6.4.1
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4 changed files with 98 additions and 121 deletions
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@ -488,26 +488,30 @@ module lemma2∙4∙12 where
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```
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definition2∙6∙1 : {A B : Set l} {x y : A × B}
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→ (p : x ≡ y)
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→ (Σ.fst x ≡ Σ.fst y) × (Σ.snd x ≡ Σ.snd y)
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definition2∙6∙1 p = ap Σ.fst p , ap Σ.snd p
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→ (fst x ≡ fst y) × (snd x ≡ snd y)
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definition2∙6∙1 p = ap fst p , ap snd p
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```
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### Theorem 2.6.2
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```
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module theorem2∙6∙2 where
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pair-≡ : {A B : Set l} {x y : A × B} → (Σ.fst x ≡ Σ.fst y) × (Σ.snd x ≡ Σ.snd y) → x ≡ y
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private
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variable
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A B : Set l
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x y : A × B
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pair-≡ : (fst x ≡ fst y) × (snd x ≡ snd y) → x ≡ y
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pair-≡ (refl , refl) = refl
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theorem2∙6∙2 : {A B : Set l} {x y : A × B}
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→ isequiv (definition2∙6∙1 {A = A} {B = B} {x = x} {y = y})
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theorem2∙6∙2 {A} {B} {x} {y} = qinv-to-isequiv (mkQinv pair-≡ backward forward)
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where
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backward : (definition2∙6∙1 ∘ pair-≡) ∼ id
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backward (refl , refl) = refl
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backward : ((definition2∙6∙1 {A = A} {B = B} {x = x} {y = y}) ∘ pair-≡) ∼ id
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backward (refl , refl) = refl
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forward : (pair-≡ ∘ definition2∙6∙1) ∼ id
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forward refl = refl
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forward : (pair-≡ ∘ (definition2∙6∙1 {A = A} {B = B} {x = x} {y = y})) ∼ id
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forward refl = refl
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theorem2∙6∙2 : isequiv (definition2∙6∙1 {A = A} {B = B} {x = x} {y = y})
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theorem2∙6∙2 = qinv-to-isequiv (mkQinv pair-≡ backward forward)
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pair-≃ : {A B : Set l} {x y : A × B} → (x ≡ y) ≃ ((fst x ≡ fst y) × (snd x ≡ snd y))
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pair-≃ = definition2∙6∙1 , theorem2∙6∙2
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@ -84,24 +84,17 @@ isSet A = (x y : A) → (p q : x ≡ y) → p ≡ q
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where
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open theorem2∙6∙2
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open axiom2∙10∙3
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open isequiv
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p' : (xa ≡ ya) × (xb ≡ yb)
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p' = definition2∙6∙1 p
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p' = theorem2∙6∙2 .h-id p
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q' = theorem2∙6∙2 .h-id q
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q' : (xa ≡ ya) × (xb ≡ yb)
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q' = definition2∙6∙1 q
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pr1 : ap fst p ≡ ap fst q
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pr1 = A-set xa ya (ap fst p) (ap fst q)
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pr2 : ap snd p ≡ ap snd q
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pr2 = B-set xb yb (ap snd p) (ap snd q)
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lol : {A B : Set l} {x y : A × B} → (x ≡ y) ≡ ((fst x ≡ fst y) × (snd x ≡ snd y))
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lol = ua pair-≃
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convert : (p : (xa , xb) ≡ (ya , yb)) → (xa ≡ ya) × (xb ≡ yb)
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convert p = definition2∙6∙1 p
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goal : p ≡ q -- (p : (xa, xb)≡(ya, yb)) (q : (xa, xb)≡(ya, yb))
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-- (p': xa≡ya × xb≡yb)
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goal2 = p' ≡ q'
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goal = {! ap (λ z → ?) goal2 !}
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goal = sym p' ∙ ap pair-≡ (pair-≡ (pr1 , pr2)) ∙ q'
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```
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### Example 3.1.6
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@ -433,19 +426,12 @@ module section3∙7 where
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∣_∣ : {A : Set l} → A → ∥ A ∥
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trunc-witness : {A : Set l} → isProp (∥ A ∥)
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rec-∥_∥ : (A : Set l) → {B : Set l}
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→ isProp B
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→ (f : A → B)
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→ Σ (∥ A ∥ → B) (λ g → (a : A) → g (∣ a ∣) ≡ f a)
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rec-trunc : (A : Set) → {B : Set} → isProp B → (f : A → B) → ∥ A ∥ → B
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trunc-rec : (A : Set) → {B : Set} → isProp B → (f : A → B)
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→ ∥ A ∥ → B
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trunc-rec-1 : {A : Set} → {B : Set} → (Bprop : isProp B) → (f : A → B)
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→ (a : A) → trunc-rec A Bprop f (∣ a ∣) ≡ f a
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rec-trunc-1 : {A : Set} → {B : Set} → (Bprop : isProp B) → (f : A → B) → (a : A) → rec-trunc A Bprop f (∣ a ∣) ≡ f a
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{-# REWRITE rec-trunc-1 #-}
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open section3∙7 public
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{-# REWRITE trunc-rec-1 #-}
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```
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### Definition 3.7.1
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@ -503,7 +489,7 @@ lemma3∙9∙1 : {P : Set} → isProp P → P ≃ ∥ P ∥
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lemma3∙9∙1 {P} Pprop = lemma3∙3∙3 Pprop witness ∣_∣ rec-func
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where
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rec-func : ∥ P ∥ → P
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rec-func = trunc-rec P Pprop id
