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