Move lemmas about equivalences to that module
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@ -42,91 +42,16 @@ module _ (ℓ : Level) where
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IsPreCategory.isIdentity isPreCat {A} {B} = isIdentity {A} {B}
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IsPreCategory.arrowsAreSets isPreCat {A} {B} = arrowsAreSets {A} {B}
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open IsPreCategory isPreCat hiding (_<<<_)
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isIso = TypeIsomorphism
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module _ {hA hB : hSet ℓ} where
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open Σ hA renaming (fst to A ; snd to sA)
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open Σ hB renaming (fst to B ; snd to sB)
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lem1 : (f : A → B) → isSet A → isSet B → isProp (isIso f)
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lem1 f sA sB = res
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where
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module _ (x y : isIso f) where
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module x = Σ x renaming (fst to inverse ; snd to areInverses)
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module y = Σ y renaming (fst to inverse ; snd to areInverses)
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-- I had a lot of difficulty using the corresponding proof where
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-- AreInverses is defined. This is sadly a bit anti-modular. The
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-- reason for my troubles is probably related to the type of objects
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-- being hSet's rather than sets.
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p : ∀ {f} g → isProp (AreInverses {A = A} {B} f g)
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p {f} g xx yy i = ve-re , re-ve
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where
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ve-re : g ∘ f ≡ idFun _
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ve-re = arrowsAreSets {A = hA} {B = hA} _ _ (fst xx) (fst yy) i
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re-ve : f ∘ g ≡ idFun _
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re-ve = arrowsAreSets {A = hB} {B = hB} _ _ (snd xx) (snd yy) i
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1eq : x.inverse ≡ y.inverse
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1eq = begin
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x.inverse ≡⟨⟩
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x.inverse ∘ idFun _ ≡⟨ cong (λ φ → x.inverse ∘ φ) (sym (snd y.areInverses)) ⟩
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x.inverse ∘ (f ∘ y.inverse) ≡⟨⟩
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(x.inverse ∘ f) ∘ y.inverse ≡⟨ cong (λ φ → φ ∘ y.inverse) (fst x.areInverses) ⟩
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idFun _ ∘ y.inverse ≡⟨⟩
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y.inverse ∎
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2eq : (λ i → AreInverses f (1eq i)) [ x.areInverses ≡ y.areInverses ]
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2eq = lemPropF p 1eq
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res : x ≡ y
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res i = 1eq i , 2eq i
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module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
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lem2 : ((x : A) → isProp (P x)) → (p q : Σ A P)
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→ (p ≡ q) ≃ (fst p ≡ fst q)
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lem2 pA p q = fromIsomorphism _ _ iso
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where
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f : ∀ {p q} → p ≡ q → fst p ≡ fst q
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f e i = fst (e i)
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g : ∀ {p q} → fst p ≡ fst q → p ≡ q
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g {p} {q} = lemSig pA p q
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ve-re : (e : p ≡ q) → (g ∘ f) e ≡ e
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ve-re = pathJ (\ q (e : p ≡ q) → (g ∘ f) e ≡ e)
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(\ i j → p .fst , propSet (pA (p .fst)) (p .snd) (p .snd) (λ i → (g {p} {p} ∘ f) (λ i₁ → p) i .snd) (λ i → p .snd) i j ) q
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re-ve : (e : fst p ≡ fst q) → (f {p} {q} ∘ g {p} {q}) e ≡ e
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re-ve e = refl
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inv : AreInverses (f {p} {q}) (g {p} {q})
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inv = funExt ve-re , funExt re-ve
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iso : (p ≡ q) ≅ (fst p ≡ fst q)
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iso = f , g , inv
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module _ {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
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lem4 : isSet A → isSet B → (f : A → B)
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→ isEquiv A B f ≃ isIso f
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lem4 sA sB f =
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let
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obv : isEquiv A B f → isIso f
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obv = toIso A B
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inv : isIso f → isEquiv A B f
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inv = fromIso A B
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re-ve : (x : isEquiv A B f) → (inv ∘ obv) x ≡ x
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re-ve = inverse-from-to-iso A B
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ve-re : (x : isIso f) → (obv ∘ inv) x ≡ x
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ve-re = inverse-to-from-iso A B sA sB
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iso : isEquiv A B f ≅ isIso f
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iso = obv , inv , funExt re-ve , funExt ve-re
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in fromIsomorphism _ _ iso
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open IsPreCategory isPreCat
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module _ {hA hB : Object} where
