Atoms, Coatoms, and Simple Lattices #

This module defines atoms, which are minimal non-⊥ elements in bounded lattices, simple lattices, which are lattices with only two elements, and related ideas.

Main definitions #

Atoms and Coatoms #

is_atom a indicates that the only element below a is ⊥.
is_coatom a indicates that the only element above a is ⊤.

Atomic and Atomistic Lattices #

is_atomic indicates that every element other than ⊥ is above an atom.
is_coatomic indicates that every element other than ⊤ is below a coatom.
is_atomistic indicates that every element is the Sup of a set of atoms.
is_coatomistic indicates that every element is the Inf of a set of coatoms.

Simple Lattices #

is_simple_lattice indicates that a bounded lattice has only two elements, ⊥ and ⊤.
is_simple_lattice.bounded_distrib_lattice
Given an instance of is_simple_lattice, we provide the following definitions. These are not made global instances as they contain data :

Main results #

is_atom_dual_iff_is_coatom and is_coatom_dual_iff_is_atom express the (definitional) duality of is_atom and is_coatom.
is_simple_lattice_iff_is_atom_top and is_simple_lattice_iff_is_coatom_bot express the connection between atoms, coatoms, and simple lattices
is_compl.is_atom_iff_is_coatom and is_compl.is_coatom_if_is_atom: In a modular bounded lattice, a complement of an atom is a coatom and vice versa.
``is_atomic_iff_is_coatomic`: A modular complemented lattice is atomic iff it is coatomic.

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def is_atom {α : Type u_1} [order_bot α] (a : α) :

Prop

An atom of an order_bot is an element with no other element between it and ⊥, which is not ⊥.

Equations

is_atom a = (a ≠ ⊥ ∧ ∀ (b : α), b < a → b = ⊥)

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theorem eq_bot_or_eq_of_le_atom {α : Type u_1} [order_bot α] {a b : α} (ha : is_atom a) (hab : b ≤ a) :

b = ⊥ ∨ b = a

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theorem is_atom.Iic {α : Type u_1} [order_bot α] {x a : α} (ha : is_atom a) (hax : a ≤ x) :

is_atom ⟨a, hax⟩

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theorem is_atom.of_is_atom_coe_Iic {α : Type u_1} [order_bot α] {x : α} {a : ↥(set.Iic x)} (ha : is_atom a) :

is_atom ↑a

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def is_coatom {α : Type u_1} [order_top α] (a : α) :

Prop

A coatom of an order_top is an element with no other element between it and ⊤, which is not ⊤.

Equations

is_coatom a = (a ≠ ⊤ ∧ ∀ (b : α), a < b → b = ⊤)

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theorem eq_top_or_eq_of_coatom_le {α : Type u_1} [order_top α] {a b : α} (ha : is_coatom a) (hab : a ≤ b) :

b = ⊤ ∨ b = a

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theorem is_coatom.Ici {α : Type u_1} [order_top α] {x a : α} (ha : is_coatom a) (hax : x ≤ a) :

is_coatom ⟨a, hax⟩

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theorem is_coatom.of_is_coatom_coe_Ici {α : Type u_1} [order_top α] {x : α} {a : ↥(set.Ici x)} (ha : is_coatom a) :

is_coatom ↑a

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theorem is_atom.inf_eq_bot_of_ne {α : Type u_1} [semilattice_inf_bot α] {a b : α} (ha : is_atom a) (hb : is_atom b) (hab : a ≠ b) :

a ⊓ b = ⊥

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theorem is_atom.disjoint_of_ne {α : Type u_1} [semilattice_inf_bot α] {a b : α} (ha : is_atom a) (hb : is_atom b) (hab : a ≠ b) :

disjoint a b

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theorem is_coatom.sup_eq_top_of_ne {α : Type u_1} [semilattice_sup_top α] {a b : α} (ha : is_coatom a) (hb : is_coatom b) (hab : a ≠ b) :

a ⊔ b = ⊤

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@[simp]

theorem is_coatom_dual_iff_is_atom {α : Type u_1} {a : α} [order_bot α] :

is_coatom (⇑order_dual.to_dual a) ↔ is_atom a

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@[simp]

theorem is_atom_dual_iff_is_coatom {α : Type u_1} {a : α} [order_top α] :

is_atom (⇑order_dual.to_dual a) ↔ is_coatom a

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@[class]

structure is_atomic (α : Type u_1) [order_bot α] :

Prop

eq_bot_or_exists_atom_le : ∀ (b : α), b = ⊥ ∨ ∃ (a : α), is_atom a ∧ a ≤ b

A lattice is atomic iff every element other than ⊥ has an atom below it.

