Notation classes for lattice operations

In this file we introduce typeclasses and definitions for lattice operations.

Main definitions

• Sup: type class for the ⊔ notation
• Inf: type class for the ⊓ notation
• HasCompl: type class for the ᶜ notation
• Top: type class for the ⊤ notation
• Bot: type class for the ⊥ notation

Notations

• x ⊔ y: lattice join operation;
• x ⊓ y: lattice meet operation;
• xᶜ: complement in a lattice;

class HasCompl (α : Type u_1) : Type u_1

Set / lattice complement

• compl : α → α

  Set / lattice complement

Sup and Inf

class Sup (α : Type u_1) : Type u_1

Typeclass for the ⊔ (\lub) notation

• sup : α → α → α

  Least upper bound (\lub notation)

theorem Sup.ext {α : Type u_1} {x : Sup α} {y : Sup α} (sup : Sup.sup = Sup.sup) : x = y

class Inf (α : Type u_1) : Type u_1

Typeclass for the ⊓ (\glb) notation

• inf : α → α → α

  Greatest lower bound (\glb notation)

theorem Inf.ext {α : Type u_1} {x : Inf α} {y : Inf α} (inf : Inf.inf = Inf.inf) : x = y

class HImp (α : Type u_1) : Type u_1

Syntax typeclass for Heyting implication ⇨.

• himp : α → α → α

  Heyting implication ⇨

class HNot (α : Type u_1) : Type u_1

Syntax typeclass for Heyting negation ￢.

The difference between HasCompl and HNot is that the former belongs to Heyting algebras, while the latter belongs to co-Heyting algebras. They are both pseudo-complements, but compl underestimates while HNot overestimates. In boolean algebras, they are equal. See hnot_eq_compl.

• hnot : α → α

  Heyting negation ￢

class Top (α : Type u_1) : Type u_1

Typeclass for the ⊤ (\top) notation

• top : α

  The top (⊤, \top) element

theorem Top.ext {α : Type u_1} {x : Top α} {y : Top α} (top : ⊤ = ⊤) : x = y

class Bot (α : Type u_1) : Type u_1

Typeclass for the ⊥ (\bot) notation

• bot : α

  The bot (⊥, \bot) element

theorem Bot.ext {α : Type u_1} {x : Bot α} {y : Bot α} (bot : ⊥ = ⊥) : x = y

@[instance 100]

instance top_nonempty (α : Type u_1) [Top α] : Nonempty α

Equations

• ⋯ = ⋯

@[instance 100]

instance bot_nonempty (α : Type u_1) [Bot α] : Nonempty α

Equations

• ⋯ = ⋯