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open Σ hA renaming (fst to A ; snd to sA)
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open Σ hB renaming (fst to B ; snd to sB)
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-- lem3 and the equivalence from lem4
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step0 : Σ (A → B) (isEquiv A B) ≃ Σ (A → B) isIso
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step0 = equivSig (lem4 sA sB)
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-- lem2 with propIsSet
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step2 : (hA ≡ hB) ≃ (A ≡ B)
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step2 = lem2 (λ A → isSetIsProp) hA hB
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univ≃ : (hA ≡ hB) ≃ (hA ≊ hB)
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univ≃ = step2 ⊙ univalence ⊙ step0
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univ≃
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= equivSigProp (λ A → isSetIsProp)
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⊙ univalence
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⊙ equivSig {P = isEquiv A B} {Q = TypeIsomorphism} (equiv≃iso sA sB)
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univalent : Univalent
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univalent = univalenceFrom≃ univ≃
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@ -119,7 +119,7 @@ record RawCategory (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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--
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-- [HoTT §9.1.4]
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idToIso : (A B : Object) → A ≡ B → A ≊ B
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idToIso A B eq = transp (\ i → A ≊ eq i) (idIso A)
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idToIso A B eq = subst eq (idIso A)
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Univalent : Set (ℓa ⊔ ℓb)
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Univalent = {A B : Object} → isEquiv (A ≡ B) (A ≊ B) (idToIso A B)
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@ -348,7 +348,7 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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coe refl f ≡⟨ id-coe ⟩
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f ≡⟨ sym rightIdentity ⟩
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f <<< identity ≡⟨ cong (f <<<_) (sym subst-neutral) ⟩
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f <<< _ ∎) a' p
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f <<< _ ≡⟨ {!!} ⟩ _ ∎) a' p
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module _ {b' : Object} (p : b ≡ b') where
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private
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@ -431,28 +431,17 @@ module _ {ℓa ℓb : Level} (ℂ : RawCategory ℓa ℓb) where
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coe-dom : {f : Arrow A X} → coe p-dom f ≡ f <<< ι~
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coe-dom {f} = begin
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coe p-dom f
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≡⟨ 9-1-9 p refl f ⟩
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fst (idToIso _ _ refl) <<< f <<< fst (snd (idToIso _ _ p))
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≡⟨ cong (λ φ → φ <<< f <<< fst (snd (idToIso _ _ p))) subst-neutral ⟩
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identity <<< f <<< fst (snd (idToIso _ _ p))
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≡⟨ cong (λ φ → identity <<< f <<< φ) (cong (λ x → (fst (snd x))) lem) ⟩
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identity <<< f <<< ι~
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≡⟨ cong (_<<< ι~) leftIdentity ⟩
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coe p-dom f ≡⟨ 9-1-9-left f p ⟩
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f <<< fst (snd (idToIso _ _ (isoToId iso))) ≡⟨⟩
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f <<< fst (snd (idToIso _ _ p)) ≡⟨ cong (f <<<_) (cong (fst ∘ snd) lem) ⟩
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f <<< ι~ ∎
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coe-cod : {f : Arrow X A} → coe p-cod f ≡ ι <<< f
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coe-cod {f} = begin
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coe p-cod f
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≡⟨ 9-1-9 refl p f ⟩
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fst (idToIso _ _ p) <<< f <<< fst (snd (idToIso _ _ refl))
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≡⟨ cong (λ φ → fst (idToIso _ _ p) <<< f <<< φ) subst-neutral ⟩
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fst (idToIso _ _ p) <<< f <<< identity
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≡⟨ cong (λ φ → φ <<< f <<< identity) (cong fst lem) ⟩
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ι <<< f <<< identity
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≡⟨ sym isAssociative ⟩
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ι <<< (f <<< identity)
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≡⟨ cong (ι <<<_) rightIdentity ⟩
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≡⟨ 9-1-9-right f p ⟩
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fst (idToIso _ _ p) <<< f
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≡⟨ cong (λ φ → φ <<< f) (cong fst lem) ⟩
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ι <<< f ∎
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module _ {f : Arrow A X} {g : Arrow B X} (q : PathP (λ i → p-dom i) f g) where
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@ -7,7 +7,6 @@ open import Cat.Equivalence
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open import Cat.Category
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module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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open Category ℂ
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module _ (A B : Object) where
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@ -18,8 +17,6 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
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fst : ℂ [ object , A ]
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snd : ℂ [ object , B ]
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-- FIXME Not sure this is actually a proposition - so this name is
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-- misleading.