Instances

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@[class]

structure is_coatomic (α : Type u_1) [order_top α] :

Prop

eq_top_or_exists_le_coatom : ∀ (b : α), b = ⊤ ∨ ∃ (a : α), is_coatom a ∧ b ≤ a

A lattice is coatomic iff every element other than ⊤ has a coatom above it.

Instances

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@[simp]

theorem is_coatomic_dual_iff_is_atomic {α : Type u_1} [order_bot α] :

is_coatomic (order_dual α) ↔ is_atomic α

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@[simp]

theorem is_atomic_dual_iff_is_coatomic {α : Type u_1} [order_top α] :

is_atomic (order_dual α) ↔ is_coatomic α

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@[instance]

def is_atomic.is_coatomic_dual {α : Type u_1} [order_bot α] [is_atomic α] :

is_coatomic (order_dual α)

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@[instance]

def is_atomic.set.Iic.is_atomic {α : Type u_1} [order_bot α] [is_atomic α] {x : α} :

is_atomic ↥(set.Iic x)

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@[instance]

def is_coatomic.is_coatomic {α : Type u_1} [order_top α] [is_coatomic α] :

is_atomic (order_dual α)

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@[instance]

def is_coatomic.set.Ici.is_coatomic {α : Type u_1} [order_top α] [is_coatomic α] {x : α} :

is_coatomic ↥(set.Ici x)

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theorem is_atomic_iff_forall_is_atomic_Iic {α : Type u_1} [order_bot α] :

is_atomic α ↔ ∀ (x : α), is_atomic ↥(set.Iic x)

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theorem is_coatomic_iff_forall_is_coatomic_Ici {α : Type u_1} [order_top α] :

is_coatomic α ↔ ∀ (x : α), is_coatomic ↥(set.Ici x)

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@[class]

structure is_atomistic (α : Type u_1) [complete_lattice α] :

Prop

eq_Sup_atoms : ∀ (b : α), ∃ (s : set α), b = Sup s ∧ ∀ (a : α), a ∈ s → is_atom a

A lattice is atomistic iff every element is a Sup of a set of atoms.

Instances

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@[class]

structure is_coatomistic (α : Type u_1) [complete_lattice α] :

Prop

eq_Inf_coatoms : ∀ (b : α), ∃ (s : set α), b = Inf s ∧ ∀ (a : α), a ∈ s → is_coatom a

A lattice is coatomistic iff every element is an Inf of a set of coatoms.

Instances

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@[simp]

theorem is_coatomistic_dual_iff_is_atomistic {α : Type u_1} [complete_lattice α] :

is_coatomistic (order_dual α) ↔ is_atomistic α

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@[simp]

theorem is_atomistic_dual_iff_is_coatomistic {α : Type u_1} [complete_lattice α] :

is_atomistic (order_dual α) ↔ is_coatomistic α

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@[instance]

def is_atomistic.is_coatomistic_dual {α : Type u_1} [complete_lattice α] [h : is_atomistic α] :

is_coatomistic (order_dual α)

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@[instance]

def is_atomistic.is_atomic {α : Type u_1} [complete_lattice α] [is_atomistic α] :

is_atomic α

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@[simp]

theorem Sup_atoms_le_eq {α : Type u_1} [complete_lattice α] [is_atomistic α] (b : α) :

Sup {a : α | is_atom a ∧ a ≤ b} = b

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@[simp]

theorem Sup_atoms_eq_top {α : Type u_1} [complete_lattice α] [is_atomistic α] :

Sup {a : α | is_atom a} = ⊤

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theorem le_iff_atom_le_imp {α : Type u_1} [complete_lattice α] [is_atomistic α] {a b : α} :

a ≤ b ↔ ∀ (c : α), is_atom c → c ≤ a → c ≤ b

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@[instance]

def is_coatomistic.is_atomistic_dual {α : Type u_1} [complete_lattice α] [h : is_coatomistic α] :

is_atomistic (order_dual α)

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@[instance]

def is_coatomistic.is_coatomic {α : Type u_1} [complete_lattice α] [is_coatomistic α] :

is_coatomic α

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@[class]

structure is_simple_lattice (α : Type u_2) [bounded_lattice α] :

Prop

to_nontrivial : nontrivial α
eq_bot_or_eq_top : ∀ (a : α), a = ⊥ ∨ a = ⊤

A lattice is simple iff it has only two elements, ⊥ and ⊤.

