74 lines
2.1 KiB
Text
74 lines
2.1 KiB
Text
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;
|