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examples/misc.quox

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rhiannon morris 2023-04-03 16:07:51 +02:00
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def0 Pred : 0.★₀ → ★₁ = λ A ⇒ 0.A → ★₀;
def0 All : 0.(A : ★₀) → 0.(Pred A) → ★₁ =
λ A P ⇒ 1.(x : A) → P x;
defω cong :
0.(A : ★₀) → 0.(P : Pred A) →
1.(p : All A P) →
0.(x : A) → 0.(y : A) → 1.(xy : x ≡ y : A) →
Eq [i ⇒ P (xy @i)] (p x) (p y) =
λ A P p x y xy ⇒ δ i ⇒ p (xy @i);
def0 eq-f :
0.(A : ★₀) → 0.(P : Pred A) →
0.(p : All A P) → 0.(q : All A P) →
0.A → ★₀ =
λ A P p q x ⇒ p x ≡ q x : P x;
defω funext :
0.(A : ★₀) → 0.(P : Pred A) →
0.(p : All A P) → 0.(q : All A P) →
1.(All A (eq-f A P p q)) →
p ≡ q : All A P =
λ A P p q eq ⇒ δ i ⇒ λ x ⇒ eq x @i;