test
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- [Front](./front.md)
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- [Front](./front.md)
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# HoTT Book
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- [Chapter 1](./generated/HottBook.Chapter1.md)
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- [Chapter 2](./generated/HottBook.Chapter2.md)
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- [Chapter 3](./generated/HottBook.Chapter3.md)
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- [Chapter 4](./generated/HottBook.Chapter4.md)
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- [Chapter 5](./generated/HottBook.Chapter5.md)
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# HoTT Book (Cubical formulation)
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# HoTT Book (Cubical formulation)
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- [Chapter 1](./generated/CubicalHott.Chapter1.md)
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- [Chapter 1](./generated/CubicalHott.Chapter1.md)
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22
src/CEKCoinductive.agda
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src/CEKCoinductive.agda
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{-# OPTIONS --guardedness #-}
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module CEKCoinductive where
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open import Agda.Primitive
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private
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variable
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l l1 l2 l3 : Level
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E : Set → Set
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R : Set
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data ITreeF (ITree : Set) : Set where
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Ret : (r : R) → ITreeF ITree
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Tau : (t : ITree) → ITreeF ITree
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Vis : {A : Set l1} → (e : E A) → (k : A → ITreeF E R) → ITreeF E R
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record ITree : Set where
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coinductive
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field
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_observe : ITreeF ITree
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@ -6,6 +6,21 @@ open import CubicalHott.Chapter1
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open import CubicalHott.Chapter2
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open import CubicalHott.Chapter2
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```
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```
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## 6.3 The interval
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```
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data Iv : Type where
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0I : Iv
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1I : Iv
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seg : 0I ≡ 1I
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```
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### Lemma 6.3.1
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```
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lemma6∙3∙1 : isContr Iv
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```
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## 6.4 Circles and spheres
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## 6.4 Circles and spheres
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```
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```
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@ -18,36 +33,36 @@ data S¹ : Type where
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```
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```
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lemma6∙4∙1 : loop ≢ refl
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lemma6∙4∙1 : loop ≢ refl
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lemma6∙4∙1 p =
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-- lemma6∙4∙1 p =
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{! !}
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-- {! !}
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where
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-- where
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open ≡-Reasoning
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-- open ≡-Reasoning
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f : {A : Type} (x : A) → (p : x ≡ x) → S¹ → A
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-- f : {A : Type} (x : A) → (p : x ≡ x) → S¹ → A
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f x p base = x
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-- f x p base = x
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f x p (loop i) = p i
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-- f x p (loop i) = p i
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bad : {A : Type} (x : A) → (p : x ≡ x) → p ≡ refl
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-- bad : {A : Type} (x : A) → (p : x ≡ x) → p ≡ refl
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bad x p = begin
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-- bad x p = begin
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p ≡⟨ {! !} ⟩
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-- p ≡⟨ {! !} ⟩
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p ≡⟨ {! !} ⟩
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-- p ≡⟨ {! !} ⟩
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refl ∎
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-- refl ∎
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```
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```
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### Lemma 6.4.2
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### Lemma 6.4.2
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```
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```
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lemma6∙4∙2 : Σ ((x : S¹) → x ≡ x) (λ y → y ≢ (λ x → refl))
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lemma6∙4∙2 : Σ ((x : S¹) → x ≡ x) (λ y → y ≢ (λ x → refl))
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lemma6∙4∙2 = f , g
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-- lemma6∙4∙2 = f , g
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where
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-- where
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f : (x : S¹) → x ≡ x
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-- f : (x : S¹) → x ≡ x
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f base = loop
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-- f base = loop
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f (loop i) i₁ = loop {! !}
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-- f (loop i) i₁ = loop {! !}
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g : f ≢ (λ x → refl)
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-- g : f ≢ (λ x → refl)
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g p = let
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-- g p = let
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z = happlyd p
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-- z = happlyd p
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z2 = z base
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-- z2 = z base
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z3 = lemma6∙4∙1 z2
