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halfway through DeBruijn

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This commit is contained in:
wadler 2018-02-24 16:31:35 +01:00
parent b0226ccb41
commit edfbdd1a0d
1 changed files with 15 additions and 13 deletions
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28
src/extra/DeBruijn.lagda
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@ -57,16 +57,16 @@ env (Γ , A) = env Γ × type A
Church : Type
Church = (o ⇒ o) ⇒ o ⇒ o
plus : Exp ε (Church ⇒ Church ⇒ Church)
plus : ∀ {Γ : Env} → Exp Γ (Church ⇒ Church ⇒ Church)
plus = (abs (abs (abs (abs (app (app (var (S (S (S Z)))) (var (S Z))) (app (app (var (S (S Z))) (var (S Z))) (var Z)))))))
one : Exp ε Church
one : ∀ {Γ : Env} → Exp Γ Church
one = (abs (abs (app (var (S Z)) (var Z))))
two : Exp ε Church
two : ∀ {Γ : Env} → Exp Γ Church
two = (app (app plus one) one)
four : Exp ε Church
four : ∀ {Γ : Env} → Exp Γ Church
four = (app (app plus two) two)
\end{code}
@ -232,18 +232,20 @@ M ∎ = reflexive
ex₁ : (app (abs (var Z)) (abs (var Z))) ⟶* (abs (var Z))
ex₁ =
begin
(app (abs (var Z)) (abs (var Z)))
⟶⟨ β ⟩
(app (abs {Γ = ε} {A = o ⇒ o} (var Z)) (abs (var Z)))
⟶⟨ β Fun ⟩
(abs (var Z))
∎
\end{code}
ex₁ : (app (app plus one) one) ⟶ (abs (abs (app (var (S Z)) (app (var (S Z)) (var Z)))))
ex₁ =
\begin{code}
ex₂ : (app {Γ = ε} (app plus one) one) ⟶* (abs (abs (app (app one (var (S Z))) (app (app one (var (S Z))) (var Z)))))
ex₂ =
begin
(app (app plus one) one)
⟶⟨ ξ_1 β ⟩
(app (abs (abs (abs (app (app one) (var (S Z))) (app (app (var (S (S Z))) (var (S Z))) (var Z))))) one)
⟶⟨ β ⟩
(abs (abs (app (app one) (var (S Z))) (app (app one (var (S Z))) (var Z))))
⟶⟨ ⟩
⟶⟨ ξ₁ (β Fun) ⟩
(app (abs (abs (abs (app (app one (var (S Z))) (app (app (var (S (S Z))) (var (S Z))) (var Z)))))) one)
⟶⟨ β Fun ⟩
(abs (abs (app (app one (var (S Z))) (app (app one (var (S Z))) (var Z)))))
∎
\end{code}