(Generalized) Boolean algebras

A Boolean algebra is a bounded distributive lattice with a complement operator. Boolean algebras generalize the (classical) logic of propositions and the lattice of subsets of a set.

Generalized Boolean algebras may be less familiar, but they are essentially Boolean algebras which do not necessarily have a top element (⊤) (and hence not all elements may have complements). One example in mathlib is Finset α, the type of all finite subsets of an arbitrary (not-necessarily-finite) type α.

GeneralizedBooleanAlgebra α is defined to be a distributive lattice with bottom (⊥) admitting a relative complement operator, written using "set difference" notation as x \ y (sdiff x y). For convenience, the BooleanAlgebra type class is defined to extend GeneralizedBooleanAlgebra so that it is also bundled with a \ operator.

(A terminological point: x \ y is the complement of y relative to the interval [⊥, x]. We do not yet have relative complements for arbitrary intervals, as we do not even have lattice intervals.)

Main declarations

• GeneralizedBooleanAlgebra: a type class for generalized Boolean algebras
• BooleanAlgebra: a type class for Boolean algebras.
• Prop.booleanAlgebra: the Boolean algebra instance on Prop

Implementation notes

The sup_inf_sdiff and inf_inf_sdiff axioms for the relative complement operator in GeneralizedBooleanAlgebra are taken from Wikipedia.

[Stone's paper introducing generalized Boolean algebras][Stone1935] does not define a relative complement operator a \ b for all a, b. Instead, the postulates there amount to an assumption that for all a, b : α where a ≤ b, the equations x ⊔ a = b and x ⊓ a = ⊥ have a solution x. Disjoint.sdiff_unique proves that this x is in fact b \ a.

References

• https://en.wikipedia.org/wiki/Boolean_algebra_(structure)#Generalizations
• [Postulates for Boolean Algebras and Generalized Boolean Algebras, M.H. Stone][Stone1935]
• [Lattice Theory: Foundation, George Grätzer][Gratzer2011]

Tags

generalized Boolean algebras, Boolean algebras, lattices, sdiff, compl

Generalized Boolean algebras

Some of the lemmas in this section are from:

• [Lattice Theory: Foundation, George Grätzer][Gratzer2011]
• https://ncatlab.org/nlab/show/relative+complement
• https://people.math.gatech.edu/~mccuan/courses/4317/symmetricdifference.pdf

class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice , SDiff , Bot : Type u

A generalized Boolean algebra is a distributive lattice with ⊥ and a relative complement operation \ (called sdiff, after "set difference") satisfying (a ⊓ b) ⊔ (a \ b) = a and (a ⊓ b) ⊓ (a \ b) = ⊥, i.e. a \ b is the complement of b in a.

This is a generalization of Boolean algebras which applies to Finset α for arbitrary (not-necessarily-Fintype) α.

• sup : α → α → α
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
• le_sup_left : ∀ (a b : α), a ≤ a ⊔ b
• le_sup_right : ∀ (a b : α), b ≤ a ⊔ b
• sup_le : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c
• inf : α → α → α
• inf_le_left : ∀ (a b : α), a ⊓ b ≤ a
• inf_le_right : ∀ (a b : α), a ⊓ b ≤ b
• le_inf : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c
• le_sup_inf : ∀ (x y z : α), (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
• sdiff : α → α → α
• bot : α
• sup_inf_sdiff : ∀ (a b : α), a ⊓ b ⊔ a \ b = a

  For any a, b, (a ⊓ b) ⊔ (a / b) = a

• inf_inf_sdiff : ∀ (a b : α), a ⊓ b ⊓ a \ b = ⊥

  For any a, b, (a ⊓ b) ⊓ (a / b) = ⊥

Instances

• BooleanAlgebra.toGeneralizedBooleanAlgebra
• Finset.instGeneralizedBooleanAlgebra
• Pi.instGeneralizedBooleanAlgebra
• Prod.instGeneralizedBooleanAlgebra

theorem GeneralizedBooleanAlgebra.sup_inf_sdiff {α : Type u} [self : GeneralizedBooleanAlgebra α] (a : α) (b : α) : a ⊓ b ⊔ a \ b = a

