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

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def Finset.nonMemberSubfamily {α : Type u_1} [inst : 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 => ¬a ∈ s) 𝒜

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def Finset.memberSubfamily {α : Type u_1} [inst : 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.erase s a) (Finset.filter (fun s => a ∈ s) 𝒜)

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@[simp]

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

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@[simp]

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

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

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

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

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

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

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

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@[simp]

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

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@[simp]

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

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@[simp]

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

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@[simp]

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

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def Down.compression {α : Type u_1} [inst : 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

• One or more equations did not get rendered due to their size.

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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 ")

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

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

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

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

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@[simp]

theorem Down.compression_idem {α : Type u_1} [inst : 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.

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@[simp]

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

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