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rec-func = rec-trunc P Pprop id
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witness : isProp ∥ P ∥
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witness = trunc-witness
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@ -673,11 +659,11 @@ module lemma3∙11∙9 where
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g p = a , p
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forward : (p : P a) → transport P (sym (aContr a)) p ≡ p
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forward p = y (sym (aContr a))
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forward p = admit
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where
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y : (q : a ≡ a) → transport P q p ≡ p
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y q = {! !}
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postulate
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-- TODO
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admit : transport P (sym (aContr a)) p ≡ p
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backward : (g ∘ f) ∼ id
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backward (x , p) =
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@ -95,16 +95,19 @@ lemma4∙1∙1 {A} {B} f e @ (mkQinv g _ _) = goal
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### Lemma 4.1.2
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```
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open section3∙7
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lemma4∙1∙2 : {A : Set} {a : A} (q : a ≡ a)
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→ isSet (a ≡ a)
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→ ((x : A) → ∥ a ≡ x ∥)
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→ (g : (x : A) → ∥ a ≡ x ∥)
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→ ((p : a ≡ a) → p ∙ q ≡ q ∙ p)
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→ Σ ((x : A) → x ≡ x) (λ f → f a ≡ q)
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lemma4∙1∙2 {A} {a} q prop1 g prop3 = (λ x → {! !}) , {! !}
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where
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allsets : (x y : A) → isSet (x ≡ y)
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allsets x .x refl refl refl refl = refl
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allsets x y p q r s = {! trunc-witness ? ? !}
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ctx = let
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f = rec-trunc A {! !} {! g !}
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in {! !}
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B : (x : A) → Set
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B x = Σ (x ≡ x) (λ r → (s : a ≡ x) → r ≡ (sym s) ∙ q ∙ s)
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@ -188,6 +191,20 @@ fib : ∀ {A B} → (f : A → B) → (y : B) → Set
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fib {A = A} f y = Σ A (λ x → f x ≡ y)
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```
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### Lemma 4.2.5
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```
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lemma4∙2∙5 : {A B : Set}
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→ (f : A → B)
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→ (y : B)
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→ ((x , p) (x' , p') : fib f y)
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→ ((x , p) ≡ (x' , p')) ≃ Σ (x ≡ x') (λ γ → ap f γ ∙ p' ≡ p)
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lemma4∙2∙5 f y (x , p) (x' , p') = {! !} , {! !}
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where
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ff : (x , p) ≡ (x' , p') → Σ (x ≡ x') (λ γ → ap f γ ∙ p' ≡ p)
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ff q = ap fst q , {! !} -- (_ : f x ≡ f x') ∙ p' ≡ p
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```
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### Definition 4.2.7
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```
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@ -33,35 +33,28 @@ syntax dep-path P p u v = u ≡[ P , p ] v
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```
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module Interval where
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data #I : Set where
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#0 : #I
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#1 : #I
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I : Set
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I = #I
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0I 1I : I
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0I = #0
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1I = #1
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postulate
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I : Set
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0I 1I : I
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seg : 0I ≡ 1I
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rec-Interval : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → I → B
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rec-Interval b₀ b₁ s #0 = b₀
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rec-Interval b₀ b₁ s #1 = b₁
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rec-Interval : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → I → B
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rec-Interval-0I : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → rec-Interval b₀ b₁ s 0I ≡ b₀
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rec-Interval-1I : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → rec-Interval b₀ b₁ s 1I ≡ b₁
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{-# REWRITE rec-Interval-0I #-}
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{-# REWRITE rec-Interval-1I #-}
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postulate
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rec-Interval-3 : {B : Set} → (b₀ b₁ : B) → (s : b₀ ≡ b₁) → apd (rec-Interval b₀ b₁ s) seg ≡ s
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{-# REWRITE rec-Interval-3 #-}
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{-# REWRITE rec-Interval-3 #-}
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rec-Interval-d : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → (x : I) → B x
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rec-Interval-d b₀ b₁ s #0 = b₀
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rec-Interval-d b₀ b₁ s #1 = b₁
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rec-Interval-d : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → (x : I) → B x
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rec-Interval-d-0I : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → rec-Interval-d {B = B} b₀ b₁ s 0I ≡ b₀
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rec-Interval-d-1I : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → rec-Interval-d {B = B} b₀ b₁ s 1I ≡ b₁
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{-# REWRITE rec-Interval-d-0I #-}
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{-# REWRITE rec-Interval-d-1I #-}
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postulate
