.elim_none {β : Sort u_1} {α : Type u_2} (x : β) (f : αβ) :
    none.elim x f = x
    @[simp]
    theorem Option.elim_some {β : Sort u_1} {α : Type u_2} (x : β) (f : αβ) (a : α) :
    (some a).elim x f = f a
    @[simp]
    theorem Option.getD_map {α : Type u_1} {β : Type u_2} (f : αβ) (x : α) (o : Option α) :
    (Option.map f o).getD (f x) = f (o.getD x)
    noncomputable def Option.choice (α : Type u_1) :

    An arbitrary some a with a : α if α is nonempty, and otherwise none.

    Equations
    Instances For
      theorem Option.choice_eq {α : Type u_1} [Subsingleton α] (a : α) :
      @[simp]
      theorem Option.toList_some {α : Type u_1} (a : α) :
      (some a).toList = [a]
      @[simp]
      theorem Option.toList_none (α : Type u_1) :
      none.toList = []
      @[simp]
      theorem Option.some_or {α✝ : Type u_1} {a : α✝} {o : Option α✝} :
      (some a).or o = some a
      @[simp]
      theorem Option.none_or {α✝ : Type u_1} {o : Option α✝} :
      none.or o = o
      @[deprecated Option.some_or]
      theorem Option.or_some {α✝ : Type u_1} {a : α✝} {o : Option α✝} :
      (some a).or o = some a
      @[simp]
      theorem Option.or_some' {α : Type u_1} {a : α} {o : Option α} :
      o.or (some a) = some (o.getD a)

      This will be renamed to or_some once the existing deprecated lemma is removed.

      theorem Option.or_eq_bif {α✝ : Type u_1} {o o' : Option α✝} :
      o.or o' = bif o.isSome then o else o'
      @[simp]
      theorem Option.isSome_or {α✝ : Type u_1} {o o' : Option α✝} :
      (o.or o').isSome = (o.isSome || o'.isSome)
      @[simp]
      theorem Option.isNone_or {α✝ : Type u_1} {o o' : Option α✝} :
      (o.or o').isNone = (o.isNone && o'.isNone)
      @[simp]
      theorem Option.or_eq_none {α✝ : Type u_1} {o o' : Option α✝} :
      o.or o' = none o = none o' = none
      @[simp]
      theorem Option.or_eq_some {α✝ : Type u_1} {o o' : Option α✝} {a : α✝} :
      o.or o' = some a o = some a o = none o' = some a
      theorem Option.or_assoc {α✝ : Type u_1} {o₁ o₂ o₃ : Option α✝} :
      (o₁.or o₂).or o₃ = o₁.or (o₂.or o₃)
      @[simp]
      theorem Option.or_none {α✝ : Type u_1} {o : Option α✝} :
      o.or none = o
      @[simp]
      theorem Option.or_self {α✝ : Type u_1} {o : Option α✝} :
      o.or o = o
      theorem Option.or_eq_orElse {α✝ : Type u_1} {o o' : Option α✝} :
      o.or o' = o.orElse fun (x : Unit) => o'
      theorem Option.map_or {α✝ α✝¹ : Type u_1} {f : α✝α✝¹} {o o' : Option α✝} :
      f <$> o.or o' = (f <$> o).or (f <$> o')
      theorem Option.map_or' {α✝ : Type u_1} {o o' : Option α✝} {α✝¹ : Type u_2} {f : α✝α✝¹} :
      Option.map f (o.or o') = (Option.map f o).or (Option.map f o')
      theorem Option.or_of_isSome {α : Type u_1} {o o' : Option α} (h : o.isSome = true) :
      o.or o' = o
      theorem Option.or_of_isNone {α : Type u_1} {o o' : Option α} (h : o.isNone = true) :
      o.or o' = o'

