De Morgan types

Content created by Fredrik Bakke.

Created on 2025-01-07.
Last modified on 2025-08-14.

module logic.de-morgan-types where
Imports 
open import foundation.cartesian-product-types
open import foundation.contractible-types
open import foundation.coproduct-types
open import foundation.decidable-types
open import foundation.dependent-pair-types
open import foundation.disjunction
open import foundation.double-negation
open import foundation.empty-types
open import foundation.function-types
open import foundation.logical-equivalences
open import foundation.negation
open import foundation.propositional-truncations
open import foundation.retracts-of-types
open import foundation.truncation-levels
open import foundation.truncations
open import foundation.unit-type
open import foundation.universe-levels

open import foundation-core.decidable-propositions
open import foundation-core.equivalences
open import foundation-core.propositions

open import logic.irrefutable-types
open import logic.propositionally-decidable-types

Idea

In classical logic, i.e., logic where we assume the law of excluded middle, the De Morgan laws refers to the pair of logical equivalences

  ¬ (P ∨ Q) ⇔ (¬ P) ∧ (¬ Q)
  ¬ (P ∧ Q) ⇔ (¬ P) ∨ (¬ Q).

Out of these in total four logical implications, all but one are validated in constructive mathematics. The odd one out is

  ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q).

Indeed, this would state that we could constructively deduce from a proof that not both of P and Q are true, which of P and Q that is false. This logical law is what we refer to as De Morgan’s Law. If a type P is such that for every other type Q, the De Morgan implication

  ¬ (P ∧ Q) ⇒ (¬ P) ∨ (¬ Q)

holds, we say P is De Morgan.

Equivalently, a type is De Morgan iff its negation is decidable. Since this is a small condition, it is frequently more convenient to use and is what we take as the main definition.

Definition

The small condition of being a De Morgan type

I.e., types whose negation is decidable.

module _
  {l : Level} (A : UU l)
  where

  is-de-morgan : UU l
  is-de-morgan = is-decidable (¬ A)

  is-prop-is-de-morgan : is-prop is-de-morgan
  is-prop-is-de-morgan = is-prop-is-decidable is-prop-neg

  is-de-morgan-Prop : Prop l
  is-de-morgan-Prop = is-decidable-Prop (neg-type-Prop A)

The subuniverse of De Morgan types

We use the decidability of the negation condition to define the subuniverse of De Morgan types.

De-Morgan-Type : (l : Level) → UU (lsuc l)
De-Morgan-Type l = Σ (UU l) (is-de-morgan)

module _
  {l : Level} (A : De-Morgan-Type l)
  where

  type-De-Morgan-Type : UU l
  type-De-Morgan-Type = pr1 A

  is-de-morgan-type-De-Morgan-Type : is-de-morgan type-De-Morgan-Type
  is-de-morgan-type-De-Morgan-Type = pr2 A

Types that satisfy De Morgan’s law

satisfies-de-morgans-law-type-Level :
  {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2)
satisfies-de-morgans-law-type-Level l2 A =
  (B : UU l2) → ¬ (A × B) → disjunction-type (¬ A) (¬ B)

satisfies-de-morgans-law-type : {l1 : Level} (A : UU l1) → UUω
satisfies-de-morgans-law-type A =
  {l2 : Level} (B : UU l2) → ¬ (A × B) → disjunction-type (¬ A) (¬ B)

is-prop-satisfies-de-morgans-law-type-Level :
  {l1 l2 : Level} {A : UU l1} →
  is-prop (satisfies-de-morgans-law-type-Level l2 A)
is-prop-satisfies-de-morgans-law-type-Level {A = A} =
  is-prop-Π (λ B → is-prop-Π (λ p → is-prop-disjunction-type (¬ A) (¬ B)))

satisfies-de-morgans-law-type-Prop :
  {l1 : Level} (l2 : Level) (A : UU l1) → Prop (l1 ⊔ lsuc l2)
satisfies-de-morgans-law-type-Prop l2 A =
  ( satisfies-de-morgans-law-type-Level l2 A ,
    is-prop-satisfies-de-morgans-law-type-Level)

