44 lines
1.6 KiB
Plaintext
44 lines
1.6 KiB
Plaintext
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load "misc.quox"
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def0 Irr1 : (A : ★) → (A → ★) → ★ =
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λ A P ⇒ (x : A) → (p q : P x) → p ≡ q : P x
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def0 Sub : (A : ★) → (P : A → ★) → ★ =
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λ A P ⇒ (x : A) × [0. P x]
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def0 SubDup : (A : ★) → (P : A → ★) → Sub A P → ★ =
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λ A P s ⇒ Dup A (fst s)
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-- (x! : [ω.A]) × [0. x! ≡ [fst s] : [ω.A]]
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def subdup-to-dup :
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0.(A : ★) → 0.(P : A → ★) → 0.(Irr1 A P) →
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0.(s : Sub A P) → SubDup A P s → Dup (Sub A P) s =
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λ A P pirr s sd ⇒
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case sd return Dup (Sub A P) s of { (sω, ss0) ⇒
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case ss0 return Dup (Sub A P) s of { [ss0] ⇒
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case sω
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return sω' ⇒ 0.(sω' ≡ [fst s] : [ω.A]) → Dup (Sub A P) s
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of { [s!] ⇒ λ ss' ⇒
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let ω.p : [0.P (fst s)] = revive0 (P (fst s)) (snd s);
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0.ss : s! ≡ fst s : A = boxω-inj A s! (fst s) ss' in
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([(s!, coe (𝑖 ⇒ [0.P (ss @𝑖)]) @1 @0 p)],
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[δ 𝑗 ⇒ [(ss @𝑗, coe (𝑖 ⇒ [0.P (ss @𝑖)]) @1 @𝑗 p)]])
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} ss0
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}}
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def subdup : 0.(A : ★) → 0.(P : A → ★) → 0.(Irr1 A P) →
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((x : A) → Dup A x) →
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(s : Sub A P) → SubDup A P s =
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λ A P pirr dup s ⇒
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case s return s' ⇒ SubDup A P s' of { (x, p) ⇒
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drop0 (P x) (Dup A x) p (dup x)
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}
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def dup : 0.(A : ★) → 0.(P : A → ★) → 0.(Irr1 A P) →
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((x : A) → Dup A x) →
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(s : Sub A P) → Dup (Sub A P) s =
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λ A P pirr dup s ⇒ subdup-to-dup A P pirr s (subdup A P pirr dup s)
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def forget : 0.(A : ★) → 0.(P : A → ★) → Sub A P → A =
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λ A P s ⇒ case s return A of { (x, p) ⇒ drop0 (P x) A p x }
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