# Documentation

This file defines shadows of a set family. The shadow of a set family is the set family of sets we get by removing any element from any set of the original family. If one pictures Finset α as a big hypercube (each dimension being membership of a given element), then taking the shadow corresponds to projecting each finset down once in all available directions.

## Main definitions #

• Finset.shadow: The shadow of a set family. Everything we can get by removing a new element from some set.
• Finset.upShadow: The upper shadow of a set family. Everything we can get by adding an element to some set.

## Notation #

We define notation in locale FinsetFamily:

• ∂ 𝒜: Shadow of 𝒜.
• ∂⁺ 𝒜: Upper shadow of 𝒜.

We also maintain the convention that a, b : α are elements of the ground type, s, t : Finset α are finsets, and 𝒜, ℬ : Finset (Finset α) are finset families.

## Tags #

def Finset.shadow {α : Type u_1} [] (𝒜 : Finset ()) :

The shadow of a set family 𝒜 is all sets we can get by removing one element from any set in 𝒜, and the (k times) iterated shadow (shadow^[k]) is all sets we can get by removing k elements from any set in 𝒜.

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The shadow of a set family 𝒜 is all sets we can get by removing one element from any set in 𝒜, and the (k times) iterated shadow (shadow^[k]) is all sets we can get by removing k elements from any set in 𝒜.

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@[simp]
theorem Finset.shadow_empty {α : Type u_1} [] :

The shadow of the empty set is empty.

@[simp]
theorem Finset.shadow_iterate_empty {α : Type u_1} [] (k : ) :
@[simp]
theorem Finset.shadow_singleton_empty {α : Type u_1} [] :
theorem Finset.shadow_monotone {α : Type u_1} [] :

theorem Finset.mem_shadow_iff {α : Type u_1} [] {𝒜 : Finset ()} {t : } :
t 𝒜.shadow s𝒜, as, s.erase a = t

t is in the shadow of 𝒜 iff there is a s ∈ 𝒜 from which we can remove one element to get t.

theorem Finset.erase_mem_shadow {α : Type u_1} [] {𝒜 : Finset ()} {s : } {a : α} (hs : s 𝒜) (ha : a s) :
theorem Finset.mem_shadow_iff_exists_sdiff {α : Type u_1} [] {𝒜 : Finset ()} {t : } :
t 𝒜.shadow s𝒜, t s (s \ t).card = 1

t ∈ ∂𝒜 iff t is exactly one element less than something from 𝒜.

See also Finset.mem_shadow_iff_exists_mem_card_add_one.

theorem Finset.mem_shadow_iff_insert_mem {α : Type u_1} [] {𝒜 : Finset ()} {t : } :
t 𝒜.shadow at, insert a t 𝒜

t is in the shadow of 𝒜 iff we can add an element to it so that the resulting finset is in 𝒜.

theorem Finset.mem_shadow_iff_exists_mem_card_add_one {α : Type u_1} [] {𝒜 : Finset ()} {t : } :
t 𝒜.shadow s𝒜, t s s.card = t.card + 1

s ∈ ∂ 𝒜 iff s is exactly one element less than something from 𝒜.

See also Finset.mem_shadow_iff_exists_sdiff.

theorem Finset.mem_shadow_iterate_iff_exists_card {α : Type u_1} [] {𝒜 : Finset ()} {t : } {k : } :
t Finset.shadow^[k] 𝒜 ∃ (u : ), u.card = k Disjoint t u t u 𝒜
theorem Finset.mem_shadow_iterate_iff_exists_sdiff {α : Type u_1} [] {𝒜 : Finset ()} {t : } {k : } :
t Finset.shadow^[k] 𝒜 s𝒜, t s (s \ t).card = k

t ∈ ∂^k 𝒜 iff t is exactly k elements less than something from 𝒜.

See also Finset.mem_shadow_iff_exists_mem_card_add.

theorem Finset.mem_shadow_iterate_iff_exists_mem_card_add {α : Type u_1} [] {𝒜 : Finset ()} {t : } {k : } :
t Finset.shadow^[k] 𝒜 s𝒜, t s s.card = t.card + k

t ∈ ∂^k 𝒜 iff t is exactly k elements less than something in 𝒜.

