Colexigraphic order

We define the colex order for finite sets, and give a couple of important lemmas and properties relating to it.

The colex ordering likes to avoid large values: If the biggest element of t is bigger than all elements of s, then s < t.

In the special case of ℕ, it can be thought of as the "binary" ordering. That is, order s based on $∑_{i ∈ s} 2^i$. It's defined here on Finset α for any linear order α.

In the context of the Kruskal-Katona theorem, we are interested in how colex behaves for sets of a fixed size. For example, for size 3, the colex order on ℕ starts 012, 013, 023, 123, 014, 024, 124, 034, 134, 234, ...

Main statements

• Colex order properties - linearity, decidability and so on.
• Finset.Colex.forall_lt_mono: if s < t in colex, and everything in t is < a, then everything in s is < a. This confirms the idea that an enumeration under colex will exhaust all sets using elements < a before allowing a to be included.
• Finset.toColex_image_le_toColex_image: Strictly monotone functions preserve colex.
• Finset.geomSum_le_geomSum_iff_toColex_le_toColex: Colex for α = ℕ is the same as binary. This also proves binary expansions are unique.

See also

Related files are:

• Data.List.Lex: Lexicographic order on lists.
• Data.Pi.Lex: Lexicographic order on Πₗ i, α i.
• Data.PSigma.Order: Lexicographic order on Σ' i, α i.
• Data.Sigma.Order: Lexicographic order on Σ i, α i.
• Data.Prod.Lex: Lexicographic order on α × β.

TODO

• Generalise Colex.initSeg so that it applies to ℕ.

References

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

Tags

colex, colexicographic, binary

theorem Finset.Colex.ext {α : Type u_3} (x : Finset.Colex α) (y : Finset.Colex α) (ofColex : x.ofColex = y.ofColex) : x = y

theorem Finset.Colex.ext_iff {α : Type u_3} (x : Finset.Colex α) (y : Finset.Colex α) : x = y ↔ x.ofColex = y.ofColex

structure Finset.Colex (α : Type u_3) : Type u_3

Type synonym of Finset α equipped with the colexicographic order rather than the inclusion order.

• toColex :: (
• ofColex : Finset α

  ofColex is the "identity" function between Finset.Colex α and Finset α.

• )

instance Finset.instInhabitedColex {α : Type u_1} : Inhabited (Finset.Colex α)

Equations

• Finset.instInhabitedColex = { default := { ofColex := ∅ } }

@[simp]

theorem Finset.toColex_ofColex {α : Type u_1} (s : Finset.Colex α) : { ofColex := s.ofColex } = s

theorem Finset.ofColex_toColex {α : Type u_1} (s : Finset α) : { ofColex := s }.ofColex = s

theorem Finset.toColex_inj {α : Type u_1} {s : Finset α} {t : Finset α} : { ofColex := s } = { ofColex := t } ↔ s = t

@[simp]

theorem Finset.ofColex_inj {α : Type u_1} {s : Finset.Colex α} {t : Finset.Colex α} : s.ofColex = t.ofColex ↔ s = t

theorem Finset.toColex_ne_toColex {α : Type u_1} {s : Finset α} {t : Finset α} : { ofColex := s } ≠ { ofColex := t } ↔ s ≠ t

theorem Finset.ofColex_ne_ofColex {α : Type u_1} {s : Finset.Colex α} {t : Finset.Colex α} : s.ofColex ≠ t.ofColex ↔ s ≠ t

theorem Finset.toColex_injective {α : Type u_1} : Function.Injective Finset.Colex.toColex

theorem Finset.ofColex_injective {α : Type u_1} : Function.Injective Finset.Colex.ofColex

instance Finset.Colex.instLE {α : Type u_1} [PartialOrder α] : LE (Finset.Colex α)

Equations

• Finset.Colex.instLE = { le := fun (s t : Finset.Colex α) => ∀ ⦃a : α⦄, a ∈ s.ofColex → a ∉ t.ofColex → ∃ b ∈ t.ofColex, b ∉ s.ofColex ∧ a ≤ b }

instance Finset.Colex.instPartialOrder {α : Type u_1} [PartialOrder α] : PartialOrder (Finset.Colex α)

