Disjointness and complements

This file defines Disjoint, Codisjoint, and the IsCompl predicate.

Main declarations

• Disjoint x y: two elements of a lattice are disjoint if their inf is the bottom element.
• Codisjoint x y: two elements of a lattice are codisjoint if their join is the top element.
• IsCompl x y: In a bounded lattice, predicate for "x is a complement of y". Note that in a non distributive lattice, an element can have several complements.
• ComplementedLattice α: Typeclass stating that any element of a lattice has a complement.

def Disjoint {α : Type u_1} [PartialOrder α] [OrderBot α] (a b : α) : Prop

Two elements of a lattice are disjoint if their inf is the bottom element. (This generalizes disjoint sets, viewed as members of the subset lattice.)

Note that we define this without reference to ⊓, as this allows us to talk about orders where the infimum is not unique, or where implementing Inf would require additional Decidable arguments.

Equations

• Disjoint a b = ∀ ⦃x : α⦄, x ≤ a → x ≤ b → x ≤ ⊥

Instances For

• Finset.decidableDisjoint

@[simp]

theorem disjoint_of_subsingleton {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} [Subsingleton α] : Disjoint a b

theorem disjoint_comm {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} : Disjoint a b ↔ Disjoint b a

theorem Disjoint.symm {α : Type u_1} [PartialOrder α] [OrderBot α] ⦃a b : α⦄ : Disjoint a b → Disjoint b a

theorem symmetric_disjoint {α : Type u_1} [PartialOrder α] [OrderBot α] : Symmetric Disjoint

@[simp]

theorem disjoint_bot_left {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α} : Disjoint ⊥ a

@[simp]

theorem disjoint_bot_right {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α} : Disjoint a ⊥

theorem Disjoint.mono {α : Type u_1} [PartialOrder α] [OrderBot α] {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d) : Disjoint b d → Disjoint a c

theorem Disjoint.mono_left {α : Type u_1} [PartialOrder α] [OrderBot α] {a b c : α} (h : a ≤ b) : Disjoint b c → Disjoint a c

theorem Disjoint.mono_right {α : Type u_1} [PartialOrder α] [OrderBot α] {a b c : α} : b ≤ c → Disjoint a c → Disjoint a b

@[simp]

theorem disjoint_self {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α} : Disjoint a a ↔ a = ⊥

theorem Disjoint.eq_bot_of_self {α : Type u_1} [PartialOrder α] [OrderBot α] {a : α} : Disjoint a a → a = ⊥

Alias of the forward direction of disjoint_self.

theorem Disjoint.ne {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (ha : a ≠ ⊥) (hab : Disjoint a b) : a ≠ b

theorem Disjoint.eq_bot_of_le {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b) (h : a ≤ b) : a = ⊥

theorem Disjoint.eq_bot_of_ge {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b) : b ≤ a → b = ⊥

theorem Disjoint.eq_iff {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b) : a = b ↔ a = ⊥ ∧ b = ⊥

theorem Disjoint.ne_iff {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (hab : Disjoint a b) : a ≠ b ↔ a ≠ ⊥ ∨ b ≠ ⊥

theorem disjoint_of_le_iff_left_eq_bot {α : Type u_1} [PartialOrder α] [OrderBot α] {a b : α} (h : a ≤ b) : Disjoint a b ↔ a = ⊥

@[simp]

theorem disjoint_top {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α} : Disjoint a ⊤ ↔ a = ⊥

@[simp]

theorem top_disjoint {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α} : Disjoint ⊤ a ↔ a = ⊥

theorem disjoint_iff_inf_le {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} : Disjoint a b ↔ a ⊓ b ≤ ⊥

theorem disjoint_iff {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} : Disjoint a b ↔ a ⊓ b = ⊥

theorem Disjoint.le_bot {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} : Disjoint a b → a ⊓ b ≤ ⊥

theorem Disjoint.eq_bot {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} : Disjoint a b → a ⊓ b = ⊥

theorem disjoint_assoc {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} : Disjoint (a ⊓ b) c ↔ Disjoint a (b ⊓ c)

theorem disjoint_left_comm {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} : Disjoint a (b ⊓ c) ↔ Disjoint b (a ⊓ c)

theorem disjoint_right_comm {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} : Disjoint (a ⊓ b) c ↔ Disjoint (a ⊓ c) b

theorem Disjoint.inf_left {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b) : Disjoint (a ⊓ c) b

