split <;> simp [Option.map]
case h
α : Type u_1
β b✝ : Type u_2
f : α ⇀ β
b : b✝
a : α
⊢ Option.map (fun x => b) (f a) = if ∃ b, f a = some b then some b else none
case h.inl
α : Type u_1
β b✝ : Type u_2
f : α ⇀ β
b : b✝
a : α
h✝ : ∃ b, f a = some b
⊢ (match f a with
    | some x => some b
    | none => none) =
    some b

case h.inr
α : Type u_1
β b✝ : Type u_2
f : α ⇀ β
b : b✝
a : α
h✝ : ¬∃ b, f a = some b
⊢ (match f a with
    | some x => some b
    | none => none) =
    none