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resources/VanDoornDissertation/dissertation.pdf
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@ -1,5 +1,5 @@
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\RequirePackage{fix-cm}
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\def\OPTpagesize{4.8in,7.9in} % Page size
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\def\OPTpagesize{6in,9in} % Page size
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\documentclass[12pt]{report}
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\usepackage[hyphens]{url}
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\usepackage[
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@ -16,8 +16,8 @@ transportconst : {A : Type l} {x y : A}
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→ (B : Type l2)
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→ (p : x ≡ y)
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→ (b : B)
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→ subst ? p ? ≡ ?
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transportconst B p b = ?
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→ subst (λ _ → B) p b ≡ b
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transportconst B p b i = {! !}
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```
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### Definition 2.4.1
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@ -58,7 +58,6 @@ mkIsEquiv : {A B : Type} {f : A → B}
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mkIsEquiv {B = B} {f = f} g forward backward = record { equiv-proof = eqv-prf } where
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postulate
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helper : (y : B) → (z : fiber f y) → (g y , forward y) ≡ z
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-- helper y (x , p) = Σ-≡,≡→≡ (ap g (sym p) ∙ backward x , {! !})
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eqv-prf : (y : B) → isContr (fiber f y)
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eqv-prf y = (g y , forward y) , helper y
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@ -67,22 +66,22 @@ mkIsEquiv {B = B} {f = f} g forward backward = record { equiv-proof = eqv-prf }
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### Theorem 2.11.3
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```
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theorem2∙11∙3 : {A B : Type} {f g : A → B} {a a' : A}
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→ (p : a ≡ a')
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→ (q : f a ≡ g a)
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→ subst (λ x → f x ≡ g x) p q ≡ sym (ap f p) ∙ q ∙ ap g p
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theorem2∙11∙3 refl q = sym (unitR q)
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-- theorem2∙11∙3 : {A B : Type} {f g : A → B} {a a' : A}
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-- → (p : a ≡ a')
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-- → (q : f a ≡ g a)
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-- → subst (λ x → f x ≡ g x) p q ≡ sym (ap f p) ∙ q ∙ ap g p
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-- theorem2∙11∙3 refl q = sym (unitR q)
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```
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### Theorem 2.11.5
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```
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postulate
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theorem2∙11∙5 : {A : Type} {a a' : A}
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→ (p : a ≡ a')
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→ (q : a ≡ a)
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→ (r : a' ≡ a')
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→ (transport (λ x → x ≡ x) p q ≡ r) ≃ (q ∙ p ≡ p ∙ r)
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-- postulate
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-- theorem2∙11∙5 : {A : Type} {a a' : A}
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-- → (p : a ≡ a')
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-- → (q : a ≡ a)
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-- → (r : a' ≡ a')
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-- → (transport (λ x → x ≡ x) p q ≡ r) ≃ (q ∙ p ≡ p ∙ r)
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-- theorem2∙11∙5 {a = a} refl q r = f , mkIsEquiv g {! !} {! !} where
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-- f : (q ≡ r) → (q ∙ refl ≡ r)
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-- f refl = unitR q
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@ -4,6 +4,9 @@ module CubicalHottBook.Prelude where
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open import Cubical.Foundations.Function public
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open import Data.Product.Properties public
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open import Cubical.Foundations.Prelude public
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open import Cubical.Data.Equality public using (id)
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open import Cubical.Foundations.Equiv public
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module Inductive where
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open import Cubical.Data.Equality public
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open Inductive public using (id)
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@ -480,6 +480,4 @@ module Pushout where
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Pushout : {A B C : Set} → (f : C → A) → (g : C → B) → Set
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syntax Pushout A B C = A ⊔[ C ] B
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```
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## asdf
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```
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4
src/VanDoornDissertation/HoTT/LongExactSequence.agda
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src/VanDoornDissertation/HoTT/LongExactSequence.agda
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{-# OPTIONS --cubical --safe #-}
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module VanDoornDissertation.HoTT.LongExactSequence where
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