Equations

• Finset.Colex.instPartialOrder = PartialOrder.mk ⋯

theorem Finset.Colex.le_def {α : Type u_1} [PartialOrder α] {s : Finset.Colex α} {t : Finset.Colex α} : s ≤ t ↔ ∀ ⦃a : α⦄, a ∈ s.ofColex → a ∉ t.ofColex → ∃ b ∈ t.ofColex, b ∉ s.ofColex ∧ a ≤ b

theorem Finset.Colex.toColex_le_toColex {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} : { ofColex := s } ≤ { ofColex := t } ↔ ∀ ⦃a : α⦄, a ∈ s → a ∉ t → ∃ b ∈ t, b ∉ s ∧ a ≤ b

theorem Finset.Colex.toColex_lt_toColex {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} : { ofColex := s } < { ofColex := t } ↔ s ≠ t ∧ ∀ ⦃a : α⦄, a ∈ s → a ∉ t → ∃ b ∈ t, b ∉ s ∧ a ≤ b

theorem Finset.Colex.toColex_mono {α : Type u_1} [PartialOrder α] : Monotone Finset.Colex.toColex

If s ⊆ t, then s ≤ t in the colex order. Note the converse does not hold, as inclusion does not form a linear order.

theorem Finset.Colex.toColex_strictMono {α : Type u_1} [PartialOrder α] : StrictMono Finset.Colex.toColex

If s ⊂ t, then s < t in the colex order. Note the converse does not hold, as inclusion does not form a linear order.

theorem Finset.Colex.toColex_le_toColex_of_subset {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} (h : s ⊆ t) : { ofColex := s } ≤ { ofColex := t }

If s ⊆ t, then s ≤ t in the colex order. Note the converse does not hold, as inclusion does not form a linear order.

theorem Finset.Colex.toColex_lt_toColex_of_ssubset {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} (h : s ⊂ t) : { ofColex := s } < { ofColex := t }

If s ⊂ t, then s < t in the colex order. Note the converse does not hold, as inclusion does not form a linear order.

instance Finset.Colex.instOrderBot {α : Type u_1} [PartialOrder α] : OrderBot (Finset.Colex α)

Equations

• Finset.Colex.instOrderBot = OrderBot.mk ⋯

@[simp]

theorem Finset.Colex.toColex_empty {α : Type u_1} [PartialOrder α] : { ofColex := ∅ } = ⊥

@[simp]

theorem Finset.Colex.ofColex_bot {α : Type u_1} [PartialOrder α] : ⊥.ofColex = ∅

theorem Finset.Colex.forall_le_mono {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} {a : α} (hst : { ofColex := s } ≤ { ofColex := t }) (ht : ∀ b ∈ t, b ≤ a) (b : α) : b ∈ s → b ≤ a

If s ≤ t in colex, and all elements in t are small, then all elements in s are small.

theorem Finset.Colex.forall_lt_mono {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} {a : α} (hst : { ofColex := s } ≤ { ofColex := t }) (ht : ∀ b ∈ t, b < a) (b : α) : b ∈ s → b < a

If s ≤ t in colex, and all elements in t are small, then all elements in s are small.

theorem Finset.Colex.toColex_le_singleton {α : Type u_1} [PartialOrder α] {s : Finset α} {a : α} : { ofColex := s } ≤ { ofColex := {a} } ↔ ∀ b ∈ s, b ≤ a ∧ (a ∈ s → b = a)

s ≤ {a} in colex iff all elements of s are strictly less than a, except possibly a in which case s = {a}.

theorem Finset.Colex.toColex_lt_singleton {α : Type u_1} [PartialOrder α] {s : Finset α} {a : α} : { ofColex := s } < { ofColex := {a} } ↔ ∀ b ∈ s, b < a

s < {a} in colex iff all elements of s are strictly less than a.