For any a, b, (a ⊓ b) ⊔ (a / b) = a

theorem GeneralizedBooleanAlgebra.inf_inf_sdiff {α : Type u} [self : GeneralizedBooleanAlgebra α] (a : α) (b : α) : a ⊓ b ⊓ a \ b = ⊥

For any a, b, (a ⊓ b) ⊓ (a / b) = ⊥

@[simp]

theorem sup_inf_sdiff {α : Type u} [GeneralizedBooleanAlgebra α] (x : α) (y : α) : x ⊓ y ⊔ x \ y = x

@[simp]

theorem inf_inf_sdiff {α : Type u} [GeneralizedBooleanAlgebra α] (x : α) (y : α) : x ⊓ y ⊓ x \ y = ⊥

@[simp]

theorem sup_sdiff_inf {α : Type u} [GeneralizedBooleanAlgebra α] (x : α) (y : α) : x \ y ⊔ x ⊓ y = x

@[simp]

theorem inf_sdiff_inf {α : Type u} [GeneralizedBooleanAlgebra α] (x : α) (y : α) : x \ y ⊓ (x ⊓ y) = ⊥

@[instance 100]

instance GeneralizedBooleanAlgebra.toOrderBot {α : Type u} [GeneralizedBooleanAlgebra α] : OrderBot α

Equations

• GeneralizedBooleanAlgebra.toOrderBot = let __spread.0 := GeneralizedBooleanAlgebra.toBot; OrderBot.mk ⋯

theorem disjoint_inf_sdiff {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : Disjoint (x ⊓ y) (x \ y)

theorem sdiff_unique {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (s : x ⊓ y ⊔ z = x) (i : x ⊓ y ⊓ z = ⊥) : x \ y = z

@[simp]

theorem sdiff_inf_sdiff {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ y ⊓ y \ x = ⊥

theorem disjoint_sdiff_sdiff {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : Disjoint (x \ y) (y \ x)

@[simp]

theorem inf_sdiff_self_right {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x ⊓ y \ x = ⊥

@[simp]

theorem inf_sdiff_self_left {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : y \ x ⊓ x = ⊥

@[instance 100]

instance GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra {α : Type u} [GeneralizedBooleanAlgebra α] : GeneralizedCoheytingAlgebra α

Equations

• GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra = let __spread.0 := inst; let __spread.1 := GeneralizedBooleanAlgebra.toOrderBot; GeneralizedCoheytingAlgebra.mk ⋯

theorem disjoint_sdiff_self_left {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : Disjoint (y \ x) x

theorem disjoint_sdiff_self_right {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : Disjoint x (y \ x)

theorem le_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : x ≤ y \ z ↔ x ≤ y ∧ Disjoint x z

@[simp]

theorem sdiff_eq_left {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ y = x ↔ Disjoint x y

theorem Disjoint.sdiff_eq_of_sup_eq {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hi : Disjoint x z) (hs : x ⊔ z = y) : y \ x = z

theorem Disjoint.sdiff_unique {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hd : Disjoint x z) (hz : z ≤ y) (hs : y ≤ x ⊔ z) : y \ x = z

theorem disjoint_sdiff_iff_le {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y) : Disjoint z (y \ x) ↔ z ≤ x

theorem le_iff_disjoint_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y) : z ≤ x ↔ Disjoint z (y \ x)

theorem inf_sdiff_eq_bot_iff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y) : z ⊓ y \ x = ⊥ ↔ z ≤ x

theorem le_iff_eq_sup_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hz : z ≤ y) (hx : x ≤ y) : x ≤ z ↔ y = z ⊔ y \ x

theorem sdiff_sup {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : y \ (x ⊔ z) = y \ x ⊓ y \ z

theorem sdiff_eq_sdiff_iff_inf_eq_inf {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : y \ x = y \ z ↔ y ⊓ x = y ⊓ z

theorem sdiff_eq_self_iff_disjoint {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ y = x ↔ Disjoint y x

theorem sdiff_eq_self_iff_disjoint' {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ y = x ↔ Disjoint x y

theorem sdiff_lt {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] (hx : y ≤ x) (hy : y ≠ ⊥) : x \ y < x

