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Michael Zhang 2024-07-11 01:01:49 -05:00
parent 95ee861349
commit 8cac88e879

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@ -315,17 +315,53 @@ example3∙6∙2 {A} {B} allProps = λ f g → funext λ x → allProps x (f x)
```
module section3∙7 where
postulate
∥_∥ : Set → Set
_ : {A : Set} → (a : A) → ∥ A ∥
witness : {A : Set} → (x y : ∥ A ∥) → x ≡ y → Set
∥_∥ : Set l → Set l
_ : {A : Set l} → (a : A) → ∥ A ∥
witness : {A : Set l} → (x y : ∥ A ∥) → x ≡ y → Set l
rec-∥_∥ : (A : Set) → {B : Set} → isProp B → (f : A → B)
rec-∥_∥ : (A : Set l) → {B : Set l} → isProp B → (f : A → B)
→ Σ (∥ A ∥ → B) (λ g → (a : A) → g ( a ) ≡ f a)
open section3∙7 public
```
### Definition 3.7.1
```
module definition3∙7∙1 where
= 𝟙
_∧_ = _×_
_⇒_ : (P Q : Set l) → Set l
P ⇒ Q = P → Q
_⇔_ = _≡_
-- ¬_ : (P : Set l) → Set l
-- ¬ P = P → ⊥
__ : (P Q : Set l) → Set l
P Q = ∥ P + Q ∥
forall' : (A : Set l) → (P : A → Set l) → Set l
forall' A P = (x : A) → P x
exists' : (A : Set l) → (P : A → Set l) → Set l
exists' A P = ∥ Σ A P ∥
```
## 3.8 The axiom of choice
### Definition 3.8.1
```
module axiom-of-choice where
private
variable
X : Set
A : X → Set
P : (x : X) → A x → Set
postulate
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)) ∥
```
## 3.9 The principle of unique choice
### Lemma 3.9.1