theorem Disjoint.inf_left' {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b) : Disjoint (c ⊓ a) b

theorem Disjoint.inf_right {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b) : Disjoint a (b ⊓ c)

theorem Disjoint.inf_right' {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} (c : α) (h : Disjoint a b) : Disjoint a (c ⊓ b)

theorem Disjoint.of_disjoint_inf_of_le {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} (h : Disjoint (a ⊓ b) c) (hle : a ≤ c) : Disjoint a b

theorem Disjoint.of_disjoint_inf_of_le' {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b c : α} (h : Disjoint (a ⊓ b) c) (hle : b ≤ c) : Disjoint a b

theorem Disjoint.right_lt_sup_of_left_ne_bot {α : Type u_1} [SemilatticeSup α] [OrderBot α] {a b : α} (h : Disjoint a b) (ha : a ≠ ⊥) : b < a ⊔ b

@[simp]

theorem disjoint_sup_left {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} : Disjoint (a ⊔ b) c ↔ Disjoint a c ∧ Disjoint b c

@[simp]

theorem disjoint_sup_right {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} : Disjoint a (b ⊔ c) ↔ Disjoint a b ∧ Disjoint a c

theorem Disjoint.sup_left {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (ha : Disjoint a c) (hb : Disjoint b c) : Disjoint (a ⊔ b) c

theorem Disjoint.sup_right {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (hb : Disjoint a b) (hc : Disjoint a c) : Disjoint a (b ⊔ c)

theorem Disjoint.left_le_of_le_sup_right {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (h : a ≤ b ⊔ c) (hd : Disjoint a c) : a ≤ b

theorem Disjoint.left_le_of_le_sup_left {α : Type u_1} [DistribLattice α] [OrderBot α] {a b c : α} (h : a ≤ c ⊔ b) (hd : Disjoint a c) : a ≤ b

def Codisjoint {α : Type u_1} [PartialOrder α] [OrderTop α] (a b : α) : Prop

Two elements of a lattice are codisjoint if their sup is the top element.

Note that we define this without reference to ⊔, as this allows us to talk about orders where the supremum is not unique, or where implement Sup would require additional Decidable arguments.

Equations

• Codisjoint a b = ∀ ⦃x : α⦄, a ≤ x → b ≤ x → ⊤ ≤ x

Instances For

• Finset.decidableCodisjoint

theorem codisjoint_comm {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} : Codisjoint a b ↔ Codisjoint b a

@[deprecated codisjoint_comm (since := "2024-11-23")]

theorem Codisjoint_comm {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} : Codisjoint a b ↔ Codisjoint b a

Alias of codisjoint_comm.

theorem Codisjoint.symm {α : Type u_1} [PartialOrder α] [OrderTop α] ⦃a b : α⦄ : Codisjoint a b → Codisjoint b a

theorem symmetric_codisjoint {α : Type u_1} [PartialOrder α] [OrderTop α] : Symmetric Codisjoint

@[simp]

theorem codisjoint_top_left {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α} : Codisjoint ⊤ a

@[simp]

theorem codisjoint_top_right {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α} : Codisjoint a ⊤

theorem Codisjoint.mono {α : Type u_1} [PartialOrder α] [OrderTop α] {a b c d : α} (h₁ : a ≤ b) (h₂ : c ≤ d) : Codisjoint a c → Codisjoint b d

theorem Codisjoint.mono_left {α : Type u_1} [PartialOrder α] [OrderTop α] {a b c : α} (h : a ≤ b) : Codisjoint a c → Codisjoint b c

theorem Codisjoint.mono_right {α : Type u_1} [PartialOrder α] [OrderTop α] {a b c : α} : b ≤ c → Codisjoint a b → Codisjoint a c

@[simp]

theorem codisjoint_self {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α} : Codisjoint a a ↔ a = ⊤

theorem Codisjoint.eq_top_of_self {α : Type u_1} [PartialOrder α] [OrderTop α] {a : α} : Codisjoint a a → a = ⊤

Alias of the forward direction of codisjoint_self.