theorem Finset.Colex.singleton_le_toColex {α : Type u_1} [PartialOrder α] {s : Finset α} {a : α} : { ofColex := {a} } ≤ { ofColex := s } ↔ ∃ x ∈ s, a ≤ x

{a} ≤ s in colex iff s contains an element greated than or equal to a.

theorem Finset.Colex.singleton_le_singleton {α : Type u_1} [PartialOrder α] {a : α} {b : α} : { ofColex := {a} } ≤ { ofColex := {b} } ↔ a ≤ b

Colex is an extension of the base order.

theorem Finset.Colex.singleton_lt_singleton {α : Type u_1} [PartialOrder α] {a : α} {b : α} : { ofColex := {a} } < { ofColex := {b} } ↔ a < b

Colex is an extension of the base order.

instance Finset.Colex.instDecidableEq {α : Type u_1} [DecidableEq α] : DecidableEq (Finset.Colex α)

Equations

• Finset.Colex.instDecidableEq s t = decidable_of_iff' (s.ofColex = t.ofColex) ⋯

instance Finset.Colex.instDecidableLE {α : Type u_1} [PartialOrder α] [DecidableEq α] [DecidableRel fun (x x_1 : α) => x ≤ x_1] : DecidableRel fun (x x_1 : Finset.Colex α) => x ≤ x_1

Equations

• Finset.Colex.instDecidableLE s t = decidable_of_iff' (∀ ⦃a : α⦄, a ∈ s.ofColex → a ∉ t.ofColex → ∃ b ∈ t.ofColex, b ∉ s.ofColex ∧ a ≤ b) ⋯

instance Finset.Colex.instDecidableLT {α : Type u_1} [PartialOrder α] [DecidableEq α] [DecidableRel fun (x x_1 : α) => x ≤ x_1] : DecidableRel fun (x x_1 : Finset.Colex α) => x < x_1

Equations

• Finset.Colex.instDecidableLT = decidableLTOfDecidableLE

theorem Finset.Colex.le_iff_sdiff_subset_lowerClosure {α : Type u_1} [PartialOrder α] {s : Finset.Colex α} {t : Finset.Colex α} : s ≤ t ↔ ↑s.ofColex \ ↑t.ofColex ⊆ ↑(lowerClosure (↑t.ofColex \ ↑s.ofColex))

theorem Finset.Colex.toColex_sdiff_le_toColex_sdiff {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} {u : Finset α} [DecidableEq α] (hus : u ⊆ s) (hut : u ⊆ t) : { ofColex := s \ u } ≤ { ofColex := t \ u } ↔ { ofColex := s } ≤ { ofColex := t }

The colexigraphic order is insensitive to removing the same elements from both sets.

theorem Finset.Colex.toColex_sdiff_lt_toColex_sdiff {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} {u : Finset α} [DecidableEq α] (hus : u ⊆ s) (hut : u ⊆ t) : { ofColex := s \ u } < { ofColex := t \ u } ↔ { ofColex := s } < { ofColex := t }

The colexigraphic order is insensitive to removing the same elements from both sets.

@[simp]

theorem Finset.Colex.toColex_sdiff_le_toColex_sdiff' {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} [DecidableEq α] : { ofColex := s \ t } ≤ { ofColex := t \ s } ↔ { ofColex := s } ≤ { ofColex := t }

@[simp]

theorem Finset.Colex.toColex_sdiff_lt_toColex_sdiff' {α : Type u_1} [PartialOrder α] {s : Finset α} {t : Finset α} [DecidableEq α] : { ofColex := s \ t } < { ofColex := t \ s } ↔ { ofColex := s } < { ofColex := t }

instance Finset.Colex.instLinearOrder {α : Type u_1} [LinearOrder α] : LinearOrder (Finset.Colex α)

