Set family operations #

This file defines a few binary operations on Set α for use in set family combinatorics.

Main declarations #

s ⊻ t: Set of elements of the form a ⊔ b where a ∈ s, b ∈ t.
s ⊼ t: Set of elements of the form a ⊓ b where a ∈ s, b ∈ t.

Notation #

We define the following notation in locale SetFamily:

s ⊻ t
s ⊼ t

References #

[B. Bollobás, Combinatorics][bollobas1986]

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class HasSups (α : Type u_4) :

Type u_4

sups : α → α → α

Notation typeclass for pointwise supremum ⊻.

Instances

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class HasInfs (α : Type u_4) :

Type u_4

infs : α → α → α

Notation typeclass for pointwise infimum ⊼.

Instances

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def «term_⊻_» :

Lean.TrailingParserDescr

Instances For

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def «term_⊼_» :

Lean.TrailingParserDescr

Instances For

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def Set.hasSups {α : Type u_2} [SemilatticeSup α] :

HasSups (Set α)

s ⊻ t is the set of elements of the form a ⊔ b where a ∈ s, b ∈ t.

Instances For

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@[simp]

theorem Set.mem_sups {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} {c : α} :

c ∈ s ⊻ t ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ a ⊔ b = c

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theorem Set.sup_mem_sups {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} {a : α} {b : α} :

a ∈ s → b ∈ t → a ⊔ b ∈ s ⊻ t

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theorem Set.sups_subset {α : Type u_2} [SemilatticeSup α] {s₁ : Set α} {s₂ : Set α} {t₁ : Set α} {t₂ : Set α} :

s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ ⊻ t₁ ⊆ s₂ ⊻ t₂

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theorem Set.sups_subset_left {α : Type u_2} [SemilatticeSup α] {s : Set α} {t₁ : Set α} {t₂ : Set α} :

t₁ ⊆ t₂ → s ⊻ t₁ ⊆ s ⊻ t₂

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theorem Set.sups_subset_right {α : Type u_2} [SemilatticeSup α] {s₁ : Set α} {s₂ : Set α} {t : Set α} :

s₁ ⊆ s₂ → s₁ ⊻ t ⊆ s₂ ⊻ t

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theorem Set.image_subset_sups_left {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} {b : α} :

b ∈ t → (fun a => a ⊔ b) '' s ⊆ s ⊻ t

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theorem Set.image_subset_sups_right {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} {a : α} :

a ∈ s → (fun x x_1 => x ⊔ x_1) a '' t ⊆ s ⊻ t

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theorem Set.forall_sups_iff {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} {p : α → Prop} :

((c : α) → c ∈ s ⊻ t → p c) ↔ (a : α) → a ∈ s → (b : α) → b ∈ t → p (a ⊔ b)

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@[simp]

theorem Set.sups_subset_iff {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} {u : Set α} :

s ⊻ t ⊆ u ↔ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ⊔ b ∈ u

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@[simp]

theorem Set.sups_nonempty {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} :

Set.Nonempty (s ⊻ t) ↔ Set.Nonempty s ∧ Set.Nonempty t

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theorem Set.Nonempty.sups {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} :

Set.Nonempty s → Set.Nonempty t → Set.Nonempty (s ⊻ t)

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theorem Set.Nonempty.of_sups_left {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} :

Set.Nonempty (s ⊻ t) → Set.Nonempty s

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theorem Set.Nonempty.of_sups_right {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} :

Set.Nonempty (s ⊻ t) → Set.Nonempty t

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@[simp]

theorem Set.empty_sups {α : Type u_2} [SemilatticeSup α] {t : Set α} :

∅ ⊻ t = ∅

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@[simp]

theorem Set.sups_empty {α : Type u_2} [SemilatticeSup α] {s : Set α} :

s ⊻ ∅ = ∅

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@[simp]

theorem Set.sups_eq_empty {α : Type u_2} [SemilatticeSup α] {s : Set α} {t : Set α} :

s ⊻ t = ∅ ↔ s = ∅ ∨ t = ∅

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@[simp]

theorem Set.singleton_sups {α : Type u_2} [SemilatticeSup α] {t : Set α} {a : α} :

