Documentation

Mathlib.Combinatorics.SetFamily.Compression.Down

Down-compressions #

This file defines down-compression.

Down-compressing π’œ : Finset (Finset Ξ±) along a : Ξ± means removing a from the elements of π’œ, when the resulting set is not already in π’œ.

Main declarations #

Notation #

𝓓 a π’œ is notation for Down.compress a π’œ in locale SetFamily.

References #

Tags #

compression, down-compression

source
def Finset.nonMemberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
Finset (Finset Ξ±)

Elements of π’œ that do not contain a.

Instances For
    source
    def Finset.memberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
    Finset (Finset Ξ±)

    Image of the elements of π’œ which contain a under removing a. Finsets that do not contain a such that insert a s ∈ π’œ.

    Instances For
      source
      @[simp]
      theorem Finset.mem_nonMemberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±} {a : Ξ±} :
      s ∈ Finset.nonMemberSubfamily a π’œ ↔ s ∈ π’œ ∧ Β¬a ∈ s
      source
      @[simp]
      theorem Finset.mem_memberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±} {a : Ξ±} :
      s ∈ Finset.memberSubfamily a π’œ ↔ insert a s ∈ π’œ ∧ Β¬a ∈ s
      source
      theorem Finset.nonMemberSubfamily_inter {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) (ℬ : Finset (Finset Ξ±)) :
      Finset.nonMemberSubfamily a (π’œ ∩ ℬ) = Finset.nonMemberSubfamily a π’œ ∩ Finset.nonMemberSubfamily a ℬ
      source
      theorem Finset.memberSubfamily_inter {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) (ℬ : Finset (Finset Ξ±)) :
      Finset.memberSubfamily a (π’œ ∩ ℬ) = Finset.memberSubfamily a π’œ ∩ Finset.memberSubfamily a ℬ
      source
      theorem Finset.nonMemberSubfamily_union {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) (ℬ : Finset (Finset Ξ±)) :
      Finset.nonMemberSubfamily a (π’œ βˆͺ ℬ) = Finset.nonMemberSubfamily a π’œ βˆͺ Finset.nonMemberSubfamily a ℬ
      source
      theorem Finset.memberSubfamily_union {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) (ℬ : Finset (Finset Ξ±)) :
      Finset.memberSubfamily a (π’œ βˆͺ ℬ) = Finset.memberSubfamily a π’œ βˆͺ Finset.memberSubfamily a ℬ
      source
      theorem Finset.card_memberSubfamily_add_card_nonMemberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
      Finset.card (Finset.memberSubfamily a π’œ) + Finset.card (Finset.nonMemberSubfamily a π’œ) = Finset.card π’œ
      source
      theorem Finset.memberSubfamily_union_nonMemberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
      Finset.memberSubfamily a π’œ βˆͺ Finset.nonMemberSubfamily a π’œ = Finset.image (fun s => Finset.erase s a) π’œ
      source
      @[simp]
      theorem Finset.memberSubfamily_memberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {a : Ξ±} :
      Finset.memberSubfamily a (Finset.memberSubfamily a π’œ) = βˆ…
      source
      @[simp]
      theorem Finset.memberSubfamily_nonMemberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {a : Ξ±} :
      Finset.memberSubfamily a (Finset.nonMemberSubfamily a π’œ) = βˆ…
      source
      @[simp]
      theorem Finset.nonMemberSubfamily_memberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {a : Ξ±} :
      Finset.nonMemberSubfamily a (Finset.memberSubfamily a π’œ) = Finset.memberSubfamily a π’œ
      source
      @[simp]
      theorem Finset.nonMemberSubfamily_nonMemberSubfamily {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {a : Ξ±} :
      Finset.nonMemberSubfamily a (Finset.nonMemberSubfamily a π’œ) = Finset.nonMemberSubfamily a π’œ
      source
      def Down.compression {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
      Finset (Finset Ξ±)

      a-down-compressing π’œ means removing a from the elements of π’œ that contain it, when the resulting Finset is not already in π’œ.

      Instances For
        source
        def FinsetFamily.term𝓓 :
        Lean.ParserDescr

        a-down-compressing π’œ means removing a from the elements of π’œ that contain it, when the resulting Finset is not already in π’œ.

        Instances For
          source
          theorem Down.mem_compression {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±} {a : Ξ±} :
          s ∈ Down.compression a π’œ ↔ s ∈ π’œ ∧ Finset.erase s a ∈ π’œ ∨ Β¬s ∈ π’œ ∧ insert a s ∈ π’œ

          a is in the down-compressed family iff it's in the original and its compression is in the original, or it's not in the original but it's the compression of something in the original.

          source
          theorem Down.erase_mem_compression {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±} {a : Ξ±} (hs : s ∈ π’œ) :
          Finset.erase s a ∈ Down.compression a π’œ
          source
          theorem Down.erase_mem_compression_of_mem_compression {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±} {a : Ξ±} :
          s ∈ Down.compression a π’œ β†’ Finset.erase s a ∈ Down.compression a π’œ
          source
          theorem Down.mem_compression_of_insert_mem_compression {Ξ± : Type u_1} [DecidableEq Ξ±] {π’œ : Finset (Finset Ξ±)} {s : Finset Ξ±} {a : Ξ±} (h : insert a s ∈ Down.compression a π’œ) :
          s ∈ Down.compression a π’œ
          source
          @[simp]
          theorem Down.compression_idem {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
          Down.compression a (Down.compression a π’œ) = Down.compression a π’œ

          Down-compressing a family is idempotent.

          source
          @[simp]
          theorem Down.card_compression {Ξ± : Type u_1} [DecidableEq Ξ±] (a : Ξ±) (π’œ : Finset (Finset Ξ±)) :
          Finset.card (Down.compression a π’œ) = Finset.card π’œ

          Down-compressing a family doesn't change its size.