@[simp]

theorem le_sdiff_iff {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x ≤ y \ x ↔ x = ⊥

@[simp]

theorem sdiff_eq_right {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ y = y ↔ x = ⊥ ∧ y = ⊥

theorem sdiff_ne_right {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ y ≠ y ↔ x ≠ ⊥ ∨ y ≠ ⊥

theorem sdiff_lt_sdiff_right {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (h : x < y) (hz : z ≤ x) : x \ z < y \ z

theorem sup_inf_inf_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : x ⊓ y ⊓ z ⊔ y \ z = x ⊓ y ⊔ y \ z

theorem sdiff_sdiff_right {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : x \ (y \ z) = x \ y ⊔ x ⊓ y ⊓ z

theorem sdiff_sdiff_right' {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : x \ (y \ z) = x \ y ⊔ x ⊓ z

theorem sdiff_sdiff_eq_sdiff_sup {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (h : z ≤ x) : x \ (y \ z) = x \ y ⊔ z

@[simp]

theorem sdiff_sdiff_right_self {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x \ (x \ y) = x ⊓ y

theorem sdiff_sdiff_eq_self {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] (h : y ≤ x) : x \ (x \ y) = y

theorem sdiff_eq_symm {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hy : y ≤ x) (h : x \ y = z) : x \ z = y

theorem sdiff_eq_comm {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hy : y ≤ x) (hz : z ≤ x) : x \ y = z ↔ x \ z = y

theorem eq_of_sdiff_eq_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hxz : x ≤ z) (hyz : y ≤ z) (h : z \ x = z \ y) : x = y

theorem sdiff_sdiff_left' {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : (x \ y) \ z = x \ y ⊓ x \ z

theorem sdiff_sdiff_sup_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : z \ (x \ y ⊔ y \ x) = z ⊓ (z \ x ⊔ y) ⊓ (z \ y ⊔ x)

theorem sdiff_sdiff_sup_sdiff' {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : z \ (x \ y ⊔ y \ x) = z ⊓ x ⊓ y ⊔ z \ x ⊓ z \ y

theorem sdiff_sdiff_sdiff_cancel_left {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hca : z ≤ x) : (x \ y) \ (x \ z) = z \ y

theorem sdiff_sdiff_sdiff_cancel_right {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (hcb : z ≤ y) : (x \ z) \ (y \ z) = x \ y

theorem inf_sdiff {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : (x ⊓ y) \ z = x \ z ⊓ y \ z

theorem inf_sdiff_assoc {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : (x ⊓ y) \ z = x ⊓ y \ z

theorem inf_sdiff_right_comm {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : x \ z ⊓ y = (x ⊓ y) \ z

theorem inf_sdiff_distrib_left {α : Type u} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) : a ⊓ b \ c = (a ⊓ b) \ (a ⊓ c)

theorem inf_sdiff_distrib_right {α : Type u} [GeneralizedBooleanAlgebra α] (a : α) (b : α) (c : α) : a \ b ⊓ c = (a ⊓ c) \ (b ⊓ c)

theorem disjoint_sdiff_comm {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] : Disjoint (x \ z) y ↔ Disjoint x (y \ z)

theorem sup_eq_sdiff_sup_sdiff_sup_inf {α : Type u} {x : α} {y : α} [GeneralizedBooleanAlgebra α] : x ⊔ y = x \ y ⊔ y \ x ⊔ x ⊓ y

theorem sup_lt_of_lt_sdiff_left {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (h : y < z \ x) (hxz : x ≤ z) : x ⊔ y < z

theorem sup_lt_of_lt_sdiff_right {α : Type u} {x : α} {y : α} {z : α} [GeneralizedBooleanAlgebra α] (h : x < z \ y) (hyz : y ≤ z) : x ⊔ y < z

instance Prod.instGeneralizedBooleanAlgebra {α : Type u} {β : Type u_1} [GeneralizedBooleanAlgebra α] [GeneralizedBooleanAlgebra β] : GeneralizedBooleanAlgebra (α × β)

