{-# OPTIONS --cubical --safe #-}
module Cubical.Foundations.Logic where
import Cubical.Data.Empty as D
import Cubical.Data.Prod as D
import Cubical.Data.Sum as D
import Cubical.Data.Unit as D
open import Cubical.Foundations.Prelude
open import Cubical.HITs.PropositionalTruncation
open import Cubical.Foundations.HLevels using (hProp; ΣProp≡; isPropIsProp; propPi) public
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Function
open import Cubical.Relation.Nullary hiding (¬_)
infix 10 ¬_
infixr 8 _⊔_
infixr 8 _⊔′_
infixr 8 _⊓_
infixr 8 _⊓′_
infixr 6 _⇒_
infixr 4 _⇔_
infix 30 _≡ₚ_
infix 30 _≢ₚ_
infix 2 ∃[]-syntax
infix 2 ∃[∶]-syntax
infix 2 ∀[∶]-syntax
infix 2 ∀[]-syntax
infix 2 ⇒∶_⇐∶_
infix 2 ⇐∶_⇒∶_
private
variable
ℓ ℓ' ℓ'' : Level
P Q R : hProp ℓ
A B C : Type ℓ
[_] : hProp ℓ → Type ℓ
[_] = fst
∥_∥ₚ : Type ℓ → hProp ℓ
∥ A ∥ₚ = ∥ A ∥ , propTruncIsProp
_≡ₚ_ : (x y : A) → hProp _
x ≡ₚ y = ∥ x ≡ y ∥ₚ
hProp≡ : [ P ] ≡ [ Q ] → P ≡ Q
hProp≡ p = ΣProp≡ (\ _ → isPropIsProp) p
_⇒_ : (A : hProp ℓ) → (B : hProp ℓ') → hProp _
A ⇒ B = ([ A ] → [ B ]) , propPi λ _ → B .snd
⇔toPath : [ P ⇒ Q ] → [ Q ⇒ P ] → P ≡ Q
⇔toPath {P = P} {Q = Q} P⇒Q Q⇒P = hProp≡ (isoToPath
(iso P⇒Q Q⇒P (λ b → Q .snd (P⇒Q (Q⇒P b)) b) λ a → P .snd (Q⇒P (P⇒Q a)) a))
pathTo⇒ : P ≡ Q → [ P ⇒ Q ]
pathTo⇒ p x = subst fst p x
pathTo⇐ : P ≡ Q → [ Q ⇒ P ]
pathTo⇐ p x = subst fst (sym p) x
substₚ : {x y : A} (B : A → hProp ℓ) → [ x ≡ₚ y ⇒ B x ⇒ B y ]
substₚ {x = x} {y = y} B = elimPropTrunc (λ _ → propPi λ _ → B y .snd) (subst (fst ∘ B))
⇒∶_⇐∶_ : [ P ⇒ Q ] → [ Q ⇒ P ] → P ≡ Q
⇒∶_⇐∶_ = ⇔toPath
⇐∶_⇒∶_ : [ Q ⇒ P ] → [ P ⇒ Q ] → P ≡ Q
⇐∶ g ⇒∶ f = ⇔toPath f g
⊥ : hProp _
⊥ = D.⊥ , λ ()
⊤ : hProp _
⊤ = D.Unit , (λ _ _ _ → D.tt)
¬_ : hProp ℓ → hProp _
¬ A = ([ A ] → D.⊥) , propPi λ _ → D.isProp⊥
_≢ₚ_ : (x y : A) → hProp _
x ≢ₚ y = ¬ x ≡ₚ y
_⊔′_ : Type ℓ → Type ℓ' → Type _
A ⊔′ B = ∥ A D.⊎ B ∥
_⊔_ : hProp ℓ → hProp ℓ' → hProp _
P ⊔ Q = ∥ [ P ] D.⊎ [ Q ] ∥ₚ
inl : A → A ⊔′ B
inl x = ∣ D.inl x ∣
inr : B → A ⊔′ B
inr x = ∣ D.inr x ∣