See also Finset.mem_shadow_iterate_iff_exists_sdiff.

theorem Set.Sized.shadow {α : Type u_1} [] {𝒜 : Finset ()} {r : } (h𝒜 : Set.Sized r 𝒜) :

The shadow of a family of r-sets is a family of r - 1-sets.

theorem Set.Sized.shadow_iterate {α : Type u_1} [] {𝒜 : Finset ()} {k : } {r : } (h𝒜 : Set.Sized r 𝒜) :
Set.Sized (r - k) (Finset.shadow^[k] 𝒜)

The k-th shadow of a family of r-sets is a family of r - k-sets.

theorem Finset.sized_shadow_iff {α : Type u_1} [] {𝒜 : Finset ()} {r : } (h : 𝒜) :
Set.Sized r 𝒜.shadow Set.Sized (r + 1) 𝒜
theorem Finset.exists_subset_of_mem_shadow {α : Type u_1} [] {𝒜 : Finset ()} {t : } (hs : t 𝒜.shadow) :
s𝒜, t s

Being in the shadow of 𝒜 means we have a superset in 𝒜.

def Finset.upShadow {α : Type u_1} [] [] (𝒜 : Finset ()) :

The upper shadow of a set family 𝒜 is all sets we can get by adding one element to any set in 𝒜, and the (k times) iterated upper shadow (upShadow^[k]) is all sets we can get by adding k elements from any set in 𝒜.

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The upper shadow of a set family 𝒜 is all sets we can get by adding one element to any set in 𝒜, and the (k times) iterated upper shadow (upShadow^[k]) is all sets we can get by adding k elements from any set in 𝒜.

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@[simp]
theorem Finset.upShadow_empty {α : Type u_1} [] [] :

The upper shadow of the empty set is empty.

theorem Finset.upShadow_monotone {α : Type u_1} [] [] :

theorem Finset.mem_upShadow_iff {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } :
t 𝒜.upShadow s𝒜, as, insert a s = t

t is in the upper shadow of 𝒜 iff there is a s ∈ 𝒜 from which we can remove one element to get t.

theorem Finset.insert_mem_upShadow {α : Type u_1} [] [] {𝒜 : Finset ()} {s : } {a : α} (hs : s 𝒜) (ha : as) :
theorem Finset.mem_upShadow_iff_exists_sdiff {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } :
t 𝒜.upShadow s𝒜, s t (t \ s).card = 1

t is in the upper shadow of 𝒜 iff t is exactly one element more than something from 𝒜.

See also Finset.mem_upShadow_iff_exists_mem_card_add_one.

theorem Finset.mem_upShadow_iff_erase_mem {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } :
t 𝒜.upShadow at, t.erase a 𝒜

t is in the upper shadow of 𝒜 iff we can remove an element from it so that the resulting finset is in 𝒜.

theorem Finset.mem_upShadow_iff_exists_mem_card_add_one {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } :
t 𝒜.upShadow s𝒜, s t t.card = s.card + 1

t is in the upper shadow of 𝒜 iff t is exactly one element less than something from 𝒜.

See also Finset.mem_upShadow_iff_exists_sdiff.

theorem Finset.mem_upShadow_iterate_iff_exists_card {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } {k : } :
t Finset.upShadow^[k] 𝒜 ∃ (u : ), u.card = k u t t \ u 𝒜
theorem Finset.mem_upShadow_iterate_iff_exists_sdiff {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } {k : } :
t Finset.upShadow^[k] 𝒜 s𝒜, s t (t \ s).card = k

t is in the upper shadow of 𝒜 iff t is exactly k elements less than something from 𝒜.

See also Finset.mem_upShadow_iff_exists_mem_card_add.

theorem Finset.mem_upShadow_iterate_iff_exists_mem_card_add {α : Type u_1} [] [] {𝒜 : Finset ()} {t : } {k : } :
t Finset.upShadow^[k] 𝒜 s𝒜, s t t.card = s.card + k

t ∈ ∂⁺^k 𝒜 iff t is exactly k elements less than something in 𝒜.

See also Finset.mem_upShadow_iterate_iff_exists_sdiff.

theorem Set.Sized.upShadow {α : Type u_1} [] [] {𝒜 : Finset ()} {r : } (h𝒜 : Set.Sized r 𝒜) :

The upper shadow of a family of r-sets is a family of r + 1-sets.

theorem Finset.exists_subset_of_mem_upShadow {α : Type u_1} [] [] {𝒜 : Finset ()} {s : } (hs : s 𝒜.upShadow) :
t𝒜, t s

Being in the upper shadow of 𝒜 means we have a superset in 𝒜.

theorem Finset.mem_upShadow_iff_exists_mem_card_add {α : Type u_1} [] [] {𝒜 : Finset ()} {s : } {k : } :
s Finset.upShadow^[k] 𝒜 t𝒜, t s t.card + k = s.card

t ∈ ∂^k 𝒜 iff t is exactly k elements more than something in 𝒜.

@[simp]
theorem Finset.shadow_compls {α : Type u_1} [] [] {𝒜 : Finset ()} :