Equations

• Prod.instGeneralizedBooleanAlgebra = GeneralizedBooleanAlgebra.mk ⋯ ⋯

instance Pi.instGeneralizedBooleanAlgebra {ι : Type u_2} {α : ι → Type u_3} [(i : ι) → GeneralizedBooleanAlgebra (α i)] : GeneralizedBooleanAlgebra ((i : ι) → α i)

Equations

• Pi.instGeneralizedBooleanAlgebra = GeneralizedBooleanAlgebra.mk ⋯ ⋯

Boolean algebras

class BooleanAlgebra (α : Type u) extends DistribLattice , HasCompl , SDiff , HImp , Top , Bot : Type u

A Boolean algebra is a bounded distributive lattice with a complement operator ᶜ such that x ⊓ xᶜ = ⊥ and x ⊔ xᶜ = ⊤. For convenience, it must also provide a set difference operation \ and a Heyting implication ⇨ satisfying x \ y = x ⊓ yᶜ and x ⇨ y = y ⊔ xᶜ.

This is a generalization of (classical) logic of propositions, or the powerset lattice.

Since BoundedOrder, OrderBot, and OrderTop are mixins that require LE to be present at define-time, the extends mechanism does not work with them. Instead, we extend using the underlying Bot and Top data typeclasses, and replicate the order axioms of those classes here. A "forgetful" instance back to BoundedOrder is provided.

• sup : α → α → α
• le : α → α → Prop
• lt : α → α → Prop
• le_refl : ∀ (a : α), a ≤ a
• le_trans : ∀ (a b c : α), a ≤ b → b ≤ c → a ≤ c
• lt_iff_le_not_le : ∀ (a b : α), a < b ↔ a ≤ b ∧ ¬b ≤ a
• le_antisymm : ∀ (a b : α), a ≤ b → b ≤ a → a = b
• le_sup_left : ∀ (a b : α), a ≤ a ⊔ b
• le_sup_right : ∀ (a b : α), b ≤ a ⊔ b
• sup_le : ∀ (a b c : α), a ≤ c → b ≤ c → a ⊔ b ≤ c
• inf : α → α → α
• inf_le_left : ∀ (a b : α), a ⊓ b ≤ a
• inf_le_right : ∀ (a b : α), a ⊓ b ≤ b
• le_inf : ∀ (a b c : α), a ≤ b → a ≤ c → a ≤ b ⊓ c
• le_sup_inf : ∀ (x y z : α), (x ⊔ y) ⊓ (x ⊔ z) ≤ x ⊔ y ⊓ z
• compl : α → α
• sdiff : α → α → α
• himp : α → α → α
• top : α
• bot : α
• inf_compl_le_bot : ∀ (x : α), x ⊓ xᶜ ≤ ⊥

  The infimum of x and xᶜ is at most ⊥

• top_le_sup_compl : ∀ (x : α), ⊤ ≤ x ⊔ xᶜ

  The supremum of x and xᶜ is at least ⊤

• le_top : ∀ (a : α), a ≤ ⊤

  ⊤ is the greatest element

• bot_le : ∀ (a : α), ⊥ ≤ a

  ⊥ is the least element

• sdiff_eq : ∀ (x y : α), x \ y = x ⊓ yᶜ

  x \ y is equal to x ⊓ yᶜ

• himp_eq : ∀ (x y : α), x ⇨ y = y ⊔ xᶜ

  x ⇨ y is equal to y ⊔ xᶜ

Instances

• Bool.instBooleanAlgebra
• Finset.booleanAlgebra
• OrderDual.instBooleanAlgebra
• PUnit.instBooleanAlgebra
• Pi.instBooleanAlgebra
• Prod.instBooleanAlgebra
• Prop.instBooleanAlgebra
• Set.instBooleanAlgebraSet

theorem BooleanAlgebra.inf_compl_le_bot {α : Type u} [self : BooleanAlgebra α] (x : α) : x ⊓ xᶜ ≤ ⊥