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record IsProduct (raw : RawProduct) : Set (ℓa ⊔ ℓb) where
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open RawProduct raw public
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field
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@ -7,7 +7,10 @@ open import Cubical.PathPrelude hiding (inverse)
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open import Cubical.PathPrelude using (isEquiv ; isContr ; fiber) public
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open import Cubical.GradLemma hiding (isoToPath)
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open import Cat.Prelude using (lemPropF ; setPi ; lemSig ; propSet ; Preorder ; equalityIsEquivalence ; propSig ; id-coe)
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open import Cat.Prelude using
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( lemPropF ; setPi ; lemSig ; propSet
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; Preorder ; equalityIsEquivalence ; propSig ; id-coe
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; Setoid )
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import Cubical.Univalence as U
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@ -327,6 +330,48 @@ preorder≅ ℓ = record
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k = pathJ D (trans id-coe id-coe) B (sym p)
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in k
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setoid≅ : (ℓ : Level) → Setoid _ _
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setoid≅ ℓ = record
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{ Carrier = Set ℓ
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; _≈_ = _≅_
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; isEquivalence = record
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{ refl = idFun _ , idFun _ , (funExt λ _ → refl) , (funExt λ _ → refl)
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; sym = symmetryIso
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; trans = composeIso
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}
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}
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setoid≃ : (ℓ : Level) → Setoid _ _
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setoid≃ ℓ = record
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{ Carrier = Set ℓ
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; _≈_ = _≃_
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; isEquivalence = record
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{ refl = idEquiv
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; sym = Equivalence.symmetry
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; trans = λ x x₁ → Equivalence.compose x x₁
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}
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}
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-- If the second component of a pair is propositional, then equality of such
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-- pairs is equivalent to equality of their first components.
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module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
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equivSigProp : ((x : A) → isProp (P x)) → {p q : Σ A P}
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→ (p ≡ q) ≃ (fst p ≡ fst q)
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equivSigProp pA {p} {q} = fromIsomorphism _ _ iso
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where
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f : ∀ {p q} → p ≡ q → fst p ≡ fst q
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f = cong fst
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g : ∀ {p q} → fst p ≡ fst q → p ≡ q
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g = lemSig pA _ _
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ve-re : (e : p ≡ q) → (g ∘ f) e ≡ e
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ve-re = pathJ (\ q (e : p ≡ q) → (g ∘ f) e ≡ e)
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(\ i j → p .fst , propSet (pA (p .fst)) (p .snd) (p .snd) (λ i → (g {p} {p} ∘ f) (λ i₁ → p) i .snd) (λ i → p .snd) i j ) q
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re-ve : (e : fst p ≡ fst q) → (f {p} {q} ∘ g {p} {q}) e ≡ e
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re-ve e = refl
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inv : AreInverses (f {p} {q}) (g {p} {q})
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inv = funExt ve-re , funExt re-ve
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iso : (p ≡ q) ≅ (fst p ≡ fst q)
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iso = f , g , inv
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module _ {ℓ : Level} {A B : Set ℓ} where
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isoToPath : (A ≅ B) → (A ≡ B)
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@ -347,6 +392,24 @@ module _ {ℓ : Level} {A B : Set ℓ} where
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aux : (A U.≃ B) ≃ (A ≃ B)
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aux = fromIsomorphism _ _ (doEta , deEta , funExt (λ{ (U.con _ _) → refl}) , refl)
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-- Equivalence is equivalent to isomorphism when the equivalence (resp.
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-- isomorphism) acts on sets.
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module _ (sA : isSet A) (sB : isSet B) where
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equiv≃iso : (f : A → B) → isEquiv A B f ≃ Isomorphism f
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equiv≃iso f =
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let
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obv : isEquiv A B f → Isomorphism f
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obv = toIso A B
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inv : Isomorphism f → isEquiv A B f
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inv = fromIso A B
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re-ve : (x : isEquiv A B f) → (inv ∘ obv) x ≡ x
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re-ve = inverse-from-to-iso A B
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ve-re : (x : Isomorphism f) → (obv ∘ inv) x ≡ x
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ve-re = inverse-to-from-iso A B sA sB
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iso : isEquiv A B f ≅ Isomorphism f
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iso = obv , inv , funExt re-ve , funExt ve-re
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in fromIsomorphism _ _ iso
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-- A few results that I have not generalized to work with both the eta and no-eta variable of ≃
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module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
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-- Equality on sigma's whose second component is a proposition is equivalent
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@ -438,6 +501,8 @@ module _ {ℓa ℓb : Level} {A : Set ℓa} {B : Set ℓb} where
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in fromIsomorphism _ _ iso
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module _ {ℓa ℓb : Level} {A : Set ℓa} {P : A → Set ℓb} where
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-- Equivalence of pairs whose first components are identitical can be obtained
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-- from an equivalence of their seecond components.
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equivSig : {ℓc : Level} {Q : A → Set ℓc}
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→ ((a : A) → P a ≃ Q a) → Σ A P ≃ Σ A Q
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equivSig {Q = Q} eA = res
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@ -76,7 +76,9 @@ module _ {ℓa ℓb : Level} {A : Set ℓa} {B : A → Set ℓb} {a b : Σ A B}
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snd (Σ≡ i) = snd≡ i
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import Relation.Binary
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open Relation.Binary public using (Preorder ; Transitive ; IsEquivalence ; Rel)
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open Relation.Binary public using
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( Preorder ; Transitive ; IsEquivalence ; Rel
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; Setoid )
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equalityIsEquivalence : {ℓ : Level} {A : Set ℓ} → IsEquivalence {A = A} _≡_
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IsEquivalence.refl equalityIsEquivalence = refl
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