Instances

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theorem is_simple_lattice_iff_is_simple_lattice_order_dual {α : Type u_1} [bounded_lattice α] :

is_simple_lattice α ↔ is_simple_lattice (order_dual α)

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@[instance]

def order_dual.is_simple_lattice {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

is_simple_lattice (order_dual α)

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@[simp]

theorem is_atom_top {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

is_atom ⊤

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@[simp]

theorem is_coatom_bot {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

is_coatom ⊥

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@[instance]

def is_simple_lattice.bounded_distrib_lattice {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

bounded_distrib_lattice α

A simple bounded_lattice is also distributive.

Equations

is_simple_lattice.bounded_distrib_lattice = {sup := bounded_lattice.sup infer_instance, le := bounded_lattice.le infer_instance, lt := bounded_lattice.lt infer_instance, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := bounded_lattice.inf infer_instance, inf_le_left := _, inf_le_right := _, le_inf := _, le_sup_inf := _, bot := bounded_lattice.bot infer_instance, bot_le := _, top := bounded_lattice.top infer_instance, le_top := _}

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@[instance]

def is_simple_lattice.is_atomic {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

is_atomic α

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@[instance]

def is_simple_lattice.is_coatomic {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

is_coatomic α

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def is_simple_lattice.order_iso_bool {α : Type u_1} [decidable_eq α] [bounded_lattice α] [is_simple_lattice α] :

α ≃o bool

Every simple lattice is order-isomorphic to bool.

Equations

is_simple_lattice.order_iso_bool = {to_equiv := {to_fun := λ (x : α), to_bool (x = ⊤), inv_fun := λ (x : bool), cond x ⊤ ⊥, left_inv := _, right_inv := _}, map_rel_iff' := _}

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@[instance]

def is_simple_lattice.fintype {α : Type u_1} [decidable_eq α] [bounded_lattice α] [is_simple_lattice α] :

fintype α

Equations

is_simple_lattice.fintype = fintype.of_equiv bool is_simple_lattice.order_iso_bool.to_equiv.symm

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def is_simple_lattice.boolean_algebra {α : Type u_1} [decidable_eq α] [bounded_lattice α] [is_simple_lattice α] :

boolean_algebra α

A simple bounded_lattice is also a boolean_algebra.

Equations

is_simple_lattice.boolean_algebra = {sup := bounded_distrib_lattice.sup is_simple_lattice.bounded_distrib_lattice, le := bounded_distrib_lattice.le is_simple_lattice.bounded_distrib_lattice, lt := bounded_distrib_lattice.lt is_simple_lattice.bounded_distrib_lattice, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := bounded_distrib_lattice.inf is_simple_lattice.bounded_distrib_lattice, inf_le_left := _, inf_le_right := _, le_inf := _, le_sup_inf := _, bot := bounded_distrib_lattice.bot is_simple_lattice.bounded_distrib_lattice, bot_le := _, sdiff := λ (x y : α), ite (x = ⊤ ∧ y = ⊥) ⊤ ⊥, sup_inf_sdiff := _, inf_inf_sdiff := _, top := bounded_distrib_lattice.top is_simple_lattice.bounded_distrib_lattice, le_top := _, compl := λ (x : α), ite (x = ⊥) ⊤ ⊥, inf_compl_le_bot := _, top_le_sup_compl := _, sdiff_eq := _}

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def is_simple_lattice.complete_lattice {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

complete_lattice α

A simple bounded_lattice is also complete.

Equations

is_simple_lattice.complete_lattice = {sup := bounded_lattice.sup infer_instance, le := bounded_lattice.le infer_instance, lt := bounded_lattice.lt infer_instance, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := bounded_lattice.inf infer_instance, inf_le_left := _, inf_le_right := _, le_inf := _, top := bounded_lattice.top infer_instance, le_top := _, bot := bounded_lattice.bot infer_instance, bot_le := _, Sup := λ (s : set α), ite (⊤ ∈ s) ⊤ ⊥, le_Sup := _, Sup_le := _, Inf := λ (s : set α), ite (⊥ ∈ s) ⊥ ⊤, Inf_le := _, le_Inf := _}

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def is_simple_lattice.complete_boolean_algebra {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] :

complete_boolean_algebra α

A simple bounded_lattice is also a complete_boolean_algebra.

