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Some miscellaneous separated/stable type facts #1030

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merged 5 commits into from
Sep 4, 2023
Merged

Some miscellaneous separated/stable type facts #1030

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d698387
Some miscellaneous separated/stable type facts
dolio Aug 8, 2023
a108c18
Add Separated for Π and ×
dolio Aug 9, 2023
be91b37
Some tweaks in C.R.N.Properties
dolio Sep 2, 2023
1799a5a
Use Discrete→Separated for separatedBool
dolio Sep 2, 2023
58b19e6
Merge branch 'master' into pr-1030
MatthiasHu Sep 4, 2023
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3 changes: 3 additions & 0 deletions Cubical/Data/Bool/Properties.agda
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Expand Up @@ -406,3 +406,6 @@ Iso-⊤⊎⊤-Bool .leftInv (inl tt) = refl
Iso-⊤⊎⊤-Bool .leftInv (inr tt) = refl
Iso-⊤⊎⊤-Bool .rightInv true = refl
Iso-⊤⊎⊤-Bool .rightInv false = refl

separatedBool : Separated Bool
separatedBool = Discrete→Separated _≟_
24 changes: 24 additions & 0 deletions Cubical/Relation/Nullary/Properties.agda
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Expand Up @@ -16,6 +16,7 @@ open import Cubical.Foundations.Equiv
open import Cubical.Functions.Fixpoint

open import Cubical.Data.Empty as ⊥
open import Cubical.Data.Sigma.Base using (_×_)

open import Cubical.Relation.Nullary.Base
open import Cubical.HITs.PropositionalTruncation.Base
Expand All @@ -24,6 +25,7 @@ private
variable
ℓ : Level
A B : Type ℓ
P : A -> Type ℓ

-- Functions with a section preserve discreteness.
sectionDiscrete
Expand All @@ -48,6 +50,17 @@ Stable¬ ¬¬¬a a = ¬¬¬a ¬¬a
where
¬¬a = λ ¬a → ¬a a

StableΠ : (∀ x → Stable (P x)) -> Stable (∀ x → P x)
StableΠ Ps e x = Ps x λ k → e λ f → k (f x)

Stable→ : Stable B → Stable (A → B)
Stable→ Bs = StableΠ (λ _ → Bs)

Stable× : Stable A -> Stable B -> Stable (A × B)
Stable× As Bs e = λ where
.fst → As λ k → e (k ∘ fst)
.snd → Bs λ k → e (k ∘ snd)

fromYes : A → Dec A → A
fromYes _ (yes a) = a
fromYes a (no _) = a
Expand Down Expand Up @@ -171,6 +184,17 @@ Separated→PStable≡ st x y = Stable→PStable (st x y)
Separated→isSet : Separated A → isSet A
Separated→isSet = PStable≡→isSet ∘ Separated→PStable≡

SeparatedΠ : (∀ x → Separated (P x)) -> Separated ((x : A) -> P x)
SeparatedΠ Ps f g e i x = Ps x (f x) (g x) (λ k → e (k ∘ cong (λ f → f x))) i

Separated→ : Separated B -> Separated (A → B)
Separated→ Bs = SeparatedΠ (λ _ → Bs)

Separated× : Separated A -> Separated B -> Separated (A × B)
Separated× As Bs p q e i = λ where
.fst → As (fst p) (fst q) (λ k → e λ r → k (cong fst r)) i
.snd → Bs (snd p) (snd q) (λ k → e λ r → k (cong snd r)) i

-- Proof of Hedberg's theorem: a type with decidable equality is an h-set
Discrete→Separated : Discrete A → Separated A
Discrete→Separated d x y = Dec→Stable (d x y)
Expand Down