feat(Mathlib/Data/Finite/Option): option type is finite iff type is finite

Mathlib.lean

@@ -4032,6 +4032,7 @@ public import Mathlib.Data.FinEnum
 public import Mathlib.Data.FinEnum.Option
 public import Mathlib.Data.Finite.Card
 public import Mathlib.Data.Finite.Defs
+public import Mathlib.Data.Finite.Option
 public import Mathlib.Data.Finite.Perm
 public import Mathlib.Data.Finite.Prod
 public import Mathlib.Data.Finite.Set

Mathlib/Data/Finite/Option.lean

@@ -0,0 +1,34 @@
+/-
+Copyright (c) 2026 Alex Brodbelt. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Alex Brodbelt, Eric Wieser
+-/
+module
+
+public import Mathlib.Basic.Finite.Defs
+
+import Mathlib.Data.Fintype.Option
+import Mathlib.Logic.Equiv.Fin.Basic
+
+/-!
+# Finiteness conditions for `Option` types
+
+This file shows that `Option α` is finite iff `α` is. Similarly, `Option α` is infinite iff `α` is.
+-/
+
+public section
+
+/-- `Option α` is finite if and only if the underlying type `α` is finite. -/
+@[simp]
+theorem Option.finite_iff {α : Type*} : Finite (Option α) ↔ Finite α where
+  mpr _ := inferInstance
+  mp
+  | @Finite.intro _ 0 e => (e none).elim0
+  | @Finite.intro _ (n + 1) e => ⟨(e.trans (finSuccEquiv n)).removeNone⟩
+
+/-- `Option α` is infinite if and only if the underlying type `α` is infinite. -/
+@[simp]
+theorem Option.infinite_iff {α : Type*} : Infinite (Option α) ↔ Infinite α := by
+  rw [← not_finite_iff_infinite, ← not_finite_iff_infinite, Option.finite_iff]
+
+end