73 lines
1.6 KiB
Agda
73 lines
1.6 KiB
Agda
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{-# OPTIONS --cubical #-}
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open import Cubical.Foundations.Prelude
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open import Cubical.Foundations.Equiv
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open import Cubical.Foundations.Isomorphism
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open import Cubical.Foundations.Univalence
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-- Path induction
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path-ind : {A : Set}
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→ (C : (x y : A) → x ≡ y → Set)
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→ (c : (x : A) → C x x refl)
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→ (x y : A) → (p : x ≡ y) → C x y p
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path-ind C c x y p = ?
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-- Lemma 2.1.1
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-- TODO: Not sure if there's actually a way to express inductive principle of
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-- identity in code?
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lemma211 : {A : Set} → (x y : A) → (x ≡ y) → (y ≡ x)
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lemma211 x y p = (λ i → p (~ i))
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D : {A : Set} → (x y : A) → x ≡ y → Set
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D x y p = y ≡ x
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d : {A : Set} → (x : A) → D x x refl
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d x = refl
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-- Lemma 2.1.2
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lemma212 : {A : Set} → {x y z : A} → (x ≡ y) → (y ≡ z) → (x ≡ z)
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lemma212 p q = p ∙ q
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-- (x ≡ z) i0 = x
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-- (x ≡ z) i1 = z
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--
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-- if i = i0 then
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-- p i0 = x
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-- q i0 = y
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--
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-- if i = i1 then
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-- p i1 = y
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-- q i1 = z
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--
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-- overall
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-- p i = i0 ∧ i
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-- q i = i1 ∨ i
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-- Lemma 2.11.1
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ap : {A B : Set} (f : A → B) {x y : A} → x ≡ y → f x ≡ f y
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ap f p i = f (p i)
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lemma2111 : {A B : Set} → (f : A → B) → (ie : isEquiv f) → {a a′ : A}
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→ isEquiv (ap f)
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lemma2111 f ie .equiv-proof y = ?
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-- Lemma 2.11.2
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lemma2112a : {A : Set} → (a : A) → {x₁ x₂ : A} → (p : x₁ ≡ x₂) → (q : a ≡ x₁)
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→ transport ? p ≡ q ∙ p
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lemma2112a a p q = ?
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-- Lemma 2.11.3
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lemma2113 : {A B : Set}
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→ (f g : A → B)
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→ {a a′ : A}
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→ (p : a ≡ a′)
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→ (q : f a ≡ g a)
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→ Set
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-- TODO:
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-- → transport (λ i → f a′ ≡ g a′) x ≡ (sym (ap f p)) ∙ q ∙ (ap g p)
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