The infimum of x and xᶜ is at most ⊥

theorem BooleanAlgebra.top_le_sup_compl {α : Type u} [self : BooleanAlgebra α] (x : α) : ⊤ ≤ x ⊔ xᶜ

The supremum of x and xᶜ is at least ⊤

theorem BooleanAlgebra.le_top {α : Type u} [self : BooleanAlgebra α] (a : α) : a ≤ ⊤

⊤ is the greatest element

theorem BooleanAlgebra.bot_le {α : Type u} [self : BooleanAlgebra α] (a : α) : ⊥ ≤ a

⊥ is the least element

theorem BooleanAlgebra.sdiff_eq {α : Type u} [self : BooleanAlgebra α] (x : α) (y : α) : x \ y = x ⊓ yᶜ

x \ y is equal to x ⊓ yᶜ

theorem BooleanAlgebra.himp_eq {α : Type u} [self : BooleanAlgebra α] (x : α) (y : α) : x ⇨ y = y ⊔ xᶜ

x ⇨ y is equal to y ⊔ xᶜ

@[instance 100]

instance BooleanAlgebra.toBoundedOrder {α : Type u} [h : BooleanAlgebra α] : BoundedOrder α

Equations

• BooleanAlgebra.toBoundedOrder = BoundedOrder.mk

@[reducible, inline]

abbrev GeneralizedBooleanAlgebra.toBooleanAlgebra {α : Type u} [GeneralizedBooleanAlgebra α] [OrderTop α] : BooleanAlgebra α

A bounded generalized boolean algebra is a boolean algebra.

Equations

• GeneralizedBooleanAlgebra.toBooleanAlgebra = let __spread.0 := inst✝; let __spread.1 := GeneralizedBooleanAlgebra.toOrderBot; let __spread.2 := inst; BooleanAlgebra.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

theorem inf_compl_eq_bot' {α : Type u} {x : α} [BooleanAlgebra α] : x ⊓ xᶜ = ⊥

@[simp]

theorem sup_compl_eq_top {α : Type u} {x : α} [BooleanAlgebra α] : x ⊔ xᶜ = ⊤

@[simp]

theorem compl_sup_eq_top {α : Type u} {x : α} [BooleanAlgebra α] : xᶜ ⊔ x = ⊤

theorem isCompl_compl {α : Type u} {x : α} [BooleanAlgebra α] : IsCompl x xᶜ

theorem sdiff_eq {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x \ y = x ⊓ yᶜ

theorem himp_eq {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x ⇨ y = y ⊔ xᶜ

@[instance 100]

instance BooleanAlgebra.toComplementedLattice {α : Type u} [BooleanAlgebra α] : ComplementedLattice α

Equations

• ⋯ = ⋯

@[instance 100]

instance BooleanAlgebra.toGeneralizedBooleanAlgebra {α : Type u} [BooleanAlgebra α] : GeneralizedBooleanAlgebra α

Equations

• BooleanAlgebra.toGeneralizedBooleanAlgebra = let __spread.0 := inst; GeneralizedBooleanAlgebra.mk ⋯ ⋯

@[instance 100]

instance BooleanAlgebra.toBiheytingAlgebra {α : Type u} [BooleanAlgebra α] : BiheytingAlgebra α

Equations

• BooleanAlgebra.toBiheytingAlgebra = let __spread.0 := inst; let __spread.1 := GeneralizedBooleanAlgebra.toGeneralizedCoheytingAlgebra; BiheytingAlgebra.mk ⋯ ⋯

