aoc2023/lib/list.quox

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2023-12-01 12:52:23 -05:00
load "nat.quox";
load "maybe.quox";
load "bool.quox";
namespace vec {
def0 Vec : → ★ → ★ =
λ n A ⇒
caseω n return ★ of {
zero ⇒ {nil};
succ _, 0.Tail ⇒ A × Tail
};
def elim : 0.(A : ★) → 0.(P : (n : ) → Vec n A → ★) →
P 0 'nil →
ω.((x : A) → 0.(n : ) → 0.(xs : Vec n A) →
P n xs → P (succ n) (x, xs)) →
(n : ) → (xs : Vec n A) → P n xs =
λ A P pn pc n ⇒
case n return n' ⇒ (xs' : Vec n' A) → P n' xs' of {
zero ⇒ λ nil ⇒
case nil return nil' ⇒ P 0 nil' of { 'nil ⇒ pn };
succ n, ih ⇒ λ cons ⇒
case cons return cons' ⇒ P (succ n) cons' of {
(first, rest) ⇒ pc first n rest (ih rest)
}
};
-- haha gross
def elimω : 0.(A : ★) → 0.(P : (n : ) → Vec n A → ★) →
ω.(P 0 'nil) →
ω.(ω.(x : A) → 0.(n : ) → 0.(xs : Vec n A) →
ω.(P n xs) → P (succ n) (x, xs)) →
ω.(n : ) → ω.(xs : Vec n A) → P n xs =
λ A P pn pc n ⇒
caseω n return n' ⇒ ω.(xs' : Vec n' A) → P n' xs' of {
zero ⇒ λ nil ⇒
caseω nil return nil' ⇒ P 0 nil' of { 'nil ⇒ pn };
succ n, ω.ih ⇒ λ cons ⇒
caseω cons return cons' ⇒ P (succ n) cons' of {
(first, rest) ⇒ pc first n rest (ih rest)
}
};
#[compile-scheme "(lambda% (n xs) xs)"]
def up : 0.(A : ★) → (n : ) → Vec n A → Vec¹ n A =
λ A n ⇒
case n return n' ⇒ Vec n' A → Vec¹ n' A of {
zero ⇒ λ xs ⇒
case xs return Vec¹ 0 A of { 'nil ⇒ 'nil };
succ n', f' ⇒ λ xs ⇒
case xs return Vec¹ (succ n') A of {
(first, rest) ⇒ (first, f' rest)
}
}
}
def0 Vec = vec.Vec;
namespace list {
def0 List : ★ → ★ =
λ A ⇒ (len : ) × Vec len A;
def Nil : 0.(A : ★) → List A =
λ A ⇒ (0, 'nil);
def Cons : 0.(A : ★) → A → List A → List A =
λ A x xs ⇒ case xs return List A of { (len, elems) ⇒ (succ len, x, elems) };
def elim : 0.(A : ★) → 0.(P : List A → ★) →
P (Nil A) →
ω.((x : A) → 0.(xs : List A) → P xs → P (Cons A x xs)) →
(xs : List A) → P xs =
λ A P pn pc xs ⇒
case xs return xs' ⇒ P xs' of { (len, elems) ⇒
vec.elim A (λ n xs ⇒ P (n, xs))
pn (λ x n xs ih ⇒ pc x (n, xs) ih)
len elems
};
def elimω : 0.(A : ★) → 0.(P : List A → ★) →
ω.(P (Nil A)) →
ω.(ω.(x : A) → 0.(xs : List A) → ω.(P xs) → P (Cons A x xs)) →
ω.(xs : List A) → P xs =
λ A P pn pc xs ⇒
caseω xs return xs' ⇒ P xs' of { (len, elems) ⇒
vec.elimω A (λ n xs ⇒ P (n, xs))
pn (λ x n xs ih ⇒ pc x (n, xs) ih)
len elems
};
-- [fixme] List A <: List¹ A should be automatic, imo
#[compile-scheme "(lambda (xs) xs)"]
def up : 0.(A : ★) → List A → List¹ A =
λ A xs ⇒
case xs return List¹ A of { (len, elems) ⇒
case nat.dup! len return List¹ A of { [p] ⇒
caseω p return List¹ A of { (lenω, eq0) ⇒
case eq0 return List¹ A of { [eq] ⇒
(lenω, vec.up A lenω (coe (𝑖 ⇒ Vec (eq @𝑖) A) @1 @0 elems))
}
}
}
};
def foldr : 0.(A B : ★) → B → ω.(A → B → B) → List A → B =
λ A B z f xs ⇒ elim A (λ _ ⇒ B) z (λ x _ y ⇒ f x y) xs;
def foldl : 0.(A B : ★) → B → ω.(B → A → B) → List A → B =
λ A B z f xs ⇒
foldr A (B → B) (λ b ⇒ b) (λ a g b ⇒ g (f b a)) xs z;
def map : 0.(A B : ★) → ω.(A → B) → List A → List B =
λ A B f ⇒ foldr A (List B) (Nil B) (λ x ys ⇒ Cons B (f x) ys);
-- ugh
def foldrω : 0.(A B : ★) → ω.B → ω.(ω.A → ω.B → B) → ω.(List A) → B =
λ A B z f xs ⇒ elimω A (λ _ ⇒ B) z (λ x _ y ⇒ f x y) xs;
def foldlω : 0.(A B : ★) → ω.B → ω.(ω.B → ω.A → B) → ω.(List A) → B =
λ A B z f xs ⇒
foldrω A (ω.B → B) (λ b ⇒ b) (λ a g b ⇒ g (f b a)) xs z;
def mapω : 0.(A B : ★) → ω.(ω.A → B) → ω.(List A) → List B =
λ A B f ⇒ foldrω A (List B) (Nil B) (λ x ys ⇒ Cons B (f x) ys);
def0 All : (A : ★) → (P : A → ★) → List A → ★ =
λ A P xs ⇒ foldr¹ A ★ True (λ x ps ⇒ P x × ps) (up A xs);
def append : 0.(A : ★) → List A → List A → List A =
λ A xs ys ⇒ foldr A (List A) ys (Cons A) xs;
def reverse : 0.(A : ★) → List A → List A =
λ A ⇒ foldl A (List A) (Nil A) (λ xs x ⇒ Cons A x xs);
def find : 0.(A : ★) → ω.(ω.A → Bool) → ω.(List A) → Maybe A =
λ A p ⇒
foldlω A (Maybe A) (Nothing A) (λ m x ⇒ maybe.or A m (maybe.check A p x));
postulate0 SchemeList : ★ → ★
#[compile-scheme
"(lambda (list) (cons (length list) (fold-right cons 'nil list)))"]
postulate from-scheme : 0.(A : ★) → SchemeList A → List A
#[compile-scheme
"(lambda (list)
(let loop [(acc '()) (list (cdr list))]
(if (pair? list)
(loop (cons (car list) acc) (cdr list))
(reverse acc))))"]
postulate to-scheme : 0.(A : ★) → List A → SchemeList A
}
def0 List = list.List;