rhi/quox
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add misc.refl, misc.sing, nat.minus

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rhiannon morris 2023-11-05 15:38:38 +01:00
parent e211887a34
commit 580fbc8fd8
2 changed files with 13 additions and 0 deletions
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5
examples/misc.quox
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@ -32,6 +32,8 @@ def funext :
(All A (eq-f A P p q)) → p ≡ q : All A P = (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 λ A P p q eq ⇒ δ 𝑖 ⇒ λ x ⇒ eq x @𝑖
def refl : 0.(A : ★) → (x : A) → x ≡ x : A = λ A x ⇒ δ _ ⇒ x
def sym : 0.(A : ★) → 0.(x y : A) → (x ≡ y : A) → y ≡ x : A = def sym : 0.(A : ★) → 0.(x y : A) → (x ≡ y : A) → y ≡ x : A =
λ A x y eq ⇒ δ 𝑖 ⇒ comp A (eq @0) @𝑖 { 0 𝑗 ⇒ eq @𝑗; 1 _ ⇒ eq @0 } λ A x y eq ⇒ δ 𝑖 ⇒ comp A (eq @0) @𝑖 { 0 𝑗 ⇒ eq @𝑗; 1 _ ⇒ eq @0 }
@ -51,6 +53,9 @@ def0 HEq : (A B : ★) → A → B → ★¹ =
def0 Sing : (A : ★) → A → ★ = def0 Sing : (A : ★) → A → ★ =
λ A x ⇒ (val : A) × [0. val ≡ x : A] λ A x ⇒ (val : A) × [0. val ≡ x : A]
def sing : 0.(A : ★) → (x : A) → Sing A x =
λ A x ⇒ (x, [δ _ ⇒ x])
namespace sing { namespace sing {
def val : 0.(A : ★) → 0.(x : A) → Sing A x → A = def val : 0.(A : ★) → 0.(x : A) → Sing A x → A =

8
examples/nat.quox
View file

@ -44,6 +44,14 @@ def pred-succ : ω.(n : ℕ) → pred (succ n) ≡ n : ℕ =
def0 succ-inj : (m n : ℕ) → succ m ≡ succ n : ℕ → m ≡ n : ℕ = def0 succ-inj : (m n : ℕ) → succ m ≡ succ n : ℕ → m ≡ n : ℕ =
λ m n eq ⇒ δ 𝑖 ⇒ pred (eq @𝑖); λ m n eq ⇒ δ 𝑖 ⇒ pred (eq @𝑖);
#[compile-scheme "(lambda% (m n) (max 0 (- m n)))"]
def minus : ℕ → ℕ → ℕ =
λ m n ⇒
(case n return ℕ → ℕ of {
zero ⇒ λ m ⇒ m;
succ _, f ⇒ λ m ⇒ f (pred m)
}) m;
def0 IsSucc : ℕ → ★ = def0 IsSucc : ℕ → ★ =
λ n ⇒ case n return ★ of { zero ⇒ False; succ _ ⇒ True }; λ n ⇒ case n return ★ of { zero ⇒ False; succ _ ⇒ True };