@[simp]

theorem hnot_eq_compl {α : Type u} {x : α} [BooleanAlgebra α] : ￢x = xᶜ

theorem top_sdiff {α : Type u} {x : α} [BooleanAlgebra α] : ⊤ \ x = xᶜ

theorem eq_compl_iff_isCompl {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x = yᶜ ↔ IsCompl x y

theorem compl_eq_iff_isCompl {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : xᶜ = y ↔ IsCompl x y

theorem compl_eq_comm {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : xᶜ = y ↔ yᶜ = x

theorem eq_compl_comm {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x = yᶜ ↔ y = xᶜ

@[simp]

theorem compl_compl {α : Type u} [BooleanAlgebra α] (x : α) : xᶜᶜ = x

theorem compl_comp_compl {α : Type u} [BooleanAlgebra α] : compl ∘ compl = id

@[simp]

theorem compl_involutive {α : Type u} [BooleanAlgebra α] : Function.Involutive compl

theorem compl_bijective {α : Type u} [BooleanAlgebra α] : Function.Bijective compl

theorem compl_surjective {α : Type u} [BooleanAlgebra α] : Function.Surjective compl

theorem compl_injective {α : Type u} [BooleanAlgebra α] : Function.Injective compl

@[simp]

theorem compl_inj_iff {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : xᶜ = yᶜ ↔ x = y

theorem IsCompl.compl_eq_iff {α : Type u} {x : α} {y : α} {z : α} [BooleanAlgebra α] (h : IsCompl x y) : zᶜ = y ↔ z = x

@[simp]

theorem compl_eq_top {α : Type u} {x : α} [BooleanAlgebra α] : xᶜ = ⊤ ↔ x = ⊥

@[simp]

theorem compl_eq_bot {α : Type u} {x : α} [BooleanAlgebra α] : xᶜ = ⊥ ↔ x = ⊤

@[simp]

theorem compl_inf {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : (x ⊓ y)ᶜ = xᶜ ⊔ yᶜ

@[simp]

theorem compl_le_compl_iff_le {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : yᶜ ≤ xᶜ ↔ x ≤ y

@[simp]

theorem compl_lt_compl_iff_lt {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : yᶜ < xᶜ ↔ x < y

theorem compl_le_of_compl_le {α : Type u} {x : α} {y : α} [BooleanAlgebra α] (h : yᶜ ≤ x) : xᶜ ≤ y

theorem compl_le_iff_compl_le {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : xᶜ ≤ y ↔ yᶜ ≤ x

@[simp]

theorem compl_le_self {α : Type u} {x : α} [BooleanAlgebra α] : xᶜ ≤ x ↔ x = ⊤

@[simp]

theorem compl_lt_self {α : Type u} {x : α} [BooleanAlgebra α] [Nontrivial α] : xᶜ < x ↔ x = ⊤

@[simp]

theorem sdiff_compl {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x \ yᶜ = x ⊓ y

instance OrderDual.instBooleanAlgebra {α : Type u} [BooleanAlgebra α] : BooleanAlgebra αᵒᵈ

Equations

• OrderDual.instBooleanAlgebra = let __spread.0 := OrderDual.instDistribLattice α; let __spread.1 := OrderDual.instHeytingAlgebra; BooleanAlgebra.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

@[simp]

theorem sup_inf_inf_compl {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x ⊓ y ⊔ x ⊓ yᶜ = x

theorem compl_sdiff {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : (x \ y)ᶜ = x ⇨ y

@[simp]

theorem compl_himp {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : (x ⇨ y)ᶜ = x \ y

theorem compl_sdiff_compl {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : xᶜ \ yᶜ = y \ x

@[simp]

theorem compl_himp_compl {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : xᶜ ⇨ yᶜ = y ⇨ x

theorem disjoint_compl_left_iff {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : Disjoint xᶜ y ↔ y ≤ x

theorem disjoint_compl_right_iff {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : Disjoint x yᶜ ↔ x ≤ y

theorem codisjoint_himp_self_left {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : Codisjoint (x ⇨ y) x

theorem codisjoint_himp_self_right {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : Codisjoint x (x ⇨ y)

theorem himp_le {α : Type u} {x : α} {y : α} {z : α} [BooleanAlgebra α] : x ⇨ y ≤ z ↔ y ≤ z ∧ Codisjoint x z

