Finite equipartitions

This file defines finite equipartitions, the partitions whose parts all are the same size up to a difference of 1.

Main declarations

• Finpartition.IsEquipartition: Predicate for a Finpartition to be an equipartition.

def Finpartition.IsEquipartition {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) : Prop

An equipartition is a partition whose parts are all the same size, up to a difference of 1.

theorem Finpartition.isEquipartition_iff_card_parts_eq_average {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) : Finpartition.IsEquipartition P ↔ ∀ (a : Finset α), a ∈ P.parts → Finset.card a = Finset.card s / Finset.card P.parts ∨ Finset.card a = Finset.card s / Finset.card P.parts + 1

theorem Finpartition.Set.Subsingleton.isEquipartition {α : Type u_1} [DecidableEq α] {s : Finset α} {P : Finpartition s} (h : Set.Subsingleton ↑P.parts) : Finpartition.IsEquipartition P

theorem Finpartition.IsEquipartition.card_parts_eq_average {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} {P : Finpartition s} (hP : Finpartition.IsEquipartition P) (ht : t ∈ P.parts) : Finset.card t = Finset.card s / Finset.card P.parts ∨ Finset.card t = Finset.card s / Finset.card P.parts + 1

theorem Finpartition.IsEquipartition.average_le_card_part {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} {P : Finpartition s} (hP : Finpartition.IsEquipartition P) (ht : t ∈ P.parts) : Finset.card s / Finset.card P.parts ≤ Finset.card t

theorem Finpartition.IsEquipartition.card_part_le_average_add_one {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} {P : Finpartition s} (hP : Finpartition.IsEquipartition P) (ht : t ∈ P.parts) : Finset.card t ≤ Finset.card s / Finset.card P.parts + 1

Discrete and indiscrete finpartition

theorem Finpartition.bot_isEquipartition {α : Type u_1} [DecidableEq α] (s : Finset α) : Finpartition.IsEquipartition ⊥

theorem Finpartition.top_isEquipartition {α : Type u_1} [DecidableEq α] (s : Finset α) [Decidable (s = ⊥)] : Finpartition.IsEquipartition ⊤

theorem Finpartition.indiscrete_isEquipartition {α : Type u_1} [DecidableEq α] (s : Finset α) {hs : s ≠ ∅} : Finpartition.IsEquipartition (Finpartition.indiscrete hs)