      beq #

      @[simp]
      theorem Option.none_beq_none {α : Type u_1} [BEq α] :
      (none == none) = true
      @[simp]
      theorem Option.none_beq_some {α : Type u_1} [BEq α] (a : α) :
      (none == some a) = false
      @[simp]
      theorem Option.some_beq_none {α : Type u_1} [BEq α] (a : α) :
      (some a == none) = false
      @[simp]
      theorem Option.some_beq_some {α : Type u_1} [BEq α] {a b : α} :
      (some a == some b) = (a == b)
      @[simp]
      theorem Option.reflBEq_iff {α : Type u_1} [BEq α] :
      @[simp]
      theorem Option.lawfulBEq_iff {α : Type u_1} [BEq α] :

      ite #

      @[simp]
      theorem Option.dite_none_left_eq_some {β : Type u_1} {a : β} {p : Prop} [Decidable p] {b : ¬pOption β} :
      (if h : p then none else b h) = some a ∃ (h : ¬p), b h = some a
      @[simp]
      theorem Option.dite_none_right_eq_some {α : Type u_1} {a : α} {p : Prop} [Decidable p] {b : pOption α} :
      (if h : p then b h else none) = some a ∃ (h : p), b h = some a
      @[simp]
      theorem Option.some_eq_dite_none_left {β : Type u_1} {a : β} {p : Prop} [Decidable p] {b : ¬pOption β} :
      (some a = if h : p then none else b h) ∃ (h : ¬p), some a = b h
      @[simp]
      theorem Option.some_eq_dite_none_right {α : Type u_1} {a : α} {p : Prop} [Decidable p] {b : pOption α} :
      (some a = if h : p then b h else none) ∃ (h : p), some a = b h
      @[simp]
      theorem Option.ite_none_left_eq_some {β : Type u_1} {a : β} {p : Prop} [Decidable p] {b : Option β} :
      (if p then none else b) = some a ¬p b = some a
      @[simp]
      theorem Option.ite_none_right_eq_some {α : Type u_1} {a : α} {p : Prop} [Decidable p] {b : Option α} :
      (if p then b else none) = some a p b = some a
      @[simp]
      theorem Option.some_eq_ite_none_left {β : Type u_1} {a : β} {p : Prop} [Decidable p] {b : Option β} :
      (some a = if p then none else b) ¬p some a = b
      @[simp]
      theorem Option.some_eq_ite_none_right {α : Type u_1} {a : α} {p : Prop} [Decidable p] {b : Option α} :
      (some a = if p then b else none) p some a = b
      theorem Option.mem_dite_none_left {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : ¬pOption α} :
      (x if h : p then none else l h) ∃ (h : ¬p), x l h
      theorem Option.mem_dite_none_right {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : pOption α} :
      (x if h : p then l h else none) ∃ (h : p), x l h
      theorem Option.mem_ite_none_left {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : Option α} :
      (x if p then none else l) ¬p x l
      theorem Option.mem_ite_none_right {α : Type u_1} {p : Prop} {x : α} [Decidable p] {l : Option α} :
      (x if p then l else none) p x l
      @[simp]
      theorem Option.isSome_dite {β : Type u_1} {p : Prop} [Decidable p] {b : pβ} :
      (if h : p then some (b h) else none).isSome = true p
      @[simp]
      theorem Option.isSome_ite {α✝ : Type u_1} {b : α✝} {p : Prop} [Decidable p] :
      (if p then some b else none).isSome = true p
      @[simp]
      theorem Option.isSome_dite' {β : Type u_1} {p : Prop} [Decidable p] {b : ¬pβ} :
      (if h : p then none else some (b h)).isSome = true ¬p
      @[simp]
      theorem Option.isSome_ite' {α✝ : Type u_1} {b : α✝} {p : Prop} [Decidable p] :
      (if p then none else some b).isSome = true ¬p
      @[simp]
      theorem Option.get_dite {β : Type u_1} {p : Prop} [Decidable p] (b : pβ) (w : (if h : p then some (b h) else none).isSome = true) :
      (if h : p then some (b h) else none).get w = b
      @[simp]
      theorem Option.get_ite {α✝ : Type u_1} {b : α✝} {p : Prop} [Decidable p] (h : (if p then some b else none).isSome = true) :
      (if p then some b else none).get h = b
      @[simp]
      theorem Option.get_dite' {β : Type u_1} {p : Prop} [Decidable p] (b : ¬pβ) (w : (if h : p then none else some (b h)).isSome = true) :
      (if h : p then none else some (b h)).get w = b
      @[simp]
      theorem Option.get_ite' {α✝ : Type u_1} {b : α✝} {p : Prop} [Decidable p] (h : (if p then none else some b).isSome = true) :
      (if p then none else some b).get h = b

