update
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commit
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@ -130,7 +130,7 @@ rec-ℕ C z s (suc n) = s n (rec-ℕ C z s n)
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```
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infix 3 ¬_
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¬_ : {l : Level} → (A : Set l) → Set l
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¬_ : ∀ {l : Level} (A : Set l) → Set l
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¬_ {l} A = A → ⊥ {l}
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```
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@ -344,7 +344,7 @@ lemma2∙4∙3 {A} {B} {f} {g} H {x} {y} p = let
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### Definition 2.4.6
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```
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record qinv {l : Level} {A B : Set l} (f : A → B) : Set l where
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record qinv {l1 l2 : Level} {A : Set l1} {B : Set l2} (f : A → B) : Set (l1 ⊔ l2) where
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constructor mkQinv
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field
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g : B → A
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@ -355,7 +355,7 @@ record qinv {l : Level} {A B : Set l} (f : A → B) : Set l where
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### Example 2.4.7
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```
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id-qinv : {l : Level} {A : Set l} → qinv {l} {A} id
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id-qinv : {l : Level} {A : Set l} → qinv {l} {l} {A} {A} id
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id-qinv = mkQinv id (λ _ → refl) (λ _ → refl)
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```
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@ -396,7 +396,7 @@ record isequiv {l1 l2 : Level} {A : Set l1} {B : Set l2} (f : A → B) : Set (l1
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```
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```
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qinv-to-isequiv : {l : Level} {A B : Set l}
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qinv-to-isequiv : {l1 l2 : Level} {A : Set l1} {B : Set l2}
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→ {f : A → B}
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→ qinv f
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→ isequiv f
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@ -667,25 +667,47 @@ equation2∙9∙4 {l1} {l2} {X} {x1} {x2} {A} {B} f p =
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(λ x3 f1 → refl) x1 x2 p f
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```
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### Equation 2.9.5
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```
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module equation2∙9∙5 {X : Set} {x1 x2 : X} where
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Π : (A : X → Set) → (B : (x : X) → A x → Set) → X → Set
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Π A B x = (a : A x) → B x a
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hat : {A : X → Set} → (B : (x : X) → A x → Set) → (Σ X A) → Set
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hat B (fst , snd) = B fst snd
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--- I have no idea where this goes but this definitely needs to exist
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pair-≡ : {A : Set} {B : A → Set} {a1 a2 : A} {b1 : B a1} {b2 : B a2}
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→ (p : a1 ≡ a2) → (transport B p b1 ≡ b2) → (a1 , b1) ≡ (a2 , b2)
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pair-≡ refl refl = refl
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equation2∙9∙5 : (A : X → Set)
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→ (B : (x : X) → A x → Set)
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→ (f : (a : A x1) → B x1 a)
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→ (p : x1 ≡ x2)
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→ (a : A x2)
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→ transport (Π A B) p f a ≡ transport (hat B) (sym (pair-≡ (sym p) refl)) (f (transport A (sym p) a))
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equation2∙9∙5 A B f p a =
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J (λ x1' x2' p' → (f' : (a : A x1') → B x1' a) → (a' : A x2') → transport (Π A B) p' f' a' ≡ transport (hat B) (sym (pair-≡ (sym p') refl)) (f' (transport A (sym p') a')))
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(λ x' f' a' → refl) x1 x2 p f a
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```
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### Lemma 2.9.6
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```
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-- lemma2∙9∙6 : {X : Set} {A B : X → Set} {x y : X}
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-- → (p : x ≡ y)
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-- → (f : A x → B x)
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-- → (g : A y → B y)
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-- →
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-- let P x = A x → B x in
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-- (transport P p f ≡ g) ≃ ((a : A x) → transport B p (f a) ≡ g (transport A p a))
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-- lemma2∙9∙6 {X} {A} {B} {x} {y} p f g =
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-- let
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-- P x = A x → B x
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-- module lemma2∙9∙6 {X : Set} {A B : X → Set} {x y : X} where
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-- P : (x : X) → Set
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-- P x = A x → B x
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-- F : (x1 : X) → (f1 : A x1 → B x1) → (g1 : A x1 → B x1) → (transport P refl f1 ≡ g1) → ((a : A x1) → transport B refl (f1 a) ≡ g1 (transport A refl a))
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-- F x f g p a = {! !}
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-- in
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-- J (λ x1 y1 p1 → (f1 : A x1 → B x1) → (g1 : A y1 → B y1) → (transport P p1 f1 ≡ g1) ≃ ((a : A x1) → transport B p1 (f1 a) ≡ g1 (transport A p1 a)))
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-- (λ x1 f1 g1 → F x1 f1 g1 , qinv-to-isequiv (mkQinv {! !} {! !} {! !})) x y p f g
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-- lemma2∙9∙6 : (p : x ≡ y)
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-- → (f : A x → B x)
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-- → (g : A y → B y)
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-- → (transport P p f ≡ g) ≃ ((a : A x) → transport B p (f a) ≡ g (transport A p a))
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-- lemma2∙9∙6 p f g = func , qinv-to-isequiv (mkQinv {! !} {! !} {! !})