Equations

is_simple_lattice.complete_boolean_algebra = {sup := complete_lattice.sup is_simple_lattice.complete_lattice, le := complete_lattice.le is_simple_lattice.complete_lattice, lt := complete_lattice.lt is_simple_lattice.complete_lattice, le_refl := _, le_trans := _, lt_iff_le_not_le := _, le_antisymm := _, le_sup_left := _, le_sup_right := _, sup_le := _, inf := complete_lattice.inf is_simple_lattice.complete_lattice, inf_le_left := _, inf_le_right := _, le_inf := _, le_sup_inf := _, bot := complete_lattice.bot is_simple_lattice.complete_lattice, bot_le := _, sdiff := boolean_algebra.sdiff is_simple_lattice.boolean_algebra, sup_inf_sdiff := _, inf_inf_sdiff := _, top := complete_lattice.top is_simple_lattice.complete_lattice, le_top := _, compl := boolean_algebra.compl is_simple_lattice.boolean_algebra, inf_compl_le_bot := _, top_le_sup_compl := _, sdiff_eq := _, Sup := complete_lattice.Sup is_simple_lattice.complete_lattice, le_Sup := _, Sup_le := _, Inf := complete_lattice.Inf is_simple_lattice.complete_lattice, Inf_le := _, le_Inf := _, infi_sup_le_sup_Inf := _, inf_Sup_le_supr_inf := _}

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@[instance]

def is_simple_lattice.is_atomistic {α : Type u_1} [complete_lattice α] [is_simple_lattice α] :

is_atomistic α

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@[instance]

def is_simple_lattice.is_coatomistic {α : Type u_1} [complete_lattice α] [is_simple_lattice α] :

is_coatomistic α

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theorem fintype.is_simple_lattice.univ {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] [decidable_eq α] :

finset.univ = {⊤, ⊥}

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theorem fintype.is_simple_lattice.card {α : Type u_1} [bounded_lattice α] [is_simple_lattice α] [decidable_eq α] :

fintype.card α = 2

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@[instance]

def bool.is_simple_lattice :

is_simple_lattice bool

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theorem is_simple_lattice_iff_is_atom_top {α : Type u_1} [bounded_lattice α] :

is_simple_lattice α ↔ is_atom ⊤

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theorem is_simple_lattice_iff_is_coatom_bot {α : Type u_1} [bounded_lattice α] :

is_simple_lattice α ↔ is_coatom ⊥

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theorem set.is_simple_lattice_Iic_iff_is_atom {α : Type u_1} [bounded_lattice α] {a : α} :

is_simple_lattice ↥(set.Iic a) ↔ is_atom a

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theorem set.is_simple_lattice_Ici_iff_is_coatom {α : Type u_1} [bounded_lattice α] {a : α} :

is_simple_lattice ↥(set.Ici a) ↔ is_coatom a

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@[simp]

theorem order_iso.is_atom_iff {α : Type u_1} [order_bot α] {β : Type u_2} [order_bot β] (f : α ≃o β) (a : α) :

is_atom (⇑f a) ↔ is_atom a

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@[simp]

theorem order_iso.is_coatom_iff {α : Type u_1} [order_top α] {β : Type u_2} [order_top β] (f : α ≃o β) (a : α) :

is_coatom (⇑f a) ↔ is_coatom a

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theorem order_iso.is_simple_lattice_iff {α : Type u_1} [bounded_lattice α] {β : Type u_2} [bounded_lattice β] (f : α ≃o β) :

is_simple_lattice α ↔ is_simple_lattice β

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theorem order_iso.is_simple_lattice {α : Type u_1} [bounded_lattice α] {β : Type u_2} [bounded_lattice β] [h : is_simple_lattice β] (f : α ≃o β) :

is_simple_lattice α

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theorem order_iso.is_atomic_iff {α : Type u_1} [order_bot α] {β : Type u_2} [order_bot β] (f : α ≃o β) :

is_atomic α ↔ is_atomic β

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theorem order_iso.is_coatomic_iff {α : Type u_1} [order_top α] {β : Type u_2} [order_top β] (f : α ≃o β) :

is_coatomic α ↔ is_coatomic β

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theorem is_compl.is_atom_iff_is_coatom {α : Type u_1} [bounded_lattice α] [is_modular_lattice α] {a b : α} (hc : is_compl a b) :

is_atom a ↔ is_coatom b

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theorem is_compl.is_coatom_iff_is_atom {α : Type u_1} [bounded_lattice α] [is_modular_lattice α] {a b : α} (hc : is_compl a b) :

is_coatom a ↔ is_atom b

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theorem is_coatomic_of_is_atomic_of_is_complemented_of_is_modular {α : Type u_1} [bounded_lattice α] [is_modular_lattice α] [is_complemented α] [is_atomic α] :

is_coatomic α

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theorem is_atomic_of_is_coatomic_of_is_complemented_of_is_modular {α : Type u_1} [bounded_lattice α] [is_modular_lattice α] [is_complemented α] [is_coatomic α] :

is_atomic α

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theorem is_atomic_iff_is_coatomic {α : Type u_1} [bounded_lattice α] [is_modular_lattice α] [is_complemented α] :

is_atomic α ↔ is_coatomic α