@[simp]

theorem himp_le_iff {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x ⇨ y ≤ x ↔ x = ⊤

@[simp]

theorem himp_eq_left {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x ⇨ y = x ↔ x = ⊤ ∧ y = ⊤

theorem himp_ne_right {α : Type u} {x : α} {y : α} [BooleanAlgebra α] : x ⇨ y ≠ x ↔ x ≠ ⊤ ∨ y ≠ ⊤

instance Prop.instBooleanAlgebra : BooleanAlgebra Prop

Equations

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

instance Prod.instBooleanAlgebra {α : Type u} {β : Type u_1} [BooleanAlgebra α] [BooleanAlgebra β] : BooleanAlgebra (α × β)

Equations

• Prod.instBooleanAlgebra = let __spread.0 := Prod.instDistribLattice α β; let __spread.1 := Prod.instHeytingAlgebra; BooleanAlgebra.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance Pi.instBooleanAlgebra {ι : Type u} {α : ι → Type v} [(i : ι) → BooleanAlgebra (α i)] : BooleanAlgebra ((i : ι) → α i)

Equations

• Pi.instBooleanAlgebra = let __spread.0 := Pi.instDistribLattice; let __spread.1 := Pi.instHeytingAlgebra; BooleanAlgebra.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯

instance Bool.instBooleanAlgebra : BooleanAlgebra Bool

Equations

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

@[simp]

theorem Bool.sup_eq_bor : (fun (x x_1 : Bool) => x ⊔ x_1) = or

@[simp]

theorem Bool.inf_eq_band : (fun (x x_1 : Bool) => x ⊓ x_1) = and

@[simp]

theorem Bool.compl_eq_bnot : compl = not

@[reducible, inline]

abbrev Function.Injective.generalizedBooleanAlgebra {α : Type u} {β : Type u_1} [Sup α] [Inf α] [Bot α] [SDiff α] [GeneralizedBooleanAlgebra β] (f : α → β) (hf : Function.Injective f) (map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b) (map_bot : f ⊥ = ⊥) (map_sdiff : ∀ (a b : α), f (a \ b) = f a \ f b) : GeneralizedBooleanAlgebra α

Pullback a GeneralizedBooleanAlgebra along an injection.

Equations

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

@[reducible, inline]

abbrev Function.Injective.booleanAlgebra {α : Type u} {β : Type u_1} [Sup α] [Inf α] [Top α] [Bot α] [HasCompl α] [SDiff α] [BooleanAlgebra β] (f : α → β) (hf : Function.Injective f) (map_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b) (map_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b) (map_top : f ⊤ = ⊤) (map_bot : f ⊥ = ⊥) (map_compl : ∀ (a : α), f aᶜ = (f a)ᶜ) (map_sdiff : ∀ (a b : α), f (a \ b) = f a \ f b) : BooleanAlgebra α

Pullback a BooleanAlgebra along an injection.

Equations

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

instance PUnit.instBooleanAlgebra : BooleanAlgebra PUnit.{u_2 + 1}

Equations

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

noncomputable def DistribLattice.booleanAlgebraOfComplemented (α : Type u_2) [DistribLattice α] [BoundedOrder α] [ComplementedLattice α] : BooleanAlgebra α

An alternative constructor for boolean algebras: a distributive lattice that is complemented is a boolean algebra.

This is not an instance, because it creates data using choice.

Equations

• DistribLattice.booleanAlgebraOfComplemented α = let __spread.0 := inferInstanceAs (DistribLattice α); let __spread.1 := inferInstanceAs (BoundedOrder α); BooleanAlgebra.mk ⋯ ⋯ ⋯ ⋯ ⋯ ⋯