Negation (and emptiness).

\begin{code}

{-# OPTIONS --safe --without-K #-}

module MLTT.Negation where

open import MLTT.Universes
open import MLTT.Empty
open import MLTT.Id
open import MLTT.Pi
open import MLTT.Plus
open import MLTT.Sigma

private
 _↔_ : 𝓤 ̇  𝓥 ̇  𝓤  𝓥 ̇
 A  B = (A  B) × (B  A)

¬_ : 𝓤 ̇  𝓤 ̇
¬ A = A  𝟘 {𝓤₀}

\end{code}

Notice that decidability is not a univalent proposition in general,
but nevertheless we use "is" in our chosen terminology, against a
convention adopted in some quarters that says that "is" should be used
only for concepts that are propositions.

\begin{code}

is-decidable : 𝓤 ̇  𝓤 ̇
is-decidable A = A + ¬ A

_≠_ : {X : 𝓤 ̇ }  (x y : X)  𝓤 ̇
x  y = ¬ (x  y)

has-two-distinct-points : 𝓤 ̇  𝓤 ̇
has-two-distinct-points X = Σ (x , y)  X × X , (x  y)

has-three-distinct-points : 𝓤 ̇  𝓤 ̇
has-three-distinct-points X = Σ (x , y , z)  X × X × X , (x  y) × (y  z) × (z  x)

≠-sym : {X : 𝓤 ̇ }  {x y : X}  x  y  y  x
≠-sym u r = u (r ⁻¹)

is-empty : 𝓤 ̇  𝓤 ̇
is-empty = ¬_

¬¬_ : 𝓤 ̇  𝓤 ̇
¬¬ A = ¬ (¬ A)

¬¬¬_ : 𝓤 ̇  𝓤 ̇
¬¬¬ A = ¬ (¬¬ A)

is-nonempty : 𝓤 ̇  𝓤 ̇
is-nonempty = ¬¬_

dual : {X : 𝓤 ̇ } {Y : 𝓥 ̇ } (R : 𝓦 ̇ )  (X  Y)  (Y  R)  (X  R)
dual R f p = p  f

contrapositive : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  (A  B)  ¬ B  ¬ A
contrapositive = dual _

double-contrapositive : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  (A  B)  ¬¬ A  ¬¬ B
double-contrapositive = contrapositive  contrapositive

¬¬-functor : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  (A  B)  ¬¬ A  ¬¬ B
¬¬-functor = double-contrapositive

¬¬-kleisli : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  (A  ¬¬ B)  ¬¬ A  ¬¬ B
¬¬-kleisli f ϕ h = ϕ  a  f a h)

¬¬-intro : {A : 𝓤 ̇ }  A  ¬¬ A
¬¬-intro x u = u x

three-negations-imply-one : {A : 𝓤 ̇ }  ¬¬¬ A  ¬ A
three-negations-imply-one = contrapositive ¬¬-intro

dne' : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  (A  B)  (¬¬ B  B)  ¬¬ A  B
dne' f h ϕ = h  g  ϕ  a  g (f a)))

dne : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  (A  ¬ B)  ¬¬ A  ¬ B
dne f ϕ b = ϕ  a  f a b)

double-negation-unshift : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }  ¬¬ ((x : X)  A x)  (x : X)  ¬¬ (A x)
double-negation-unshift f x g = f  h  g (h x))

dnu : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  ¬¬ (A × B)  ¬¬ A × ¬¬ B
dnu φ = (¬¬-functor pr₁ φ) , (¬¬-functor pr₂ φ)

und : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  ¬¬ A × ¬¬ B  ¬¬ (A × B)
und (φ , γ) w = γ  y  φ  x  w (x , y)))

¬¬-stable : 𝓤 ̇  𝓤 ̇
¬¬-stable A = ¬¬ A  A

¬-is-¬¬-stable : {A : 𝓤 ̇ }  ¬¬-stable (¬ A)
¬-is-¬¬-stable = three-negations-imply-one

Π-is-¬¬-stable : {A : 𝓤 ̇ } {B : A  𝓥 ̇ }
                ((a : A)  ¬¬-stable (B a))
                ¬¬-stable (Π B)
Π-is-¬¬-stable f ϕ a = f a  v  ϕ  g  v (g a)))

→-is-¬¬-stable : {A : 𝓤 ̇ } {B : 𝓥 ̇ }
                ¬¬-stable B
                ¬¬-stable (A  B)
→-is-¬¬-stable f = Π-is-¬¬-stable  _  f)

