Down-compressions

This file defines down-compression.

Down-compressing 𝒜 : Finset (Finset α) along a : α means removing a from the elements of 𝒜, when the resulting set is not already in 𝒜.

Main declarations

• Finset.nonMemberSubfamily: 𝒜.nonMemberSubfamily a is the subfamily of sets not containing a.
• Finset.memberSubfamily: 𝒜.memberSubfamily a is the image of the subfamily of sets containing a under removing a.
• Down.compression: Down-compression.

Notation

𝓓 a 𝒜 is notation for Down.compress a 𝒜 in locale SetFamily.

References

• https://github.com/b-mehta/maths-notes/blob/master/iii/mich/combinatorics.pdf

Tags

compression, down-compression

def Finset.nonMemberSubfamily {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : Finset (Finset α)

Elements of 𝒜 that do not contain a.

Equations

• Finset.nonMemberSubfamily a 𝒜 = Finset.filter (fun (s : Finset α) => a ∉ s) 𝒜

def Finset.memberSubfamily {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : Finset (Finset α)

Image of the elements of 𝒜 which contain a under removing a. Finsets that do not contain a such that insert a s ∈ 𝒜.

Equations

• Finset.memberSubfamily a 𝒜 = Finset.image (fun (s : Finset α) => s.erase a) (Finset.filter (fun (s : Finset α) => a ∈ s) 𝒜)

@[simp]

theorem Finset.mem_nonMemberSubfamily {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} : s ∈ Finset.nonMemberSubfamily a 𝒜 ↔ s ∈ 𝒜 ∧ a ∉ s

@[simp]

theorem Finset.mem_memberSubfamily {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} : s ∈ Finset.memberSubfamily a 𝒜 ↔ insert a s ∈ 𝒜 ∧ a ∉ s

theorem Finset.nonMemberSubfamily_inter {α : Type u_1} [DecidableEq α] (a : α) (𝒜 ℬ : Finset (Finset α)) : Finset.nonMemberSubfamily a (𝒜 ∩ ℬ) = Finset.nonMemberSubfamily a 𝒜 ∩ Finset.nonMemberSubfamily a ℬ

theorem Finset.memberSubfamily_inter {α : Type u_1} [DecidableEq α] (a : α) (𝒜 ℬ : Finset (Finset α)) : Finset.memberSubfamily a (𝒜 ∩ ℬ) = Finset.memberSubfamily a 𝒜 ∩ Finset.memberSubfamily a ℬ

theorem Finset.nonMemberSubfamily_union {α : Type u_1} [DecidableEq α] (a : α) (𝒜 ℬ : Finset (Finset α)) : Finset.nonMemberSubfamily a (𝒜 ∪ ℬ) = Finset.nonMemberSubfamily a 𝒜 ∪ Finset.nonMemberSubfamily a ℬ

theorem Finset.memberSubfamily_union {α : Type u_1} [DecidableEq α] (a : α) (𝒜 ℬ : Finset (Finset α)) : Finset.memberSubfamily a (𝒜 ∪ ℬ) = Finset.memberSubfamily a 𝒜 ∪ Finset.memberSubfamily a ℬ

theorem Finset.card_memberSubfamily_add_card_nonMemberSubfamily {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : (Finset.memberSubfamily a 𝒜).card + (Finset.nonMemberSubfamily a 𝒜).card = 𝒜.card

theorem Finset.memberSubfamily_union_nonMemberSubfamily {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : Finset.memberSubfamily a 𝒜 ∪ Finset.nonMemberSubfamily a 𝒜 = Finset.image (fun (s : Finset α) => s.erase a) 𝒜

@[simp]

theorem Finset.memberSubfamily_memberSubfamily {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} : Finset.memberSubfamily a (Finset.memberSubfamily a 𝒜) = ∅

@[simp]

theorem Finset.memberSubfamily_nonMemberSubfamily {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} : Finset.memberSubfamily a (Finset.nonMemberSubfamily a 𝒜) = ∅

@[simp]

theorem Finset.nonMemberSubfamily_memberSubfamily {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} : Finset.nonMemberSubfamily a (Finset.memberSubfamily a 𝒜) = Finset.memberSubfamily a 𝒜

@[simp]

theorem Finset.nonMemberSubfamily_nonMemberSubfamily {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} : Finset.nonMemberSubfamily a (Finset.nonMemberSubfamily a 𝒜) = Finset.nonMemberSubfamily a 𝒜

theorem Finset.memberSubfamily_image_insert {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} (h𝒜 : ∀ s ∈ 𝒜, a ∉ s) : Finset.memberSubfamily a (Finset.image (insert a) 𝒜) = 𝒜

@[simp]

theorem Finset.nonMemberSubfamily_image_insert {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} : Finset.nonMemberSubfamily a (Finset.image (insert a) 𝒜) = ∅

