some examples [that don't work yet]

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rhiannon morris 2023-03-31 19:31:49 +02:00
parent 64ac16c9f9
commit 13e9285bec
2 changed files with 121 additions and 0 deletions

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def0 Tag : ★₀ = {left, right};
def0 Payload : 0.(A : ★₀) → 0(B : .★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;
defω Left : 0.(A : ★₀) → 0.(B : ★₀) → 1.A → Either A B =
λ A B x ⇒ ('left, x);
defω Right : 0.(A : ★₀) → 0.(B : ★₀) → 1.B → Either A B =
λ A B x ⇒ ('right, x);
defω either-elim :
0.(A : ★₀) → 0.(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 x ⇒
case1 x return x' ⇒ P x' of {
(tag, value) ⇒
(case1 tag
return tag' ⇒
0.(eq : (tag ≡ tag' : Tag)) →
P (tag, coerce [i ⇒ Payload A B (eq i)] @0 @1 value)
of {
'left ⇒ λ eq ⇒ f value;
'right ⇒ λ eq ⇒ g value
}) (δ _ ⇒ tag)
};

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def0 Vec : 0. → 0.★₀ → ★₀ =
λ n A ⇒
caseω n return ★₀ of {
zero ⇒ {nil};
succ _, 0.Tail ⇒ A × Tail
};
def0 List : 0.★₀ → ★₀ =
λ A ⇒ (len : ) × Vec len A;
defω nil : 0.(A : ★₀) → List A =
λ A ⇒ (0, 'nil);
defω S : 1. = λ n ⇒ succ n;
defω cons : 0.(A : ★₀) → 1.A → 1.(List A) → List A =
λ A x xs ⇒
case1 xs return List A of {
(len, elems) ⇒ (succ len, x, elems)
};
{-
-- needs coercions overall,
-- and real w-types to be linear
defω list-ind :
0.(A : ★₀) →
0.(P : ω.(List A) → ★₀) →
1.(n : P (nil A)) →
ω.(c : 1.(x : A) → 0.(xs : List A) → 1.(P xs) → P (cons A x xs)) →
1.(lst : List A) → P lst =
λ A P n c lst ⇒
case1 lst return l ⇒ P l of {
(len, elems) ⇒
case1 len return len' ⇒ P (len', elems) of {
zero ⇒ n;
succ len', 1.ih ⇒
case1 elems return P (succ len', elems) of {
(first, rest) ⇒ c first rest ih
}
}
};
defω foldr :
0.(A : ★₀) → 0.(B : ★₀) →
1.(n : B) → ω.(c : 1.A → 1.B → B) →
1.(List A) → B =
λ A B n c lst ⇒ list-ind A (λ _ ⇒ B) n (λ a as b ⇒ c a b) lst;
-- ...still does
defω foldr :
0.(A : ★₀) → 0.(B : ★₀) →
ω.(n : B) → ω.(c : 1.A → 1.B → B) →
ω.(List A) → B =
λ A B n c lst ⇒
caseω lst return B of {
(len, elems) ⇒
caseω len return B of {
zero ⇒ caseω elems return B of { 'nil ⇒ n };
succ _, ω.ih ⇒
caseω elems return B of {
(first, rest) ⇒ c first ih
}
}
};
-}
defω plus : 1. → 1. =
λ m n ⇒
case1 m return of {
zero ⇒ n;
succ _, 1.mn ⇒ succ mn
};
-- case-'s qout needs to be Σz + ωΣs
def0 plus-3-3 : plus 3 3 ≡ 6 : =
δ 𝑖 ⇒ 6;
{-
defω sum : ω.(List ) → = foldr 0 plus;
defω numbers : List =
(5, (0, 1, 2, 3, 4, 'nil));
defω number-sum : sum numbers ≡ 10 : =
δ _ ⇒ 10;
-}