116 lines
3.3 KiB
Text
116 lines
3.3 KiB
Text
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 : ★ → ★ → Tag → ★ =
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λ A B tag ⇒ case tag return ★ of { 'left ⇒ A; 'right ⇒ B }
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def0 Either : ★ → ★ → ★ =
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λ A B ⇒ (tag : Tag) × Payload A B tag
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def Left : 0.(A B : ★) → A → Either A B =
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λ A B x ⇒ ('left, x)
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def Right : 0.(A B : ★) → 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.((x : A) → P (Left A B x)) →
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ω.((x : B) → P (Right A B x)) →
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(x : Either A B) → P x =
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λ A B P f g e ⇒
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case e return e' ⇒ P e' of { (t, a) ⇒
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case t return t' ⇒ (a : Payload A B t') → P (t', a)
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of { 'left ⇒ f; 'right ⇒ g } a
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}
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def elimω :
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0.(A B : ★) → 0.(P : 0.(Either A B) → ★) →
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ω.(ω.(x : A) → P (Left A B x)) →
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ω.(ω.(x : B) → P (Right A B x)) →
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ω.(x : Either A B) → P x =
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λ A B P f g e ⇒
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case fst e return t' ⇒ ω.(a : Payload A B t') → P (t', a)
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of { 'left ⇒ f; 'right ⇒ g } (snd e)
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def fold :
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0.(A B C : ★) → ω.(A → C) → ω.(B → C) → Either A B → C =
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λ A B C ⇒ elim A B (λ _ ⇒ C)
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def foldω :
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0.(A B C : ★) → ω.(ω.A → C) → ω.(ω.B → C) → ω.(Either A B) → C =
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λ A B C ⇒ elimω A B (λ _ ⇒ C)
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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 : ★ → ★ = 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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def yes-refl : 0.(A : ★) → 0.(x : A) → Dec (x ≡ x : A) =
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λ A x ⇒ Yes (x ≡ x : A) (δ 𝑖 ⇒ x)
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def0 DecEq : ★ → ★ =
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λ A ⇒ ω.(x 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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(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 ⇒ case y return y' ⇒ P (Left [0.A] [0.Not A] y') of {[y'] ⇒ f y'})
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(λ n ⇒ case n return n' ⇒ P (Right [0.A] [0.Not A] n') of {[n'] ⇒ g n'})
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def bool : 0.(A : ★) → Dec A → Bool =
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λ A ⇒ elim A (λ _ ⇒ Bool) (λ _ ⇒ 'true) (λ _ ⇒ 'false)
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def drop' : 0.(A : ★) → Dec A → True =
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λ A ⇒ elim A (λ _ ⇒ True) (λ _ ⇒ 'true) (λ _ ⇒ 'true)
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def drop : 0.(A B : ★) → Dec A → B → B =
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λ A B x y ⇒ true.drop B (drop' A x) y
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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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namespace dect {
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def0 DecT : ★ → ★ = λ A ⇒ Either A [0.Not A]
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def YesT : 0.(A : ★) → 1.A → DecT A = λ A y ⇒ Left A [0.Not A] y
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def NoT : 0.(A : ★) → 0.(Not A) → DecT A = λ A n ⇒ Right A [0.Not A] [n]
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def elim :
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0.(A : ★) → 0.(P : 0.(DecT A) → ★) →
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ω.(1.(y : A) → P (YesT A y)) →
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ω.(0.(n : Not A) → P (NoT A n)) →
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(x : DecT A) → P x =
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λ A P f g ⇒
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either.elim A [0.Not A] P
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f
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(λ n ⇒ case n return n' ⇒ P (Right A [0.Not A] n') of {[n'] ⇒ g n'})
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}
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def0 DecT = dect.DecT
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def YesT = dect.YesT
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def NoT = dect.NoT
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