Documentation

Mathlib.Order.Partition.Equipartition

Finite equipartitions #

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

Main declarations #

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

Equations
  • P.IsEquipartition = (P.parts).EquitableOn Finset.card
Instances For
    theorem Finpartition.isEquipartition_iff_card_parts_eq_average {α : Type u_1} [DecidableEq α] {s : Finset α} (P : Finpartition s) :
    P.IsEquipartition aP.parts, a.card = s.card / P.parts.card a.card = s.card / P.parts.card + 1
    theorem Finpartition.not_isEquipartition {α : Type u_1} [DecidableEq α] {s : Finset α} {P : Finpartition s} :
    ¬P.IsEquipartition aP.parts, bP.parts, b.card + 1 < a.card
    theorem Set.Subsingleton.isEquipartition {α : Type u_1} [DecidableEq α] {s : Finset α} {P : Finpartition s} (h : (P.parts).Subsingleton) :
    P.IsEquipartition
    theorem Finpartition.IsEquipartition.card_parts_eq_average {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} {P : Finpartition s} (hP : P.IsEquipartition) (ht : t P.parts) :
    t.card = s.card / P.parts.card t.card = s.card / P.parts.card + 1
    theorem Finpartition.IsEquipartition.average_le_card_part {α : Type u_1} [DecidableEq α] {s : Finset α} {t : Finset α} {P : Finpartition s} (hP : P.IsEquipartition) (ht : t P.parts) :
    s.card / P.parts.card t.card
    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 : P.IsEquipartition) (ht : t P.parts) :
    t.card s.card / P.parts.card + 1

    Discrete and indiscrete finpartition #

    theorem Finpartition.bot_isEquipartition {α : Type u_1} [DecidableEq α] (s : Finset α) :
    .IsEquipartition
    theorem Finpartition.top_isEquipartition {α : Type u_1} [DecidableEq α] (s : Finset α) [Decidable (s = )] :
    .IsEquipartition
    theorem Finpartition.indiscrete_isEquipartition {α : Type u_1} [DecidableEq α] (s : Finset α) {hs : s } :
    (Finpartition.indiscrete hs).IsEquipartition