quox/examples/either.quox
rhiannon morris 3f06e8d68b allow multiple names in a binder
e.g. "(x y : ℕ) × plus x y ≡ 10 : ℕ"

fixes #2
2023-04-19 21:37:51 +02:00

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load "misc.quox";
load "bool.quox";
namespace either {
def0 Tag : ★₀ = {left, right};
def0 Payload : 0.★₀ → 0.★₀ → 1.Tag → ★₀ =
λ A B tag ⇒ case1 tag return ★₀ of { 'left ⇒ A; 'right ⇒ B };
def0 Either : 0.★₀ → 0.★₀ → ★₀ =
λ A B ⇒ (tag : Tag) × Payload A B tag;
def Left : 0.(A B : ★₀) → 1.A → Either A B =
λ A B x ⇒ ('left, x);
def Right : 0.(A B : ★₀) → 1.B → Either A B =
λ A B x ⇒ ('right, x);
def elim' :
0.(A B : ★₀) → 0.(P : 0.(Either A B) → ★₀) →
ω.(1.(x : A) → P (Left A B x)) →
ω.(1.(x : B) → P (Right A B x)) →
1.(t : Tag) → 1.(a : Payload A B t) → P (t, a) =
λ A B P f g t ⇒
case1 t
return t' ⇒ 1.(a : Payload A B t') → P (t', a)
of { 'left ⇒ f; 'right ⇒ g };
def elim :
0.(A B : ★₀) → 0.(P : 0.(Either A B) → ★₀) →
ω.(1.(x : A) → P (Left A B x)) →
ω.(1.(x : B) → P (Right A B x)) →
1.(x : Either A B) → P x =
λ A B P f g e ⇒
case1 e return e' ⇒ P e' of { (t, a) ⇒ elim' A B P f g t a };
}
def0 Either = either.Either;
def Left = either.Left;
def Right = either.Right;
namespace dec {
def0 Dec : 0.★₀ → ★₀ = λ A ⇒ Either [0.A] [0.Not A];
def Yes : 0.(A : ★₀) → 0.A → Dec A = λ A y ⇒ Left [0.A] [0.Not A] [y];
def No : 0.(A : ★₀) → 0.(Not A) → Dec A = λ A n ⇒ Right [0.A] [0.Not A] [n];
def0 DecEq : 0.★₀ → ★₀ =
λ A ⇒ ω.(x : A) → ω.(y : A) → Dec (x ≡ y : A);
def elim :
0.(A : ★₀) → 0.(P : 0.(Dec A) → ★₀) →
ω.(0.(y : A) → P (Yes A y)) →
ω.(0.(n : Not A) → P (No A n)) →
1.(x : Dec A) → P x =
λ A P f g ⇒
either.elim [0.A] [0.Not A] P
(λ y ⇒ case0 y return y' ⇒ P (Left [0.A] [0.Not A] y') of {[y'] ⇒ f y'})
(λ n ⇒ case0 n return n' ⇒ P (Right [0.A] [0.Not A] n') of {[n'] ⇒ g n'});
def bool : 0.(A : ★₀) → 1.(Dec A) → Bool =
λ A ⇒ elim A (λ _ ⇒ Bool) (λ _ ⇒ 'true) (λ _ ⇒ 'false);
}
def0 Dec = dec.Dec;
def0 DecEq = dec.DecEq;
def Yes = dec.Yes;
def No = dec.No;