Equations

• Finset.Colex.instLinearOrder = LinearOrder.mk ⋯ Finset.Colex.instDecidableLE decidableEqOfDecidableLE Finset.Colex.instDecidableLT ⋯ ⋯ ⋯

theorem Finset.Colex.toColex_lt_toColex_iff_exists_forall_lt {α : Type u_1} [LinearOrder α] {s : Finset α} {t : Finset α} : { ofColex := s } < { ofColex := t } ↔ ∃ a ∈ t, a ∉ s ∧ ∀ b ∈ s, b ∉ t → b < a

theorem Finset.Colex.lt_iff_exists_forall_lt {α : Type u_1} [LinearOrder α] {s : Finset.Colex α} {t : Finset.Colex α} : s < t ↔ ∃ a ∈ t.ofColex, a ∉ s.ofColex ∧ ∀ b ∈ s.ofColex, b ∉ t.ofColex → b < a

theorem Finset.Colex.toColex_le_toColex_iff_max'_mem {α : Type u_1} [LinearOrder α] {s : Finset α} {t : Finset α} : { ofColex := s } ≤ { ofColex := t } ↔ ∀ (hst : s ≠ t), Finset.max' (symmDiff s t) ⋯ ∈ t

theorem Finset.Colex.le_iff_max'_mem {α : Type u_1} [LinearOrder α] {s : Finset.Colex α} {t : Finset.Colex α} : s ≤ t ↔ ∀ (h : s ≠ t), Finset.max' (symmDiff s.ofColex t.ofColex) ⋯ ∈ t.ofColex

theorem Finset.Colex.toColex_lt_toColex_iff_max'_mem {α : Type u_1} [LinearOrder α] {s : Finset α} {t : Finset α} : { ofColex := s } < { ofColex := t } ↔ ∃ (hst : s ≠ t), Finset.max' (symmDiff s t) ⋯ ∈ t

theorem Finset.Colex.lt_iff_max'_mem {α : Type u_1} [LinearOrder α] {s : Finset.Colex α} {t : Finset.Colex α} : s < t ↔ ∃ (h : s ≠ t), Finset.max' (symmDiff s.ofColex t.ofColex) ⋯ ∈ t.ofColex

theorem Finset.Colex.toColex_image_le_toColex_image {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Finset α} {t : Finset α} (hf : StrictMono f) : { ofColex := Finset.image f s } ≤ { ofColex := Finset.image f t } ↔ { ofColex := s } ≤ { ofColex := t }

Strictly monotone functions preserve the colex ordering.

theorem Finset.Colex.toColex_image_lt_toColex_image {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] {f : α → β} {s : Finset α} {t : Finset α} (hf : StrictMono f) : { ofColex := Finset.image f s } < { ofColex := Finset.image f t } ↔ { ofColex := s } < { ofColex := t }

Strictly monotone functions preserve the colex ordering.

theorem Finset.Colex.toColex_image_ofColex_strictMono {α : Type u_1} {β : Type u_2} [LinearOrder α] [LinearOrder β] {f : α → β} (hf : StrictMono f) : StrictMono fun (s : Finset.Colex α) => { ofColex := Finset.image f s.ofColex }

Initial segments

def Finset.Colex.IsInitSeg {α : Type u_1} [LinearOrder α] (𝒜 : Finset (Finset α)) (r : ℕ) : Prop

𝒜 is an initial segment of the colexigraphic order on sets of r, and that if t is below s in colex where t has size r and s is in 𝒜, then t is also in 𝒜. In effect, 𝒜 is downwards closed with respect to colex among sets of size r.

Equations

• Finset.Colex.IsInitSeg 𝒜 r = (Set.Sized r ↑𝒜 ∧ ∀ ⦃s t : Finset α⦄, s ∈ 𝒜 → { ofColex := t } < { ofColex := s } ∧ t.card = r → t ∈ 𝒜)