theorem Codisjoint.ne {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (ha : a ≠ ⊤) (hab : Codisjoint a b) : a ≠ b

theorem Codisjoint.eq_top_of_le {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b) (h : b ≤ a) : a = ⊤

theorem Codisjoint.eq_top_of_ge {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b) : a ≤ b → b = ⊤

theorem Codisjoint.eq_iff {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b) : a = b ↔ a = ⊤ ∧ b = ⊤

theorem Codisjoint.ne_iff {α : Type u_1} [PartialOrder α] [OrderTop α] {a b : α} (hab : Codisjoint a b) : a ≠ b ↔ a ≠ ⊤ ∨ b ≠ ⊤

@[simp]

theorem codisjoint_bot {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α} : Codisjoint a ⊥ ↔ a = ⊤

@[simp]

theorem bot_codisjoint {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a : α} : Codisjoint ⊥ a ↔ a = ⊤

theorem Codisjoint.ne_bot_of_ne_top {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : α} (h : Codisjoint a b) (ha : a ≠ ⊤) : b ≠ ⊥

theorem Codisjoint.ne_bot_of_ne_top' {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : α} (h : Codisjoint a b) (hb : b ≠ ⊤) : a ≠ ⊥

theorem codisjoint_iff_le_sup {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} : Codisjoint a b ↔ ⊤ ≤ a ⊔ b

theorem codisjoint_iff {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} : Codisjoint a b ↔ a ⊔ b = ⊤

theorem Codisjoint.top_le {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} : Codisjoint a b → ⊤ ≤ a ⊔ b

theorem Codisjoint.eq_top {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} : Codisjoint a b → a ⊔ b = ⊤

theorem codisjoint_assoc {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} : Codisjoint (a ⊔ b) c ↔ Codisjoint a (b ⊔ c)

theorem codisjoint_left_comm {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} : Codisjoint a (b ⊔ c) ↔ Codisjoint b (a ⊔ c)

theorem codisjoint_right_comm {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} : Codisjoint (a ⊔ b) c ↔ Codisjoint (a ⊔ c) b

theorem Codisjoint.sup_left {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b) : Codisjoint (a ⊔ c) b

theorem Codisjoint.sup_left' {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b) : Codisjoint (c ⊔ a) b

theorem Codisjoint.sup_right {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b) : Codisjoint a (b ⊔ c)

theorem Codisjoint.sup_right' {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} (c : α) (h : Codisjoint a b) : Codisjoint a (c ⊔ b)

theorem Codisjoint.of_codisjoint_sup_of_le {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} (h : Codisjoint (a ⊔ b) c) (hle : c ≤ a) : Codisjoint a b

theorem Codisjoint.of_codisjoint_sup_of_le' {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b c : α} (h : Codisjoint (a ⊔ b) c) (hle : c ≤ b) : Codisjoint a b

@[simp]

theorem codisjoint_inf_left {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} : Codisjoint (a ⊓ b) c ↔ Codisjoint a c ∧ Codisjoint b c

@[simp]

theorem codisjoint_inf_right {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} : Codisjoint a (b ⊓ c) ↔ Codisjoint a b ∧ Codisjoint a c

theorem Codisjoint.inf_left {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (ha : Codisjoint a c) (hb : Codisjoint b c) : Codisjoint (a ⊓ b) c

theorem Codisjoint.inf_right {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (hb : Codisjoint a b) (hc : Codisjoint a c) : Codisjoint a (b ⊓ c)

theorem Codisjoint.left_le_of_le_inf_right {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (h : a ⊓ b ≤ c) (hd : Codisjoint b c) : a ≤ c

theorem Codisjoint.left_le_of_le_inf_left {α : Type u_1} [DistribLattice α] [OrderTop α] {a b c : α} (h : b ⊓ a ≤ c) (hd : Codisjoint b c) : a ≤ c

theorem Disjoint.dual {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} : Disjoint a b → Codisjoint (OrderDual.toDual a) (OrderDual.toDual b)

theorem Codisjoint.dual {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} : Codisjoint a b → Disjoint (OrderDual.toDual a) (OrderDual.toDual b)

@[simp]

theorem disjoint_toDual_iff {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : α} : Disjoint (OrderDual.toDual a) (OrderDual.toDual b) ↔ Codisjoint a b

@[simp]

theorem disjoint_ofDual_iff {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : αᵒᵈ} : Disjoint (OrderDual.ofDual a) (OrderDual.ofDual b) ↔ Codisjoint a b

@[simp]

theorem codisjoint_toDual_iff {α : Type u_1} [SemilatticeInf α] [OrderBot α] {a b : α} : Codisjoint (OrderDual.toDual a) (OrderDual.toDual b) ↔ Disjoint a b

@[simp]

theorem codisjoint_ofDual_iff {α : Type u_1} [SemilatticeSup α] [OrderTop α] {a b : αᵒᵈ} : Codisjoint (OrderDual.ofDual a) (OrderDual.ofDual b) ↔ Disjoint a b

theorem Disjoint.le_of_codisjoint {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b c : α} (hab : Disjoint a b) (hbc : Codisjoint b c) : a ≤ c

structure IsCompl {α : Type u_1} [PartialOrder α] [BoundedOrder α] (x y : α) : Prop

Two elements x and y are complements of each other if x ⊔ y = ⊤ and x ⊓ y = ⊥.