⊔-elim : (P : hProp ℓ) (Q : hProp ℓ') (R : [ P ⊔ Q ] → hProp ℓ'')
→ (∀ x → [ R (inl x) ]) → (∀ y → [ R (inr y) ]) → (∀ z → [ R z ])
⊔-elim _ _ R P⇒R Q⇒R = elimPropTrunc (snd ∘ R) (D.elim-⊎ P⇒R Q⇒R)
_⊓′_ : Type ℓ → Type ℓ' → Type _
A ⊓′ B = A D.× B
_⊓_ : hProp ℓ → hProp ℓ' → hProp _
A ⊓ B = [ A ] ⊓′ [ B ] , D.hLevelProd 1 (A .snd) (B .snd)
⊓-intro : (P : hProp ℓ) (Q : [ P ] → hProp ℓ') (R : [ P ] → hProp ℓ'')
→ (∀ a → [ Q a ]) → (∀ a → [ R a ]) → (∀ (a : [ P ]) → [ Q a ⊓ R a ] )
⊓-intro _ _ _ = D.intro-×
_⇔_ : hProp ℓ → hProp ℓ' → hProp _
A ⇔ B = (A ⇒ B) ⊓ (B ⇒ A)
∀[∶]-syntax : (A → hProp ℓ) → hProp _
∀[∶]-syntax {A = A} P = (∀ x → [ P x ]) , propPi (snd ∘ P)
∀[]-syntax : (A → hProp ℓ) → hProp _
∀[]-syntax {A = A} P = (∀ x → [ P x ]) , propPi (snd ∘ P)
syntax ∀[∶]-syntax {A = A} (λ a → P) = ∀[ a ∶ A ] P
syntax ∀[]-syntax (λ a → P) = ∀[ a ] P
∃[]-syntax : (A → hProp ℓ) → hProp _
∃[]-syntax {A = A} P = ∥ Σ A (fst ∘ P) ∥ₚ
∃[∶]-syntax : (A → hProp ℓ) → hProp _
∃[∶]-syntax {A = A} P = ∥ Σ A (fst ∘ P) ∥ₚ
syntax ∃[]-syntax {A = A} (λ x → P) = ∃[ x ∶ A ] P
syntax ∃[∶]-syntax (λ x → P) = ∃[ x ] P
Decₚ : (P : hProp ℓ) → hProp ℓ
Decₚ P = Dec [ P ] , isPropDec (snd P)
∥¬A∥≡¬∥A∥ : (A : Type ℓ) → ∥ (A → D.⊥) ∥ₚ ≡ (¬ ∥ A ∥ₚ)
∥¬A∥≡¬∥A∥ _ =
⇒∶ (λ ¬A A → elimPropTrunc (λ _ → D.isProp⊥)
(elimPropTrunc (λ _ → propPi λ _ → D.isProp⊥) (λ ¬p p → ¬p p) ¬A) A)
⇐∶ λ ¬p → ∣ (λ p → ¬p ∣ p ∣) ∣
⊔-assoc : (P : hProp ℓ) (Q : hProp ℓ') (R : hProp ℓ'')
→ P ⊔ (Q ⊔ R) ≡ (P ⊔ Q) ⊔ R
⊔-assoc P Q R =
⇒∶ ⊔-elim P (Q ⊔ R) (λ _ → (P ⊔ Q) ⊔ R)
(inl ∘ inl)
(⊔-elim Q R (λ _ → (P ⊔ Q) ⊔ R) (inl ∘ inr) inr)
⇐∶ assoc2
where
assoc2 : (A ⊔′ B) ⊔′ C → A ⊔′ (B ⊔′ C)
assoc2 ∣ D.inr a ∣ = ∣ D.inr ∣ D.inr a ∣ ∣
assoc2 ∣ D.inl ∣ D.inr b ∣ ∣ = ∣ D.inr ∣ D.inl b ∣ ∣
assoc2 ∣ D.inl ∣ D.inl c ∣ ∣ = ∣ D.inl c ∣
assoc2 ∣ D.inl (squash x y i) ∣ = propTruncIsProp (assoc2 ∣ D.inl x ∣) (assoc2 ∣ D.inl y ∣) i
assoc2 (squash x y i) = propTruncIsProp (assoc2 x) (assoc2 y) i
⊔-idem : (P : hProp ℓ) → P ⊔ P ≡ P
⊔-idem P =