      pbind #

      @[simp]
      theorem Option.pbind_none {α✝ : Type u_1} {α✝¹ : Type u_2} {f : (a : α✝) → a noneOption α✝¹} :
      none.pbind f = none
      @[simp]
      theorem Option.pbind_some {α✝ : Type u_1} {a : α✝} {α✝¹ : Type u_2} {f : (a_1 : α✝) → a_1 some aOption α✝¹} :
      (some a).pbind f = f a
      @[simp]
      theorem Option.map_pbind {α : Type u_1} {β : Type u_2} {γ : Type u_3} {o : Option α} {f : (a : α) → a oOption β} {g : βγ} :
      Option.map g (o.pbind f) = o.pbind fun (a : α) (h : a o) => Option.map g (f a h)
      theorem Option.pbind_congr {α : Type u_1} {β : Type u_2} {o o' : Option α} (ho : o = o') {f : (a : α) → a oOption β} {g : (a : α) → a o'Option β} (hf : ∀ (a : α) (h : a o'), f a = g a h) :
      o.pbind f = o'.pbind g
      theorem Option.pbind_eq_none_iff {α : Type u_1} {β : Type u_2} {o : Option α} {f : (a : α) → a oOption β} :
      o.pbind f = none o = none ∃ (a : α), ∃ (h : a o), f a h = none
      theorem Option.pbind_isSome {α : Type u_1} {β : Type u_2} {o : Option α} {f : (a : α) → a oOption β} :
      ((o.pbind f).isSome = true) = ∃ (a : α), ∃ (h : a o), (f a h).isSome = true
      theorem Option.pbind_eq_some_iff {α : Type u_1} {β : Type u_2} {o : Option α} {f : (a : α) → a oOption β} {b : β} :
      o.pbind f = some b ∃ (a : α), ∃ (h : a o), f a h = some b

      pmap #

      @[simp]
      theorem Option.pmap_none {α : Type u_1} {β : Type u_2} {p : αProp} {f : (a : α) → p aβ} {h : ∀ (a : α), a nonep a} :
      Option.pmap f none h = none
      @[simp]
      theorem Option.pmap_some {α : Type u_1} {β : Type u_2} {a : α} {p : αProp} {f : (a : α) → p aβ} {h : ∀ (a_1 : α), a_1 some ap a_1} :
      Option.pmap f (some a) h = some (f a )
      @[simp]
      theorem Option.pmap_eq_none_iff {α : Type u_1} {β : Type u_2} {o : Option α} {p : αProp} {f : (a : α) → p aβ} {h : ∀ (a : α), a op a} :
      Option.pmap f o h = none o = none
      @[simp]
      theorem Option.pmap_isSome {α : Type u_1} {β : Type u_2} {p : αProp} {f : (a : α) → p aβ} {o : Option α} {h : ∀ (a : α), a op a} :
      (Option.pmap f o h).isSome = o.isSome
      @[simp]
      theorem Option.pmap_eq_some_iff {α : Type u_1} {β : Type u_2} {b : β} {p : αProp} {f : (a : α) → p aβ} {o : Option α} {h : ∀ (a : α), a op a} :
      Option.pmap f o h = some b ∃ (a : α), ∃ (h : p a), o = some a b = f a h