/-- Equivalences between `Option α` and `β` that send `none` to `x` are equivalent to
equivalences between `α` and `{y : β // y ≠ x}`. -/
def optionSubtype [DecidableEq β] (x : β) :
    { e : Option α ≃ β // e none = x } ≃ (α ≃ { y : β // y ≠ x }) where
  toFun e :=
    { toFun := fun a =>
        ⟨(e : Option α ≃ β) a, ((EquivLike.injective _).ne_iff' e.property).2 (some_ne_none _)⟩,
      invFun := fun b =>
        get _
          (ne_none_iff_isSome.1
            (((EquivLike.injective _).ne_iff'
              ((apply_eq_iff_eq_symm_apply _).1 e.property).symm).2 b.property)),
      left_inv := fun a => by
        rw [← some_inj, some_get]
        exact symm_apply_apply (e : Option α ≃ β) a,
      right_inv := fun b => by
        ext
        simp }
  invFun e :=
    ⟨{  toFun := fun a => casesOn' a x (Subtype.val ∘ e),
        invFun := fun b => if h : b = x then none else e.symm ⟨b, h⟩,
        left_inv := fun a => by
          cases a with
          | none => simp
          | some a =>
            simp only [casesOn'_some, Function.comp_apply, Subtype.coe_eta,
              symm_apply_apply, dite_eq_ite]
            exact if_neg (e a).property,
        right_inv := fun b => by
          by_cases h : b = x <;> simp [h] },
      rfl⟩
  left_inv e := by
    ext a
    cases a
    · 

Proof state:
case a.H.none
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝ : DecidableEq β
x : β
e : { e // e none = x }
⊢ ↑((fun e =>
            {
              val :=
                { toFun := fun a => casesOn' a x (Subtype.val ∘ ⇑e),
                  invFun := fun b => if h : b = x then none else some (e.symm { val := b, property := h }),
                  left_inv :=
                    (_ :
                      ∀ (a : Option α),
                        (fun b => if h : b = x then none else some (e.symm { val := b, property := h }))
                            ((fun a => casesOn' a x (Subtype.val ∘ ⇑e)) a) =
                          a),
                  right_inv :=
                    (_ :
                      ∀ (b : β),
                        (fun a => casesOn' a x (Subtype.val ∘ ⇑e))
                            ((fun b => if h : b = x then none else some (e.symm { val := b, property := h })) b) =
                          b) },
              property :=
                (_ :
                  { toFun := fun a => casesOn' a x (Subtype.val ∘ ⇑e),
                        invFun := fun b => if h : b = x then none else some (e.symm { val := b, property := h }),
                        left_inv :=
                          (_ :
                            ∀ (a : Option α),
                              (fun b => if h : b = x then none else some (e.symm { val := b, property := h }))
                                  ((fun a => casesOn' a x (Subtype.val ∘ ⇑e)) a) =
                                a),
                        right_inv :=
                          (_ :
                            ∀ (b : β),
                              (fun a => casesOn' a x (Subtype.val ∘ ⇑e))
                                  ((fun b => if h : b = x then none else some (e.symm { val := b, property := h })) b) =
                                b) }
                      none =
                    { toFun := fun a => casesOn' a x (Subtype.val ∘ ⇑e),
                        invFun := fun b => if h : b = x then none else some (e.symm { val := b, property := h }),
                        left_inv :=
                          (_ :
                            ∀ (a : Option α),
                              (fun b => if h : b = x then none else some (e.symm { val := b, property := h }))
                                  ((fun a => casesOn' a x (Subtype.val ∘ ⇑e)) a) =
                                a),
                        right_inv :=
                          (_ :
                            ∀ (b : β),
                              (fun a => casesOn' a x (Subtype.val ∘ ⇑e))
                                  ((fun b => if h : b = x then none else some (e.symm { val := b, property := h })) b) =
                                b) }
                      none) })
          ((fun e =>
              { toFun := fun a => { val := ↑e (some a), property := (_ : ↑e (some a) ≠ x) },
                invFun := fun b => Option.get ((↑e).symm ↑b) (_ : isSome ((↑e).symm ↑b) = true),
                left_inv :=
                  (_ :
                    ∀ (a : α),
                      (fun b => Option.get ((↑e).symm ↑b) (_ : isSome ((↑e).symm ↑b) = true))
                          ((fun a => { val := ↑e (some a), property := (_ : ↑e (some a) ≠ x) }) a) =
                        a),
                right_inv :=
                  (_ :
                    ∀ (b : { y // y ≠ x }),
                      (fun a => { val := ↑e (some a), property := (_ : ↑e (some a) ≠ x) })
                          ((fun b => Option.get ((↑e).symm ↑b) (_ : isSome ((↑e).symm ↑b) = true)) b) =
                        b) })
            e))
      none =
    ↑e none

Next tactic:
simpa using e.property.symm