{a} ⊻ t = (fun b => a ⊔ b) '' t

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@[simp]

theorem Set.sups_singleton {α : Type u_2} [SemilatticeSup α] {s : Set α} {b : α} :

s ⊻ {b} = (fun a => a ⊔ b) '' s

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theorem Set.singleton_sups_singleton {α : Type u_2} [SemilatticeSup α] {a : α} {b : α} :

{a} ⊻ {b} = {a ⊔ b}

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theorem Set.sups_union_left {α : Type u_2} [SemilatticeSup α] {s₁ : Set α} {s₂ : Set α} {t : Set α} :

(s₁ ∪ s₂) ⊻ t = s₁ ⊻ t ∪ s₂ ⊻ t

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theorem Set.sups_union_right {α : Type u_2} [SemilatticeSup α] {s : Set α} {t₁ : Set α} {t₂ : Set α} :

s ⊻ (t₁ ∪ t₂) = s ⊻ t₁ ∪ s ⊻ t₂

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theorem Set.sups_inter_subset_left {α : Type u_2} [SemilatticeSup α] {s₁ : Set α} {s₂ : Set α} {t : Set α} :

(s₁ ∩ s₂) ⊻ t ⊆ s₁ ⊻ t ∩ s₂ ⊻ t

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theorem Set.sups_inter_subset_right {α : Type u_2} [SemilatticeSup α] {s : Set α} {t₁ : Set α} {t₂ : Set α} :

s ⊻ (t₁ ∩ t₂) ⊆ s ⊻ t₁ ∩ s ⊻ t₂

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theorem Set.image_sups {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeSup α] [SemilatticeSup β] [SupHomClass F α β] (f : F) (s : Set α) (t : Set α) :

↑f '' (s ⊻ t) = ↑f '' s ⊻ ↑f '' t

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theorem Set.subset_sups_self {α : Type u_2} [SemilatticeSup α] {s : Set α} :

s ⊆ s ⊻ s

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theorem Set.sups_subset_self {α : Type u_2} [SemilatticeSup α] {s : Set α} :

s ⊻ s ⊆ s ↔ SupClosed s

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@[simp]

theorem Set.sups_eq_self {α : Type u_2} [SemilatticeSup α] {s : Set α} :

s ⊻ s = s ↔ SupClosed s

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theorem Set.sep_sups_le {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) (a : α) :

{b | b ∈ s ⊻ t ∧ b ≤ a} = {b | b ∈ s ∧ b ≤ a} ⊻ {b | b ∈ t ∧ b ≤ a}

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theorem Set.iUnion_image_sup_left {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) :

⋃ (a : α) (_ : a ∈ s), (fun x x_1 => x ⊔ x_1) a '' t = s ⊻ t

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theorem Set.iUnion_image_sup_right {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) :

⋃ (b : α) (_ : b ∈ t), (fun x => x ⊔ b) '' s = s ⊻ t

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@[simp]

theorem Set.image_sup_prod {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) :

Set.image2 (fun x x_1 => x ⊔ x_1) s t = s ⊻ t

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theorem Set.sups_assoc {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) (u : Set α) :

s ⊻ t ⊻ u = s ⊻ (t ⊻ u)

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theorem Set.sups_comm {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) :

s ⊻ t = t ⊻ s

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theorem Set.sups_left_comm {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) (u : Set α) :

s ⊻ (t ⊻ u) = t ⊻ (s ⊻ u)

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theorem Set.sups_right_comm {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) (u : Set α) :

s ⊻ t ⊻ u = s ⊻ u ⊻ t

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theorem Set.sups_sups_sups_comm {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) (u : Set α) (v : Set α) :

s ⊻ t ⊻ (u ⊻ v) = s ⊻ u ⊻ (t ⊻ v)

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def Set.hasInfs {α : Type u_2} [SemilatticeInf α] :

HasInfs (Set α)

s ⊼ t is the set of elements of the form a ⊓ b where a ∈ s, b ∈ t.

