BoxIntegral.TaggedPrepartition.IsHenstock.card_filter_tag_eq_le

theorem BoxIntegral.TaggedPrepartition.IsHenstock.card_filter_tag_eq_le : ∀ {ι : Type u_1} {I : BoxIntegral.Box ι} {π : BoxIntegral.TaggedPrepartition I} [inst : Fintype ι], π.IsHenstock → ∀ (x : ι → ℝ), (Finset.filter (fun J => π.tag J = x) π.boxes).card ≤ 2 ^ Fintype.card ι

In the context of a Henstock prepartition, the theorem states that for any given tagging function value `x`, the number of prepartition boxes that are tagged with `x` is at most `2` raised to the power of the cardinality of the finite type `ι`. Here, `ι` denotes the type of the indices, `I` represents a box in the prepartition, and `π` is a Henstock prepartition of `I`. The tagging function `π.tag` assigns a point in each box of the prepartition. Therefore, this theorem is about the constraint on the number of boxes that can have the same tag in a Henstock prepartition.

More concisely: In a Henstock prepartition, the number of boxes with the same tag value under the tagging function is bounded by 2 raised to the power of the number of indices.