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-- where
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-- func : (transport P p f ≡ g) → (a : A x) → transport B p (f a) ≡ g (transport A p a)
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-- func p1 a = J (λ x y p' → {! !} ≡ y (transport A {! !} a)) {! !} (transport P p f) g p1
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```
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## 2.10 Universes and the univalence axiom
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@ -710,17 +732,18 @@ idtoeqv {l} {A} {B} p = J C c A B p
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```
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module axiom2∙10∙3 where
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postulate
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-- ua-eqv : {l1 l2 : Level} {A : Set l1} {B : Set l2} → (A ≃ B) ≃ (A ≡ B)
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ua : {l : Level} {A B : Set l} → (A ≃ B) → (A ≡ B)
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-- forward : {A B : Set} → (eqv @ (f , f-eqv) : A ≃ B) → (x : A) → transport id (ua eqv) x ≡ f x
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-- forward eqv x = {! !}
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forward : {l : Level} {A B : Set l} → (eqv : A ≃ B) → (idtoeqv ∘ ua) eqv ≡ eqv
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-- forward eqv = {! !}
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-- ua-refl : {A : Set} → refl ≡ ua (lemma2∙4∙12.id-equiv A)
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-- ua-refl = {! !}
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backward : {l : Level} {A B : Set l} → (p : A ≡ B) → (ua ∘ idtoeqv) p ≡ p
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-- backward p = {! !}
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open axiom2∙10∙3
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ua-eqv : {l : Level} {A : Set l} {B : Set l} → (A ≃ B) ≃ (A ≡ B)
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ua-eqv = ua , qinv-to-isequiv (mkQinv idtoeqv backward forward)
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open axiom2∙10∙3 hiding (forward; backward)
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```
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### Lemma 2.10.5
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@ -860,7 +883,7 @@ theorem2∙11∙4 {A} {B} {f} {g} {a} {a'} p q =
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### Theorem 2.12.5
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```
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module 2∙12∙5 {l : Level} {A B : Set l} (a₀ : A) where
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module theorem2∙12∙5 {l : Level} {A B : Set l} (a₀ : A) where
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code : A + B → Set l
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code (inl a) = a₀ ≡ a
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code (inr b) = ⊥
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@ -911,7 +934,7 @@ remark2∙12∙6 p = genBot tt
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_For all $m, n : N$ we have $(m = n) \simeq \texttt{code}(m, n)$._
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```
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module 2∙13∙1 where
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module theorem2∙13∙1 where
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open ≡-Reasoning
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code : ℕ → ℕ → Set
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@ -1,3 +1,6 @@
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<details>
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<summary>Imports</summary>
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```
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module HottBook.Chapter3 where
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@ -5,12 +8,13 @@ open import Agda.Primitive
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open import HottBook.Chapter1
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open import HottBook.Chapter1Util
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open import HottBook.Chapter2
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open import HottBook.Chapter3Definition331
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open import HottBook.Chapter3Lemma333
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open import HottBook.Chapter3Definition331 public
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open import HottBook.Chapter3Lemma333 public
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open import HottBook.Util
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```
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</details>
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## 3.1 Sets and n-types
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### Definition 3.1.1
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@ -30,7 +34,7 @@ isSet A = (x y : A) → (p q : x ≡ y) → p ≡ q
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### Example 3.1.3
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```
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⊥-is-Set : isSet ⊥
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⊥-is-Set : ∀ {l} → isSet (⊥ {l})
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⊥-is-Set () () p q
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```
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@ -44,7 +48,7 @@ isSet A = (x y : A) → (p q : x ≡ y) → p ≡ q
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### Example 3.1.9
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```
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example3∙1∙9 : ¬ (isSet Set)
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example3∙1∙9 : ∀ {l : Level} → ¬_ {lsuc l} (isSet (Set l))
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example3∙1∙9 p = remark2∙12∙6 lol
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where
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open axiom2∙10∙3
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bogus : id ≡ neg
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bogus =
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let
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helper : f-path ≡ refl
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helper = p 𝟚 𝟚 f-path refl
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-- helper : f-path ≡ refl
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-- helper = p 𝟚 𝟚 f-path refl
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a = ap
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wtf : idtoeqv f-path ≡ idtoeqv refl
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wtf = ap idtoeqv helper
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-- wtf : idtoeqv f-path ≡ idtoeqv refl
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-- wtf = ap idtoeqv helper
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in {! !}
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lol : true ≡ false
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