@[simp]

theorem Finset.Colex.isInitSeg_empty {α : Type u_1} [LinearOrder α] {r : ℕ} : Finset.Colex.IsInitSeg ∅ r

theorem Finset.Colex.IsInitSeg.total {α : Type u_1} [LinearOrder α] {𝒜₁ : Finset (Finset α)} {𝒜₂ : Finset (Finset α)} {r : ℕ} (h₁ : Finset.Colex.IsInitSeg 𝒜₁ r) (h₂ : Finset.Colex.IsInitSeg 𝒜₂ r) : 𝒜₁ ⊆ 𝒜₂ ∨ 𝒜₂ ⊆ 𝒜₁

Initial segments are nested in some way. In particular, if they're the same size they're equal.

def Finset.Colex.initSeg {α : Type u_1} [LinearOrder α] [Fintype α] (s : Finset α) : Finset (Finset α)

The initial segment of the colexicographic order on sets with s.card elements and ending at s.

Equations

• Finset.Colex.initSeg s = Finset.filter (fun (t : Finset α) => s.card = t.card ∧ { ofColex := t } ≤ { ofColex := s }) Finset.univ

@[simp]

theorem Finset.Colex.mem_initSeg {α : Type u_1} [LinearOrder α] {s : Finset α} {t : Finset α} [Fintype α] : t ∈ Finset.Colex.initSeg s ↔ s.card = t.card ∧ { ofColex := t } ≤ { ofColex := s }

theorem Finset.Colex.mem_initSeg_self {α : Type u_1} [LinearOrder α] {s : Finset α} [Fintype α] : s ∈ Finset.Colex.initSeg s

@[simp]

theorem Finset.Colex.initSeg_nonempty {α : Type u_1} [LinearOrder α] {s : Finset α} [Fintype α] : (Finset.Colex.initSeg s).Nonempty

theorem Finset.Colex.isInitSeg_initSeg {α : Type u_1} [LinearOrder α] {s : Finset α} [Fintype α] : Finset.Colex.IsInitSeg (Finset.Colex.initSeg s) s.card

theorem Finset.Colex.IsInitSeg.exists_initSeg {α : Type u_1} [LinearOrder α] {𝒜 : Finset (Finset α)} {r : ℕ} [Fintype α] (h𝒜 : Finset.Colex.IsInitSeg 𝒜 r) (h𝒜₀ : 𝒜.Nonempty) : ∃ (s : Finset α), s.card = r ∧ 𝒜 = Finset.Colex.initSeg s

theorem Finset.Colex.isInitSeg_iff_exists_initSeg {α : Type u_1} [LinearOrder α] {𝒜 : Finset (Finset α)} {r : ℕ} [Fintype α] : Finset.Colex.IsInitSeg 𝒜 r ∧ 𝒜.Nonempty ↔ ∃ (s : Finset α), s.card = r ∧ 𝒜 = Finset.Colex.initSeg s

Being a nonempty initial segment of colex is equivalent to being an initSeg.

Colex on ℕ

The colexicographic order agrees with the order induced by interpreting a set of naturals as a n-ary expansion.

theorem Finset.geomSum_ofColex_strictMono {n : ℕ} (hn : 2 ≤ n) : StrictMono fun (s : Finset.Colex ℕ) => Finset.sum s.ofColex fun (k : ℕ) => n ^ k

theorem Finset.geomSum_le_geomSum_iff_toColex_le_toColex {s : Finset ℕ} {t : Finset ℕ} {n : ℕ} (hn : 2 ≤ n) : ((Finset.sum s fun (k : ℕ) => n ^ k) ≤ Finset.sum t fun (k : ℕ) => n ^ k) ↔ { ofColex := s } ≤ { ofColex := t }

For finsets of naturals of naturals, the colexicographic order is equivalent to the order induced by the n-ary expansion.

theorem Finset.geomSum_lt_geomSum_iff_toColex_lt_toColex {s : Finset ℕ} {t : Finset ℕ} {n : ℕ} (hn : 2 ≤ n) : ((Finset.sum s fun (i : ℕ) => n ^ i) < Finset.sum t fun (i : ℕ) => n ^ i) ↔ { ofColex := s } < { ofColex := t }

For finsets of naturals of naturals, the colexicographic order is equivalent to the order induced by the n-ary expansion.