×-is-¬¬-stable : {A : 𝓤 ̇ } {B : 𝓥 ̇ }
                ¬¬-stable A
                ¬¬-stable B
                ¬¬-stable (A × B)
×-is-¬¬-stable f g ϕ = f  v  ϕ  (a , b)  v a)) ,
                       g  v  ϕ  (a , b)  v b))

negation-of-implication :  {A : 𝓤 ̇ } {B : 𝓥 ̇ }
                         ¬ (A  B)
                         ¬¬ A × ¬ B
negation-of-implication u =  v  u  a  𝟘-elim (v a))) ,
                             b  u  a  b))

negation-of-implication-converse :  {A : 𝓤 ̇ } {B : 𝓥 ̇ }
                                  ¬¬ A × ¬ B
                                  ¬ (A  B)
negation-of-implication-converse (u , v) f = u  a  v (f a))

Double-negation-of-implication← : {A : 𝓤 ̇ } {B : 𝓥 ̇ }
                                  {R : 𝓦 ̇ } {S : 𝓣 ̇ } {T : 𝓣' ̇ }
                                 (((A  B)  T)  S)
                                 (((A  S)  R) × (B  T))  R
Double-negation-of-implication← f g = pr₁ g  a  f  h  pr₂ g (h a)))

Double-negation-of-implication→ : {A : 𝓤 ̇ } {B : 𝓥 ̇ }
                                  (R : 𝓦 ̇ ) {S : 𝓣 ̇ } {T : 𝓣' ̇ } {U : 𝓣' ̇ }
                                 (S  B)
                                 ((((A  S)  T) × (B  T))  U)
                                 ((A  B)  T)  U
Double-negation-of-implication→ R k f g = f ((λ h  g  a  k (h a))) ,
                                              b  g  a  b)))

double-negation-of-implication← : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  ¬¬ (A  B)  ¬ (¬¬ A × ¬ B)
double-negation-of-implication← = Double-negation-of-implication←

double-negation-of-implication→ : {A : 𝓤 ̇ } {B : 𝓥 ̇ }  ¬ (¬¬ A × ¬ B)  ¬¬ (A  B)
double-negation-of-implication→ f g = Double-negation-of-implication→ (𝟘 {𝓤₀}) 𝟘-elim f g

not-equivalent-to-own-negation' : {A : 𝓤 ̇ } {R : 𝓥 ̇ }  (A  (A  R))  R
not-equivalent-to-own-negation' (f , g) = f (g  a  f a a)) (g  a  f a a))

not-equivalent-to-own-negation : {A : 𝓤 ̇ }  ¬ (A  ¬ A)
not-equivalent-to-own-negation = not-equivalent-to-own-negation'

not-Σ-implies-Π-not : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }
                     ¬ (Σ x  X , A x)
                     (x : X)  ¬ (A x)
not-Σ-implies-Π-not = curry

Π-not-implies-not-Σ : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }
                     ((x : X)  ¬ (A x))
                     ¬ (Σ x  X , A x)
Π-not-implies-not-Σ = uncurry

Π-implies-not-Σ-not : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }
                     ((x : X)  A x)
                     ¬ (Σ x  X , ¬ (A x))
Π-implies-not-Σ-not f (x , ν) = ν (f x)

not-Π-not-not-implies-not-not-Σ-not : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }
                                     ¬ ((x : X)  ¬¬ (A x))
                                     ¬¬ (Σ x  X , ¬ (A x))
not-Π-not-not-implies-not-not-Σ-not = contrapositive not-Σ-implies-Π-not

not-Π-implies-not-not-Σ : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }
                         ((x : X)  ¬¬-stable (A x))
                         ¬ ((x : X)  A x)
                         ¬¬ (Σ x  X , ¬ (A x))
not-Π-implies-not-not-Σ f g h = g  x  f x  u  h (x , u)))

\end{code}

Notation to try to make proofs readable:

\begin{code}

contradiction : 𝓤₀ ̇
contradiction = 𝟘

have_which-is-impossible-by_ : {A : 𝓤 ̇ } {B : 𝓦 ̇ }
                              A  (A  𝟘 {𝓤₀})  B
have a which-is-impossible-by ν = 𝟘-elim (ν a)


have_which-contradicts_ : {A : 𝓤 ̇ } {B : 𝓦 ̇ }
                         (A  𝟘 {𝓤₀})  A  B
have ν which-contradicts a = 𝟘-elim (ν a)

\end{code}

Fixities:

\begin{code}

infix  50 ¬_
infix  50 ¬¬_
infix  50 ¬¬¬_
infix  0 _≠_

\end{code}