• disjoint : Disjoint x y

  If x and y are to be complementary in an order, they should be disjoint.

• codisjoint : Codisjoint x y

  If x and y are to be complementary in an order, they should be codisjoint.

Instances For

• Finset.decidableIsCompl

theorem isCompl_iff {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : α} : IsCompl a b ↔ Disjoint a b ∧ Codisjoint a b

theorem IsCompl.symm {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} (h : IsCompl x y) : IsCompl y x

theorem isCompl_comm {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} : IsCompl x y ↔ IsCompl y x

theorem IsCompl.dual {α : Type u_1} [PartialOrder α] [BoundedOrder α] {x y : α} (h : IsCompl x y) : IsCompl (OrderDual.toDual x) (OrderDual.toDual y)

theorem IsCompl.ofDual {α : Type u_1} [PartialOrder α] [BoundedOrder α] {a b : αᵒᵈ} (h : IsCompl a b) : IsCompl (OrderDual.ofDual a) (OrderDual.ofDual b)

theorem IsCompl.of_le {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h₁ : x ⊓ y ≤ ⊥) (h₂ : ⊤ ≤ x ⊔ y) : IsCompl x y

theorem IsCompl.of_eq {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h₁ : x ⊓ y = ⊥) (h₂ : x ⊔ y = ⊤) : IsCompl x y

theorem IsCompl.inf_eq_bot {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h : IsCompl x y) : x ⊓ y = ⊥

theorem IsCompl.sup_eq_top {α : Type u_1} [Lattice α] [BoundedOrder α] {x y : α} (h : IsCompl x y) : x ⊔ y = ⊤

theorem IsCompl.inf_left_le_of_le_sup_right {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b x y : α} (h : IsCompl x y) (hle : a ≤ b ⊔ y) : a ⊓ x ≤ b

theorem IsCompl.le_sup_right_iff_inf_left_le {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y a b : α} (h : IsCompl x y) : a ≤ b ⊔ y ↔ a ⊓ x ≤ b

theorem IsCompl.inf_left_eq_bot_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z) : x ⊓ y = ⊥ ↔ x ≤ z

theorem IsCompl.inf_right_eq_bot_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z) : x ⊓ z = ⊥ ↔ x ≤ y

theorem IsCompl.disjoint_left_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z) : Disjoint x y ↔ x ≤ z

theorem IsCompl.disjoint_right_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl y z) : Disjoint x z ↔ x ≤ y

theorem IsCompl.le_left_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y) : z ≤ x ↔ Disjoint z y

theorem IsCompl.le_right_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y) : z ≤ y ↔ Disjoint z x

theorem IsCompl.left_le_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y) : x ≤ z ↔ Codisjoint z y

theorem IsCompl.right_le_iff {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (h : IsCompl x y) : y ≤ z ↔ Codisjoint z x

theorem IsCompl.Antitone {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y x' y' : α} (h : IsCompl x y) (h' : IsCompl x' y') (hx : x ≤ x') : y' ≤ y

theorem IsCompl.right_unique {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (hxy : IsCompl x y) (hxz : IsCompl x z) : y = z

theorem IsCompl.left_unique {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y z : α} (hxz : IsCompl x z) (hyz : IsCompl y z) : x = y

theorem IsCompl.sup_inf {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y x' y' : α} (h : IsCompl x y) (h' : IsCompl x' y') : IsCompl (x ⊔ x') (y ⊓ y')

theorem IsCompl.inf_sup {α : Type u_1} [DistribLattice α] [BoundedOrder α] {x y x' y' : α} (h : IsCompl x y) (h' : IsCompl x' y') : IsCompl (x ⊓ x') (y ⊔ y')

theorem Prod.disjoint_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [OrderBot α] [OrderBot β] {x y : α × β} : Disjoint x y ↔ Disjoint x.fst y.fst ∧ Disjoint x.snd y.snd

theorem Prod.codisjoint_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [OrderTop α] [OrderTop β] {x y : α × β} : Codisjoint x y ↔ Codisjoint x.fst y.fst ∧ Codisjoint x.snd y.snd

theorem Prod.isCompl_iff {α : Type u_1} {β : Type u_2} [PartialOrder α] [PartialOrder β] [BoundedOrder α] [BoundedOrder β] {x y : α × β} : IsCompl x y ↔ IsCompl x.fst y.fst ∧ IsCompl x.snd y.snd