satisfies-de-morgans-law-type-Level' :
  {l1 : Level} (l2 : Level) (A : UU l1) → UU (l1 ⊔ lsuc l2)
satisfies-de-morgans-law-type-Level' l2 A =
  (B : UU l2) → ¬ (B × A) → disjunction-type (¬ B) (¬ A)

satisfies-de-morgans-law-type' : {l1 : Level} (A : UU l1) → UUω
satisfies-de-morgans-law-type' A =
  {l2 : Level} (B : UU l2) → ¬ (B × A) → disjunction-type (¬ B) (¬ A)

is-prop-satisfies-de-morgans-law-type-Level' :
  {l1 l2 : Level} {A : UU l1} →
  is-prop (satisfies-de-morgans-law-type-Level' l2 A)
is-prop-satisfies-de-morgans-law-type-Level' {A = A} =
  is-prop-Π (λ B → is-prop-Π (λ p → is-prop-disjunction-type (¬ B) (¬ A)))

satisfies-de-morgans-law-type-Prop' :
  {l1 : Level} (l2 : Level) (A : UU l1) → Prop (l1 ⊔ lsuc l2)
satisfies-de-morgans-law-type-Prop' l2 A =
  ( satisfies-de-morgans-law-type-Level' l2 A ,
    is-prop-satisfies-de-morgans-law-type-Level')

Properties

If a type satisfies De Morgan’s law then its negation is decidable

Indeed, one need only check that A and ¬ A satisfy De Morgan’s law, as then the hypothesis of the implication

   ¬ (A ∧ ¬ A) ⇒ ¬ A ∨ ¬¬ A

is true.

module _
  {l : Level} (A : UU l)
  where

  is-de-morgan-satisfies-de-morgans-law' :
    ({l' : Level} (B : UU l') → ¬ (A × B) → ¬ A + ¬ B) → is-de-morgan A
  is-de-morgan-satisfies-de-morgans-law' H = H (¬ A) (λ f → pr2 f (pr1 f))

  is-merely-decidable-neg-satisfies-de-morgans-law :
    satisfies-de-morgans-law-type A → is-merely-decidable (¬ A)
  is-merely-decidable-neg-satisfies-de-morgans-law H =
    H (¬ A) (λ f → pr2 f (pr1 f))

  is-de-morgan-satisfies-de-morgans-law :
    satisfies-de-morgans-law-type A → is-de-morgan A
  is-de-morgan-satisfies-de-morgans-law H =
    rec-trunc-Prop
      ( is-decidable-Prop (neg-type-Prop A))
      ( id)
      ( H (¬ A) (λ f → pr2 f (pr1 f)))

If the negation of a type is decidable then it satisfies De Morgan’s law

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  satisfies-de-morgans-law-is-de-morgan-left :
    is-de-morgan A → ¬ (A × B) → ¬ A + ¬ B
  satisfies-de-morgans-law-is-de-morgan-left (inl na) f =
    inl na
  satisfies-de-morgans-law-is-de-morgan-left (inr nna) f =
    inr (λ y → nna (λ x → f (x , y)))

  satisfies-de-morgans-law-is-de-morgan-right :
    is-de-morgan B → ¬ (A × B) → ¬ A + ¬ B
  satisfies-de-morgans-law-is-de-morgan-right (inl nb) f =
    inr nb
  satisfies-de-morgans-law-is-de-morgan-right (inr nnb) f =
    inl (λ x → nnb (λ y → f (x , y)))

satisfies-de-morgans-law-is-de-morgan :
  {l : Level} {A : UU l} → is-de-morgan A → satisfies-de-morgans-law-type A
satisfies-de-morgans-law-is-de-morgan x B q =
  unit-trunc-Prop (satisfies-de-morgans-law-is-de-morgan-left x q)

satisfies-de-morgans-law-is-de-morgan' :
  {l : Level} {A : UU l} → is-de-morgan A → satisfies-de-morgans-law-type' A
satisfies-de-morgans-law-is-de-morgan' x B q =
  unit-trunc-Prop (satisfies-de-morgans-law-is-de-morgan-right x q)

module _
  {l : Level} (A : De-Morgan-Type l)
  where

  satisfies-de-morgans-law-type-De-Morgan-Type :
    satisfies-de-morgans-law-type (type-De-Morgan-Type A)
  satisfies-de-morgans-law-type-De-Morgan-Type =
    satisfies-de-morgans-law-is-de-morgan (is-de-morgan-type-De-Morgan-Type A)

  satisfies-de-morgans-law-type-De-Morgan-Type' :
    satisfies-de-morgans-law-type' (type-De-Morgan-Type A)
  satisfies-de-morgans-law-type-De-Morgan-Type' =
    satisfies-de-morgans-law-is-de-morgan' (is-de-morgan-type-De-Morgan-Type A)