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rec-Interval-d-3 : {B : I → Set} → (b₀ : B 0I) → (b₁ : B 1I) → (s : b₀ ≡[ B , seg ] b₁) → apd (rec-Interval-d {B} b₀ b₁ s) seg ≡ s
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{-# REWRITE rec-Interval-d-3 #-}
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{-# REWRITE rec-Interval-d-3 #-}
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open Interval public
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```
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@ -101,25 +94,18 @@ lemma6∙3∙2 {A = A} {B = B} {f = f} {g = g} p = apd q seg
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## 6.4 Circles and sphere
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```
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private
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data #S¹ : Set where
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#base : #S¹
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S¹ : Set
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S¹ = #S¹
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base : S¹
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base = #base
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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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rec-S¹ : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → ((x : S¹) → P x)
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rec-S¹ b l #base = b
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postulate
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rec-S¹ : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → ((x : S¹) → P x)
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rec-S¹-base : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → rec-S¹ {P = P} b l base ≡ b
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{-# REWRITE rec-S¹-base #-}
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rec-S¹-loop : {l : Level} {P : S¹ → Set l} → (b : P base) → (l : b ≡[ P , loop ] b) → apd {P = P} (rec-S¹ b l) loop ≡ l
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{-# REWRITE rec-S¹-loop #-}
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-- TODO: Uncommenting this leads to a bug in the definition of z2 in lemma 6.4.1
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-- {-# REWRITE rec-S¹-loop #-}
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```
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### Lemma 6.4.1
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@ -128,54 +114,38 @@ postulate
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lemma6∙4∙1 : loop ≢ refl
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lemma6∙4∙1 loop≡refl = goal3
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where
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-- goal : (s : S¹) (p : s ≡ s) → p ≡ refl
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-- goal s p = z1 ∙ z2 ∙ z3
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-- where
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-- f : {A : Set} {x : A} {p : x ≡ x} → S¹ → A
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-- f {x = x} {p = p} = rec-S¹ x p
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-- z1 : p ≡ apd f loop
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-- z1 = refl
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-- z2 : apd f loop ≡ apd f refl
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-- z2 = ap (apd f) loop≡refl
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-- z3 : apd f refl ≡ refl
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-- z3 = refl
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f : ∀ {l} {A : Set l} {x : A} {p : x ≡ x} → S¹ → A
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f {A = A} {x = x} {p = p} = rec-S¹ {P = λ _ → A} x p
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goal2 : ∀ {l} {A : Set l} → isSet A
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goal2 x .x refl refl = refl
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goal2 {l = l} {A = A} x .x p refl = z1 ∙ z2 ∙ z3
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where
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f' : S¹ → A
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f' = f {l = l} {A = A} {x = x} {p = p}
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z1 : p ≡ apd f' loop
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z1 = sym (rec-S¹-loop x p)
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z2 : apd f' loop ≡ apd {x = base} f' refl
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z2 = ap {A = base ≡ base} (apd f') loop≡refl
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z3 : apd {x = base} f' refl ≡ refl
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z3 = refl
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goal3 : ⊥
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goal3 = example3∙1∙9 (goal2 {A = Set})
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-- where
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-- f : {l : Level} {A : Set l} {x : A} {p : x ≡ x} → (S¹ → A)
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-- f {A = A} {x = x} {p = p} s =
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-- let p' = transportconst A loop x
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-- in (rec-S¹ x (p' ∙ p)) s
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-- -- f-loop : {A : Set} {x : A} {p : x ≡ x} → apd f loop ≡ p
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-- -- f-loop {x = x} = S¹-rec-loop x ?
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-- goal : ⊥
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-- goal2 : ∀ {l} → (A : Set l) (a : A) (p : a ≡ a) → isSet A
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-- goal = example3∙1∙9 (goal2 Set 𝟙 refl)
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-- goal3 : ∀ (s : S¹) (p : s ≡ s) → p ≡ refl
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-- goal2 {l} A a p x y r s = {! !}
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-- where
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-- f' = f {A = A} {x = a} {p = p}
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-- test = apd f' loop ∙ sym (apd f' loop)
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-- z1 : p ≡ apd f' loop
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-- z1 = {! !}
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-- z2 : apd f' loop ≡ apd f' refl
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-- z2 = {! !}
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-- z3 : apd f' refl ≡ refl
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-- z3 = refl
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-- wtf = let wtf = z1 ∙ z2 ∙ z3 in {! !}
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```
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### Lemma 6.4.2
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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 = f , g
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where
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open axiom2∙9∙3
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f = rec-S¹ loop {! !}
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g : f ≡ (λ x → refl) → ⊥
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g p = {! !}
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```
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