⇒∶ (⊔-elim P P (\ _ → P) (\ x → x) (\ x → x))
⇐∶ inl
⊔-comm : (P : hProp ℓ) (Q : hProp ℓ') → P ⊔ Q ≡ Q ⊔ P
⊔-comm P Q =
⇒∶ (⊔-elim P Q (\ _ → (Q ⊔ P)) inr inl)
⇐∶ (⊔-elim Q P (\ _ → (P ⊔ Q)) inr inl)
⊔-identityˡ : (P : hProp ℓ) → ⊥ ⊔ P ≡ P
⊔-identityˡ P =
⇒∶ (⊔-elim ⊥ P (λ _ → P) (λ ()) (λ x → x))
⇐∶ inr
⊔-identityʳ : (P : hProp ℓ) → P ⊔ ⊥ ≡ P
⊔-identityʳ P = ⇔toPath (⊔-elim P ⊥ (λ _ → P) (λ x → x) λ ()) inl
⊓-assoc : (P : hProp ℓ) (Q : hProp ℓ') (R : hProp ℓ'')
→ P ⊓ Q ⊓ R ≡ (P ⊓ Q) ⊓ R
⊓-assoc _ _ _ =
⇒∶ (λ {(x D., (y D., z)) → (x D., y) D., z})
⇐∶ (λ {((x D., y) D., z) → x D., (y D., z) })
⊓-comm : (P : hProp ℓ) (Q : hProp ℓ') → P ⊓ Q ≡ Q ⊓ P
⊓-comm _ _ = ⇔toPath D.swap D.swap
⊓-idem : (P : hProp ℓ) → P ⊓ P ≡ P
⊓-idem _ = ⇔toPath D.proj₁ (λ x → x D., x)
⊓-identityˡ : (P : hProp ℓ) → ⊤ ⊓ P ≡ P
⊓-identityˡ _ = ⇔toPath D.proj₂ λ x → D.tt D., x
⊓-identityʳ : (P : hProp ℓ) → P ⊓ ⊤ ≡ P
⊓-identityʳ _ = ⇔toPath D.proj₁ λ x → x D., D.tt
⇒-⊓-distrib : (P : hProp ℓ) (Q : hProp ℓ')(R : hProp ℓ'')
→ P ⇒ (Q ⊓ R) ≡ (P ⇒ Q) ⊓ (P ⇒ R)
⇒-⊓-distrib _ _ _ =
⇒∶ (λ f → (D.proj₁ ∘ f) D., (D.proj₂ ∘ f))
⇐∶ (λ { (f D., g) x → f x D., g x})
⊓-⊔-distribˡ : (P : hProp ℓ) (Q : hProp ℓ')(R : hProp ℓ'')
→ P ⊓ (Q ⊔ R) ≡ (P ⊓ Q) ⊔ (P ⊓ R)
⊓-⊔-distribˡ P Q R =
⇒∶ (λ { (x D., a) → ⊔-elim Q R (λ _ → (P ⊓ Q) ⊔ (P ⊓ R))
(λ y → ∣ D.inl (x D., y) ∣ )
(λ z → ∣ D.inr (x D., z) ∣ ) a })
⇐∶ ⊔-elim (P ⊓ Q) (P ⊓ R) (λ _ → P ⊓ Q ⊔ R)
(λ y → D.proj₁ y D., inl (D.proj₂ y))
(λ z → D.proj₁ z D., inr (D.proj₂ z))
⊔-⊓-distribˡ : (P : hProp ℓ) (Q : hProp ℓ')(R : hProp ℓ'')
→ P ⊔ (Q ⊓ R) ≡ (P ⊔ Q) ⊓ (P ⊔ R)
⊔-⊓-distribˡ P Q R =
⇒∶ ⊔-elim P (Q ⊓ R) (λ _ → (P ⊔ Q) ⊓ (P ⊔ R) )
(D.intro-× inl inl) (D.map-× inr inr)
⇐∶ (λ { (x D., y) → ⊔-elim P R (λ _ → P ⊔ Q ⊓ R) inl
(λ z → ⊔-elim P Q (λ _ → P ⊔ Q ⊓ R) inl (λ y → inr (y D., z)) x) y })
⊓-∀-distrib : (P : A → hProp ℓ) (Q : A → hProp ℓ')
→ (∀[ a ∶ A ] P a) ⊓ (∀[ a ∶ A ] Q a) ≡ (∀[ a ∶ A ] (P a ⊓ Q a))
⊓-∀-distrib P Q =
⇒∶ (λ {(p D., q) a → p a D., q a})
⇐∶ λ pq → (D.proj₁ ∘ pq ) D., (D.proj₂ ∘ pq)