It is irrefutable that a type is De Morgan

is-irrefutable-is-de-morgan :
  {l : Level} {A : UU l} → is-irrefutable (is-de-morgan A)
is-irrefutable-is-de-morgan = is-irrefutable-is-decidable

Decidable types are De Morgan

is-de-morgan-is-decidable :
  {l : Level} {A : UU l} → is-decidable A → is-de-morgan A
is-de-morgan-is-decidable (inl x) = inr (intro-double-negation x)
is-de-morgan-is-decidable (inr x) = inl x

satisfies-de-morgans-law-is-decidable :
  {l : Level} {A : UU l} → is-decidable A → satisfies-de-morgans-law-type A
satisfies-de-morgans-law-is-decidable H =
  satisfies-de-morgans-law-is-de-morgan (is-de-morgan-is-decidable H)

satisfies-de-morgans-law-is-decidable' :
  {l : Level} {A : UU l} → is-decidable A → satisfies-de-morgans-law-type' A
satisfies-de-morgans-law-is-decidable' H =
  satisfies-de-morgans-law-is-de-morgan' (is-de-morgan-is-decidable H)

Irrefutable types are De Morgan

is-de-morgan-is-irrefutable :
  {l : Level} {A : UU l} → ¬¬ A → is-de-morgan A
is-de-morgan-is-irrefutable = inr

Contractible types are De Morgan

is-de-morgan-is-contr :
  {l : Level} {A : UU l} → is-contr A → is-de-morgan A
is-de-morgan-is-contr H =
  is-de-morgan-is-irrefutable (intro-double-negation (center H))

is-de-morgan-unit : is-de-morgan unit
is-de-morgan-unit = is-de-morgan-is-contr is-contr-unit

Empty types are De Morgan

is-de-morgan-is-empty : {l : Level} {A : UU l} → is-empty A → is-de-morgan A
is-de-morgan-is-empty = inl

is-de-morgan-empty : is-de-morgan empty
is-de-morgan-empty = is-de-morgan-is-empty id

De Morgan types are closed under logical equivalences

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  is-de-morgan-iff :
    (A → B) → (B → A) → is-de-morgan A → is-de-morgan B
  is-de-morgan-iff f g = is-decidable-iff (map-neg g) (map-neg f)

  is-de-morgan-iff' :
    (A ↔ B) → is-de-morgan A → is-de-morgan B
  is-de-morgan-iff' (f , g) = is-de-morgan-iff f g

  satisfies-de-morgans-law-iff :
    (A → B) → (B → A) →
    satisfies-de-morgans-law-type A → satisfies-de-morgans-law-type B
  satisfies-de-morgans-law-iff f g a =
    satisfies-de-morgans-law-is-de-morgan
      ( is-de-morgan-iff f g (is-de-morgan-satisfies-de-morgans-law A a))

  satisfies-de-morgans-law-iff' :
    (A ↔ B) → satisfies-de-morgans-law-type A → satisfies-de-morgans-law-type B
  satisfies-de-morgans-law-iff' (f , g) = satisfies-de-morgans-law-iff f g

De Morgan types are closed under retracts

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (R : B retract-of A)
  where

  is-de-morgan-retract-of : is-de-morgan A → is-de-morgan B
  is-de-morgan-retract-of = is-de-morgan-iff' (iff-retract' R)

  is-de-morgan-retract-of' : is-de-morgan B → is-de-morgan A
  is-de-morgan-retract-of' = is-de-morgan-iff' (iff-retract R)