@[simp]

theorem isCompl_toDual_iff {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : α} : IsCompl (OrderDual.toDual a) (OrderDual.toDual b) ↔ IsCompl a b

@[simp]

theorem isCompl_ofDual_iff {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : αᵒᵈ} : IsCompl (OrderDual.ofDual a) (OrderDual.ofDual b) ↔ IsCompl a b

theorem isCompl_bot_top {α : Type u_1} [Lattice α] [BoundedOrder α] : IsCompl ⊥ ⊤

theorem isCompl_top_bot {α : Type u_1} [Lattice α] [BoundedOrder α] : IsCompl ⊤ ⊥

theorem eq_top_of_isCompl_bot {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl x ⊥) : x = ⊤

theorem eq_top_of_bot_isCompl {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl ⊥ x) : x = ⊤

theorem eq_bot_of_isCompl_top {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl x ⊤) : x = ⊥

theorem eq_bot_of_top_isCompl {α : Type u_1} [Lattice α] [BoundedOrder α] {x : α} (h : IsCompl ⊤ x) : x = ⊥

def IsComplemented {α : Type u_1} [Lattice α] [BoundedOrder α] (a : α) : Prop

An element is complemented if it has a complement.

Equations

• IsComplemented a = ∃ (b : α), IsCompl a b

theorem isComplemented_bot {α : Type u_1} [Lattice α] [BoundedOrder α] : IsComplemented ⊥

theorem isComplemented_top {α : Type u_1} [Lattice α] [BoundedOrder α] : IsComplemented ⊤

theorem IsComplemented.sup {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α} : IsComplemented a → IsComplemented b → IsComplemented (a ⊔ b)

theorem IsComplemented.inf {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α} : IsComplemented a → IsComplemented b → IsComplemented (a ⊓ b)

class ComplementedLattice (α : Type u_2) [Lattice α] [BoundedOrder α] : Prop

A complemented bounded lattice is one where every element has a (not necessarily unique) complement.

• exists_isCompl (a : α) : ∃ (b : α), IsCompl a b

  In a ComplementedLattice, every element admits a complement.

Instances

• BooleanAlgebra.toComplementedLattice
• ComplementedLattice.instOrderDual
• Complementeds.instComplementedLattice
• IsModularLattice.complementedLattice_Ici
• IsModularLattice.complementedLattice_Iic
• IsSemisimpleModule.quotient
• IsSemisimpleModule.submodule
• IsSemisimpleRing.isSemisimpleModule
• IsSimpleOrder.instComplementedLattice
• Submodule.complementedLattice
• Subsingleton.instComplementedLattice
• instIsSemisimpleModuleDFinsupp
• instIsSemisimpleModuleForallOfFinite
• instIsSemisimpleRingForallOfFinite
• instIsSemisimpleRingProd

theorem complementedLattice_iff (α : Type u_2) [Lattice α] [BoundedOrder α] : ComplementedLattice α ↔ ∀ (a : α), ∃ (b : α), IsCompl a b

instance Subsingleton.instComplementedLattice {α : Type u_1} [Lattice α] [BoundedOrder α] [Subsingleton α] : ComplementedLattice α

instance ComplementedLattice.instOrderDual {α : Type u_1} [Lattice α] [BoundedOrder α] [ComplementedLattice α] : ComplementedLattice αᵒᵈ

@[reducible, inline]

abbrev Complementeds (α : Type u_2) [Lattice α] [BoundedOrder α] : Type u_2

The sublattice of complemented elements.