@[simp]

theorem Finset.memberSubfamily_image_erase {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {a : α} : Finset.memberSubfamily a (Finset.image (fun (x : Finset α) => x.erase a) 𝒜) = ∅

theorem Finset.image_insert_memberSubfamily {α : Type u_1} [DecidableEq α] (𝒜 : Finset (Finset α)) (a : α) : Finset.image (insert a) (Finset.memberSubfamily a 𝒜) = Finset.filter (fun (s : Finset α) => a ∈ s) 𝒜

theorem Finset.memberFamily_induction_on {α : Type u_1} [DecidableEq α] {p : Finset (Finset α) → Prop} (𝒜 : Finset (Finset α)) (empty : p ∅) (singleton_empty : p {∅}) (subfamily : ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄, p (Finset.nonMemberSubfamily a 𝒜) → p (Finset.memberSubfamily a 𝒜) → p 𝒜) : p 𝒜

Induction principle for finset families. To prove a statement for every finset family, it suffices to prove it for

• the empty finset family.
• the finset family which only contains the empty finset.
• ℬ ∪ {s ∪ {a} | s ∈ 𝒞} assuming the property for ℬ and 𝒞, where a is an element of the ground type and 𝒜 and ℬ are families of finsets not containing a. Note that instead of giving ℬ and 𝒞, the subfamily case gives you 𝒜 = ℬ ∪ {s ∪ {a} | s ∈ 𝒞}, so that ℬ = 𝒜.nonMemberSubfamily and 𝒞 = 𝒜.memberSubfamily.

This is a way of formalising induction on n where 𝒜 is a finset family on n elements.

See also Finset.family_induction_on.

theorem Finset.family_induction_on {α : Type u_1} [DecidableEq α] {p : Finset (Finset α) → Prop} (𝒜 : Finset (Finset α)) (empty : p ∅) (singleton_empty : p {∅}) (image_insert : ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄, (∀ s ∈ 𝒜, a ∉ s) → p 𝒜 → p (Finset.image (insert a) 𝒜)) (subfamily : ∀ (a : α) ⦃𝒜 : Finset (Finset α)⦄, p (Finset.filter (fun (s : Finset α) => a ∉ s) 𝒜) → p (Finset.filter (fun (s : Finset α) => a ∈ s) 𝒜) → p 𝒜) : p 𝒜

Induction principle for finset families. To prove a statement for every finset family, it suffices to prove it for

• the empty finset family.
• the finset family which only contains the empty finset.
• {s ∪ {a} | s ∈ 𝒜} assuming the property for 𝒜 a family of finsets not containing a.
• ℬ ∪ 𝒞 assuming the property for ℬ and 𝒞, where a is an element of the ground type and ℬis a family of finsets not containing a and 𝒞 a family of finsets containing a. Note that instead of giving ℬ and 𝒞, the subfamily case gives you 𝒜 = ℬ ∪ 𝒞, so that ℬ = {s ∈ 𝒜 | a ∉ s} and 𝒞 = {s ∈ 𝒜 | a ∈ s}.

This is a way of formalising induction on n where 𝒜 is a finset family on n elements.

See also Finset.memberFamily_induction_on.

def Down.compression {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : Finset (Finset α)

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

Equations

• Down.compression a 𝒜 = (Finset.filter (fun (s : Finset α) => s.erase a ∈ 𝒜) 𝒜).disjUnion (Finset.filter (fun (s : Finset α) => s ∉ 𝒜) (Finset.image (fun (s : Finset α) => s.erase a) 𝒜)) ⋯

def FinsetFamily.term𝓓 : Lean.ParserDescr

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

Equations

• FinsetFamily.term𝓓 = Lean.ParserDescr.node `FinsetFamily.term𝓓 1024 (Lean.ParserDescr.symbol "𝓓 ")

theorem Down.mem_compression {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} : s ∈ Down.compression a 𝒜 ↔ s ∈ 𝒜 ∧ s.erase a ∈ 𝒜 ∨ s ∉ 𝒜 ∧ insert a s ∈ 𝒜

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.

theorem Down.erase_mem_compression {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} (hs : s ∈ 𝒜) : s.erase a ∈ Down.compression a 𝒜

theorem Down.erase_mem_compression_of_mem_compression {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} : s ∈ Down.compression a 𝒜 → s.erase a ∈ Down.compression a 𝒜

theorem Down.mem_compression_of_insert_mem_compression {α : Type u_1} [DecidableEq α] {𝒜 : Finset (Finset α)} {s : Finset α} {a : α} (h : insert a s ∈ Down.compression a 𝒜) : s ∈ Down.compression a 𝒜

@[simp]

theorem Down.compression_idem {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : Down.compression a (Down.compression a 𝒜) = Down.compression a 𝒜

Down-compressing a family is idempotent.

@[simp]

theorem Down.card_compression {α : Type u_1} [DecidableEq α] (a : α) (𝒜 : Finset (Finset α)) : (Down.compression a 𝒜).card = 𝒜.card

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