De Morgan types are closed under equivalences

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2} (e : A ≃ B)
  where

  is-de-morgan-equiv' : is-de-morgan A → is-de-morgan B
  is-de-morgan-equiv' = is-de-morgan-iff' (iff-equiv e)

  is-de-morgan-equiv : is-de-morgan B → is-de-morgan A
  is-de-morgan-equiv = is-de-morgan-iff' (iff-equiv' e)

Equivalent types have equivalent De Morgan predicates

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  iff-is-de-morgan : A ↔ B → is-de-morgan A ↔ is-de-morgan B
  iff-is-de-morgan e = iff-is-decidable (iff-neg e)

  equiv-iff-is-de-morgan : A ↔ B → is-de-morgan A ≃ is-de-morgan B
  equiv-iff-is-de-morgan e = equiv-is-decidable (equiv-iff-neg e)

  equiv-is-de-morgan : A ≃ B → is-de-morgan A ≃ is-de-morgan B
  equiv-is-de-morgan e = equiv-iff-is-de-morgan (iff-equiv e)

The truncation of a De Morgan type is De Morgan

module _
  {l1 : Level} {A : UU l1}
  where

  is-de-morgan-trunc : {k : 𝕋} → is-de-morgan A → is-de-morgan (type-trunc k A)
  is-de-morgan-trunc {neg-two-𝕋} a =
    is-de-morgan-is-contr is-trunc-type-trunc
  is-de-morgan-trunc {succ-𝕋 k} (inl na) =
    inl (map-universal-property-trunc (empty-Truncated-Type k) na)
  is-de-morgan-trunc {succ-𝕋 k} (inr nna) =
    inr (λ nn|a| → nna (λ a → nn|a| (unit-trunc a)))

If the truncation of a type is De Morgan then the type is De Morgan

module _
  {l1 : Level} {A : UU l1}
  where

  equiv-is-de-morgan-trunc :
    {k : 𝕋} → is-de-morgan (type-trunc (succ-𝕋 k) A) ≃ is-de-morgan A
  equiv-is-de-morgan-trunc {k} =
    equiv-is-decidable
      ( map-neg unit-trunc , is-truncation-trunc (empty-Truncated-Type k))

  is-de-morgan-is-de-morgan-trunc :
    {k : 𝕋} → is-de-morgan (type-trunc (succ-𝕋 k) A) → is-de-morgan A
  is-de-morgan-is-de-morgan-trunc = map-equiv equiv-is-de-morgan-trunc

Products of De Morgan types are De Morgan

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  is-de-morgan-product : is-de-morgan A → is-de-morgan B → is-de-morgan (A × B)
  is-de-morgan-product (inl na) b =
    inl (λ ab → na (pr1 ab))
  is-de-morgan-product (inr nna) (inl nb) =
    inl (λ ab → nb (pr2 ab))
  is-de-morgan-product (inr nna) (inr nnb) =
    inr (is-irrefutable-product nna nnb)

Coproducts of De Morgan types are De Morgan

module _
  {l1 l2 : Level} {A : UU l1} {B : UU l2}
  where

  is-de-morgan-coproduct :
    is-de-morgan A → is-de-morgan B → is-de-morgan (A + B)
  is-de-morgan-coproduct (inl na) (inl nb) =
    inl (rec-coproduct na nb)
  is-de-morgan-coproduct (inl na) (inr nnb) =
    inr (λ nab → nnb (λ nb → nab (inr nb)))
  is-de-morgan-coproduct (inr nna) _ =
    inr (λ nab → nna (λ na → nab (inl na)))

  is-de-morgan-disjunction :
    is-de-morgan A → is-de-morgan B → is-de-morgan (disjunction-type A B)
  is-de-morgan-disjunction a b = is-de-morgan-trunc (is-de-morgan-coproduct a b)

The negation of a De Morgan type is De Morgan

is-de-morgan-neg : {l : Level} {A : UU l} → is-de-morgan A → is-de-morgan (¬ A)
is-de-morgan-neg = is-decidable-neg

The identity types of De Morgan types are not generally De Morgan

Consider any type A, then its suspension ΣA is De Morgan since it is inhabited. However, its identity type N ＝ S is equivalent to A, so cannot be De Morgan unless A is.

This remains to be formalized.

Recent changes