75 lines
2.1 KiB
Plaintext
75 lines
2.1 KiB
Plaintext
load "misc.quox";
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load "bool.quox";
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namespace either {
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def0 Tag : ★ = {left, right};
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def0 Payload : 0.★ → 0.★ → 1.Tag → ★ =
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λ A B tag ⇒ case1 tag return ★ of { 'left ⇒ A; 'right ⇒ B };
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def0 Either : 0.★ → 0.★ → ★ =
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λ A B ⇒ (tag : Tag) × Payload A B tag;
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def Left : 0.(A B : ★) → 1.A → Either A B =
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λ A B x ⇒ ('left, x);
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def Right : 0.(A B : ★) → 1.B → Either A B =
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λ A B x ⇒ ('right, x);
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def elim' :
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0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
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ω.(1.(x : A) → P (Left A B x)) →
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ω.(1.(x : B) → P (Right A B x)) →
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1.(t : Tag) → 1.(a : Payload A B t) → P (t, a) =
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λ A B P f g t ⇒
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case1 t
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return t' ⇒ 1.(a : Payload A B t') → P (t', a)
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of { 'left ⇒ f; 'right ⇒ g };
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def elim :
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0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
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ω.(1.(x : A) → P (Left A B x)) →
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ω.(1.(x : B) → P (Right A B x)) →
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1.(x : Either A B) → P x =
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λ A B P f g e ⇒
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case1 e return e' ⇒ P e' of { (t, a) ⇒ elim' A B P f g t a };
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}
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def0 Either = either.Either;
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def Left = either.Left;
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def Right = either.Right;
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namespace dec {
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def0 Dec : 0.★ → ★ = λ A ⇒ Either [0.A] [0.Not A];
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def Yes : 0.(A : ★) → 0.A → Dec A = λ A y ⇒ Left [0.A] [0.Not A] [y];
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def No : 0.(A : ★) → 0.(Not A) → Dec A = λ A n ⇒ Right [0.A] [0.Not A] [n];
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def0 DecEq : 0.★ → ★ =
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λ A ⇒ ω.(x : A) → ω.(y : A) → Dec (x ≡ y : A);
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def elim :
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0.(A : ★) → 0.(P : 0.(Dec A) → ★) →
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ω.(0.(y : A) → P (Yes A y)) →
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ω.(0.(n : Not A) → P (No A n)) →
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1.(x : Dec A) → P x =
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λ A P f g ⇒
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either.elim [0.A] [0.Not A] P
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(λ y ⇒ case0 y return y' ⇒ P (Left [0.A] [0.Not A] y') of {[y'] ⇒ f y'})
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(λ n ⇒ case0 n return n' ⇒ P (Right [0.A] [0.Not A] n') of {[n'] ⇒ g n'});
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def bool : 0.(A : ★) → 1.(Dec A) → Bool =
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λ A ⇒ elim A (λ _ ⇒ Bool) (λ _ ⇒ 'true) (λ _ ⇒ 'false);
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}
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def0 Dec = dec.Dec;
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def0 DecEq = dec.DecEq;
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def Yes = dec.Yes;
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def No = dec.No;
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