Instances For

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@[simp]

theorem Set.mem_infs {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} {c : α} :

c ∈ s ⊼ t ↔ ∃ a, a ∈ s ∧ ∃ b, b ∈ t ∧ a ⊓ b = c

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theorem Set.inf_mem_infs {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} {a : α} {b : α} :

a ∈ s → b ∈ t → a ⊓ b ∈ s ⊼ t

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theorem Set.infs_subset {α : Type u_2} [SemilatticeInf α] {s₁ : Set α} {s₂ : Set α} {t₁ : Set α} {t₂ : Set α} :

s₁ ⊆ s₂ → t₁ ⊆ t₂ → s₁ ⊼ t₁ ⊆ s₂ ⊼ t₂

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theorem Set.infs_subset_left {α : Type u_2} [SemilatticeInf α] {s : Set α} {t₁ : Set α} {t₂ : Set α} :

t₁ ⊆ t₂ → s ⊼ t₁ ⊆ s ⊼ t₂

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theorem Set.infs_subset_right {α : Type u_2} [SemilatticeInf α] {s₁ : Set α} {s₂ : Set α} {t : Set α} :

s₁ ⊆ s₂ → s₁ ⊼ t ⊆ s₂ ⊼ t

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theorem Set.image_subset_infs_left {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} {b : α} :

b ∈ t → (fun a => a ⊓ b) '' s ⊆ s ⊼ t

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theorem Set.image_subset_infs_right {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} {a : α} :

a ∈ s → (fun x x_1 => x ⊓ x_1) a '' t ⊆ s ⊼ t

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theorem Set.forall_infs_iff {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} {p : α → Prop} :

((c : α) → c ∈ s ⊼ t → p c) ↔ (a : α) → a ∈ s → (b : α) → b ∈ t → p (a ⊓ b)

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@[simp]

theorem Set.infs_subset_iff {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} {u : Set α} :

s ⊼ t ⊆ u ↔ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ⊓ b ∈ u

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@[simp]

theorem Set.infs_nonempty {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} :

Set.Nonempty (s ⊼ t) ↔ Set.Nonempty s ∧ Set.Nonempty t

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theorem Set.Nonempty.infs {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} :

Set.Nonempty s → Set.Nonempty t → Set.Nonempty (s ⊼ t)

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theorem Set.Nonempty.of_infs_left {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} :

Set.Nonempty (s ⊼ t) → Set.Nonempty s

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theorem Set.Nonempty.of_infs_right {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} :

Set.Nonempty (s ⊼ t) → Set.Nonempty t

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@[simp]

theorem Set.empty_infs {α : Type u_2} [SemilatticeInf α] {t : Set α} :

∅ ⊼ t = ∅

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@[simp]

theorem Set.infs_empty {α : Type u_2} [SemilatticeInf α] {s : Set α} :

s ⊼ ∅ = ∅

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@[simp]

theorem Set.infs_eq_empty {α : Type u_2} [SemilatticeInf α] {s : Set α} {t : Set α} :

s ⊼ t = ∅ ↔ s = ∅ ∨ t = ∅

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@[simp]

theorem Set.singleton_infs {α : Type u_2} [SemilatticeInf α] {t : Set α} {a : α} :

{a} ⊼ t = (fun b => a ⊓ b) '' t

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@[simp]

theorem Set.infs_singleton {α : Type u_2} [SemilatticeInf α] {s : Set α} {b : α} :

s ⊼ {b} = (fun a => a ⊓ b) '' s

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theorem Set.singleton_infs_singleton {α : Type u_2} [SemilatticeInf α] {a : α} {b : α} :

{a} ⊼ {b} = {a ⊓ b}

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theorem Set.infs_union_left {α : Type u_2} [SemilatticeInf α] {s₁ : Set α} {s₂ : Set α} {t : Set α} :

(s₁ ∪ s₂) ⊼ t = s₁ ⊼ t ∪ s₂ ⊼ t

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theorem Set.infs_union_right {α : Type u_2} [SemilatticeInf α] {s : Set α} {t₁ : Set α} {t₂ : Set α} :

s ⊼ (t₁ ∪ t₂) = s ⊼ t₁ ∪ s ⊼ t₂

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theorem Set.infs_inter_subset_left {α : Type u_2} [SemilatticeInf α] {s₁ : Set α} {s₂ : Set α} {t : Set α} :