Equations

• Complementeds α = { a : α // IsComplemented a }

Instances For

• Complementeds.hasCoeT
• Complementeds.instBoundedOrder
• Complementeds.instComplementedLattice
• Complementeds.instDistribLattice
• Complementeds.instInhabited
• Complementeds.instMax
• Complementeds.instMin

instance Complementeds.hasCoeT {α : Type u_1} [Lattice α] [BoundedOrder α] : CoeTC (Complementeds α) α

Equations

• Complementeds.hasCoeT = { coe := Subtype.val }

theorem Complementeds.coe_injective {α : Type u_1} [Lattice α] [BoundedOrder α] : Function.Injective Subtype.val

@[simp]

theorem Complementeds.coe_inj {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : Complementeds α} : ↑a = ↑b ↔ a = b

theorem Complementeds.coe_le_coe {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : Complementeds α} : ↑a ≤ ↑b ↔ a ≤ b

theorem Complementeds.coe_lt_coe {α : Type u_1} [Lattice α] [BoundedOrder α] {a b : Complementeds α} : ↑a < ↑b ↔ a < b

instance Complementeds.instBoundedOrder {α : Type u_1} [Lattice α] [BoundedOrder α] : BoundedOrder (Complementeds α)

Equations

• Complementeds.instBoundedOrder = Subtype.boundedOrder ⋯ ⋯

@[simp]

theorem Complementeds.coe_bot {α : Type u_1} [Lattice α] [BoundedOrder α] : ↑⊥ = ⊥

@[simp]

theorem Complementeds.coe_top {α : Type u_1} [Lattice α] [BoundedOrder α] : ↑⊤ = ⊤

theorem Complementeds.mk_bot {α : Type u_1} [Lattice α] [BoundedOrder α] : ⟨⊥, ⋯⟩ = ⊥

theorem Complementeds.mk_top {α : Type u_1} [Lattice α] [BoundedOrder α] : ⟨⊤, ⋯⟩ = ⊤

instance Complementeds.instInhabited {α : Type u_1} [Lattice α] [BoundedOrder α] : Inhabited (Complementeds α)

Equations

• Complementeds.instInhabited = { default := ⊥ }

instance Complementeds.instMax {α : Type u_1} [DistribLattice α] [BoundedOrder α] : Max (Complementeds α)

Equations

• Complementeds.instMax = { max := fun (a b : Complementeds α) => ⟨↑a ⊔ ↑b, ⋯⟩ }

instance Complementeds.instMin {α : Type u_1} [DistribLattice α] [BoundedOrder α] : Min (Complementeds α)

Equations

• Complementeds.instMin = { min := fun (a b : Complementeds α) => ⟨↑a ⊓ ↑b, ⋯⟩ }

@[simp]

theorem Complementeds.coe_sup {α : Type u_1} [DistribLattice α] [BoundedOrder α] (a b : Complementeds α) : ↑(a ⊔ b) = ↑a ⊔ ↑b

@[simp]

theorem Complementeds.coe_inf {α : Type u_1} [DistribLattice α] [BoundedOrder α] (a b : Complementeds α) : ↑(a ⊓ b) = ↑a ⊓ ↑b

@[simp]

theorem Complementeds.mk_sup_mk {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α} (ha : IsComplemented a) (hb : IsComplemented b) : ⟨a, ha⟩ ⊔ ⟨b, hb⟩ = ⟨a ⊔ b, ⋯⟩

@[simp]

theorem Complementeds.mk_inf_mk {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : α} (ha : IsComplemented a) (hb : IsComplemented b) : ⟨a, ha⟩ ⊓ ⟨b, hb⟩ = ⟨a ⊓ b, ⋯⟩

instance Complementeds.instDistribLattice {α : Type u_1} [DistribLattice α] [BoundedOrder α] : DistribLattice (Complementeds α)

Equations

• Complementeds.instDistribLattice = Function.Injective.distribLattice Subtype.val ⋯ ⋯ ⋯

@[simp]

theorem Complementeds.disjoint_coe {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : Complementeds α} : Disjoint ↑a ↑b ↔ Disjoint a b

@[simp]

theorem Complementeds.codisjoint_coe {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : Complementeds α} : Codisjoint ↑a ↑b ↔ Codisjoint a b

@[simp]

theorem Complementeds.isCompl_coe {α : Type u_1} [DistribLattice α] [BoundedOrder α] {a b : Complementeds α} : IsCompl ↑a ↑b ↔ IsCompl a b

instance Complementeds.instComplementedLattice {α : Type u_1} [DistribLattice α] [BoundedOrder α] : ComplementedLattice (Complementeds α)