theorem 2.15.5

This commit is contained in:
Michael Zhang 2024-07-11 01:15:28 -05:00
parent 8cac88e879
commit b7f1863955
2 changed files with 22 additions and 4 deletions

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@ -976,7 +976,7 @@ theorem2∙15∙2 {X} {A} {B} = mkIsEquiv g (λ _ → refl) g (λ _ → refl)
### Equation 2.15.4
```
equation2∙15∙4 : {X : Set} {A B : X → Set}
equation2∙15∙4 : {X : Set l} {A B : X → Set l}
→ ((x : X) → A x × B x)
→ ((x : X) → A x) × ((x : X) → B x)
equation2∙15∙4 f = ((λ x → Σ.fst (f x)) , λ x → Σ.snd (f x))
@ -985,6 +985,16 @@ equation2∙15∙4 f = ((λ x → Σ.fst (f x)) , λ x → Σ.snd (f x))
### Theorem 2.15.5
```
-- theorem2∙15∙5 : isequiv equation2∙15∙4
-- theorem2∙15∙5 = qinv-to-isequiv (mkQinv {! !} {! !} {! !})
theorem2∙15∙5 : {X : Set l} {A B : X → Set l} → isequiv (equation2∙15∙4 {X = X} {A = A} {B = B})
theorem2∙15∙5 {X = X} {A = A} {B = B} = qinv-to-isequiv (mkQinv g forward backward)
where
g : ((x : X) → A x) × ((x : X) → B x) → (x : X) → A x × B x
g (f , g) x = f x , g x
forward : (p : ((x : X) → A x) × ((x : X) → B x)) → equation2∙15∙4 (g p) ≡ p
forward p = refl
backward : (f : (x : X) → A x × B x) → g (equation2∙15∙4 f) ≡ f
backward f = funext λ x → refl
where open axiom2∙9∙3
```

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@ -348,7 +348,7 @@ module definition3∙7∙1 where
### Definition 3.8.1
```
module axiom-of-choice where
module definition3∙8∙1 where
private
variable
X : Set
@ -359,8 +359,16 @@ module axiom-of-choice where
axiom-of-choice : (prop : (x : X) → (a : A x) → isProp (P x a))
→ ((x : X) → ∥ Σ (A x) (P x) ∥)
→ ∥ Σ ((x : X) → A x) (λ g → (x : X) → P x (g x)) ∥
open definition3∙8∙1 public
```
### Lemma 3.8.2
```
module lemma3∙8∙2 where
definition3∙8∙3 : {X : Set} → (Y : X → Set) → ((x : X) → isSet (Y x)) → ((x : X) → ∥ (Y x) ∥) → ∥ ((x : X) → Y x) ∥
definition3∙8∙3 {X} Y allYSet = {! !}
```
## 3.9 The principle of unique choice