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-- z3 = lemma6∙4∙1 z2
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in z3
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-- in z3
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```
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```
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@ -724,11 +724,8 @@ module axiom2∙10∙3 where
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postulate
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postulate
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ua : {l : Level} {A B : Set l} → (A ≃ B) → (A ≡ B)
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ua : {l : Level} {A B : Set l} → (A ≃ B) → (A ≡ B)
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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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backward : {l : Level} {A B : Set l} → (p : A ≡ B) → (ua ∘ idtoeqv) p ≡ p
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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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forward : {l : Level} {A B : Set l} → (eqv : A ≃ B) → (idtoeqv ∘ ua) eqv ≡ eqv
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ua-eqv : {A B : Set l} → (A ≃ B) ≃ (A ≡ B)
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ua-eqv : {A 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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ua-eqv = ua , qinv-to-isequiv (mkQinv idtoeqv backward forward)
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@ -184,18 +184,8 @@ theorem3∙2∙2 double-neg = conclusion
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let y = what ∙ x in
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let y = what ∙ x in
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sym y ∙ foranyu
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sym y ∙ foranyu
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-- postulate
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postulate
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huhh : (Σ.fst e) (fbool u) ≡ fbool u
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huhh : (Σ.fst e) (fbool u) ≡ fbool u
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huhh =
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let
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equiv1 = ap ua (axiom2∙10∙3.forward e)
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x : {A B : Set l} → (e : A ≃ B) → (a : A) → transport id (ua e) a ≡ Σ.fst e a
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x e a =
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{! axiom2∙10∙3.forward ? !}
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in
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{! !}
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-- sym (x e (fbool u)) ∙ nextStep
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finalStep : (x : bool) → ¬ ((Σ.fst e) x ≡ x)
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finalStep : (x : bool) → ¬ ((Σ.fst e) x ≡ x)
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finalStep (lift true) p =
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finalStep (lift true) p =
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@ -310,5 +300,9 @@ lemma3∙9∙1 {P} prop = lemma3∙3∙3 prop prop2 ∣_∣ g
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concat : g (∣ g x ∣) ≡ g (∣ g y ∣)
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concat : g (∣ g x ∣) ≡ g (∣ g y ∣)
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concat = a ∙ eqProp ∙ (sym b)
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concat = a ∙ eqProp ∙ (sym b)
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in
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in
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{! prop ? !}
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admit
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where
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postulate
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-- TODO: Finish this
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admit : x ≡ y
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```
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```
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53
src/HottBook/Chapter6.lagda.md
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src/HottBook/Chapter6.lagda.md
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```
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module HottBook.Chapter6 where
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open import HottBook.Chapter1
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open import HottBook.Chapter2
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open import HottBook.Chapter3
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private
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variable
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l : Level
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```
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# Chapter 6 Higher Inductive Types
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```
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postulate
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S¹ : Set
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base : S¹
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loop : base ≡ base
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```
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## 6.2 Induction principles and dependent paths
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```
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dep-path : {A : Set} {x y : A} → (P : A → Set) → (p : x ≡ y) → (u : P x) → (v : P y) → Set
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dep-path P p u v = transport P p u ≡ v
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syntax dep-path P p u v = u ≡[ P , p ] v
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```
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## 6.3 The interval
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```
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postulate
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I : Set
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0I : I
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1I : I
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seg : 0I ≡ 1I
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record I-rec (B : Set) (b0 b1 : B) (s : b0 ≡ b1) : Set where
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field
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f : I → B
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prop1 : f 0I ≡ b0
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prop2 : f 1I ≡ b1
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prop3 : apd f seg ≡ {! s !}
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record I-ind (P : I → Set) (b0 : P 0I) (b1 : P 1I) (s : b0 ≡[ P , seg ] b1) : Set where
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field
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f : (x : I) → P x
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prop1 : f 0I ≡ b0
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prop2 : f 1I ≡ b1
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prop3 : apd f seg ≡ {! s !}
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```
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2
src/Hurewicz.agda
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2
src/Hurewicz.agda
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module Hurewicz where
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