aoc2023/lib/bool.quox
2023-12-01 18:52:23 +01:00

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load "misc.quox";
namespace bool {
def0 Bool : ★ = {true, false};
def if-dep : 0.(P : Bool → ★) → (b : Bool) → ω.(P 'true) → ω.(P 'false) → P b =
λ P b t f ⇒ case b return b' ⇒ P b' of { 'true ⇒ t; 'false ⇒ f };
def if : 0.(A : ★) → (b : Bool) → ω.A → ω.A → A =
λ A ⇒ if-dep (λ _ ⇒ A);
def0 if-same : (A : ★) → (b : Bool) → (x : A) → if A b x x ≡ x : A =
λ A b x ⇒ if-dep (λ b' ⇒ if A b' x x ≡ x : A) b (δ _ ⇒ x) (δ _ ⇒ x);
def if2 : 0.(A B : ★) → (b : Bool) → ω.A → ω.B → if¹ ★ b A B =
λ A B ⇒ if-dep (λ b ⇒ if-dep¹ (λ _ ⇒ ★) b A B);
def0 T : Bool → ★ = λ b ⇒ if¹ ★ b True False;
def dup! : (b : Bool) → [ω. Sing Bool b] =
λ b ⇒ if-dep (λ b ⇒ [ω. Sing Bool b]) b
[('true, [δ _ ⇒ 'true])]
[('false, [δ _ ⇒ 'false])];
def dup : Bool → [ω. Bool] =
λ b ⇒ appω (Sing Bool b) Bool (λ b' ⇒ sing.val Bool b b') (dup! b);
def true-not-false : Not ('true ≡ 'false : Bool) =
λ eq ⇒ coe (𝑖 ⇒ T (eq @𝑖)) 'true;
-- [todo] infix
def and : Bool → ω.Bool → Bool = λ a b ⇒ if Bool a b 'false;
def or : Bool → ω.Bool → Bool = λ a b ⇒ if Bool a 'true b;
}
def0 Bool = bool.Bool;