(s₁ ∩ s₂) ⊼ t ⊆ s₁ ⊼ t ∩ s₂ ⊼ t

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theorem Set.infs_inter_subset_right {α : Type u_2} [SemilatticeInf α] {s : Set α} {t₁ : Set α} {t₂ : Set α} :

s ⊼ (t₁ ∩ t₂) ⊆ s ⊼ t₁ ∩ s ⊼ t₂

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theorem Set.image_infs {F : Type u_1} {α : Type u_2} {β : Type u_3} [SemilatticeInf α] [SemilatticeInf β] [InfHomClass F α β] (f : F) (s : Set α) (t : Set α) :

↑f '' (s ⊼ t) = ↑f '' s ⊼ ↑f '' t

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theorem Set.subset_infs_self {α : Type u_2} [SemilatticeInf α] {s : Set α} :

s ⊆ s ⊼ s

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theorem Set.infs_self_subset {α : Type u_2} [SemilatticeInf α] {s : Set α} :

s ⊼ s ⊆ s ↔ InfClosed s

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@[simp]

theorem Set.infs_self {α : Type u_2} [SemilatticeInf α] {s : Set α} :

s ⊼ s = s ↔ InfClosed s

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theorem Set.sep_infs_le {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) (a : α) :

{b | b ∈ s ⊼ t ∧ a ≤ b} = {b | b ∈ s ∧ a ≤ b} ⊼ {b | b ∈ t ∧ a ≤ b}

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theorem Set.iUnion_image_inf_left {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) :

⋃ (a : α) (_ : a ∈ s), (fun x x_1 => x ⊓ x_1) a '' t = s ⊼ t

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theorem Set.iUnion_image_inf_right {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) :

⋃ (b : α) (_ : b ∈ t), (fun x => x ⊓ b) '' s = s ⊼ t

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@[simp]

theorem Set.image_inf_prod {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) :

Set.image2 (fun x x_1 => x ⊓ x_1) s t = s ⊼ t

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theorem Set.infs_assoc {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) (u : Set α) :

s ⊼ t ⊼ u = s ⊼ (t ⊼ u)

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theorem Set.infs_comm {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) :

s ⊼ t = t ⊼ s

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theorem Set.infs_left_comm {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) (u : Set α) :

s ⊼ (t ⊼ u) = t ⊼ (s ⊼ u)

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theorem Set.infs_right_comm {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) (u : Set α) :

s ⊼ t ⊼ u = s ⊼ u ⊼ t

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theorem Set.infs_infs_infs_comm {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) (u : Set α) (v : Set α) :

s ⊼ t ⊼ (u ⊼ v) = s ⊼ u ⊼ (t ⊼ v)

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theorem Set.sups_infs_subset_left {α : Type u_2} [DistribLattice α] (s : Set α) (t : Set α) (u : Set α) :

s ⊻ t ⊼ u ⊆ (s ⊻ t) ⊼ (s ⊻ u)

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theorem Set.sups_infs_subset_right {α : Type u_2} [DistribLattice α] (s : Set α) (t : Set α) (u : Set α) :

t ⊼ u ⊻ s ⊆ (t ⊻ s) ⊼ (u ⊻ s)

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theorem Set.infs_sups_subset_left {α : Type u_2} [DistribLattice α] (s : Set α) (t : Set α) (u : Set α) :

s ⊼ (t ⊻ u) ⊆ s ⊼ t ⊻ s ⊼ u

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theorem Set.infs_sups_subset_right {α : Type u_2} [DistribLattice α] (s : Set α) (t : Set α) (u : Set α) :

(t ⊻ u) ⊼ s ⊆ t ⊼ s ⊻ u ⊼ s

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@[simp]

theorem upperClosure_sups {α : Type u_2} [SemilatticeSup α] (s : Set α) (t : Set α) :

upperClosure (s ⊻ t) = upperClosure s ⊔ upperClosure t

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@[simp]

theorem lowerClosure_infs {α : Type u_2} [SemilatticeInf α] (s : Set α) (t : Set α) :

lowerClosure (s ⊼ t) = lowerClosure s ⊓ lowerClosure t