changed index to point to TypedDB
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wadler
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4 changed files with 116 additions and 130 deletions
2
index.md
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index.md
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@ -42,7 +42,7 @@ fixes are encouraged.
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- [Maps: Total and Partial Maps](Maps)
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- [Stlc: The Simply Typed Lambda-Calculus](Stlc)
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- [StlcProp: Properties of STLC](StlcProp)
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- [Scoped: Scoped and Typed DeBruijn representation](Scoped)
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- [TypedDB: Typed DeBruijn representation](TypedDB)
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## Backmatter
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@ -327,8 +327,10 @@ to treat variables as values, and to treat
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`λ[ x ∶ A ] N` as a value only if `N` is a value.
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Indeed, this is how Agda normalises terms.
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Formalising this approach requires a more sophisticated
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definition of substitution, which permits substituting
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closed terms for values.
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definition of substitution. Here we only
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substitute closed terms for variables, while
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the alternative requires the ability to substitute
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open terms for variables.
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## Substitution
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@ -342,11 +344,11 @@ For instance, we have
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⟹
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not · (not · true)
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where we substitute `false` for `` `x `` in the body
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where we substitute `not` for `` `f `` in the body
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of the function abstraction.
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We write substitution as `N [ x := V ]`, meaning
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substitute term `V` for free occurrences of variable `x` in term `V`,
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substitute term `V` for free occurrences of variable `x` in term `N`,
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or, more compactly, substitute `V` for `x` in `N`.
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Substitution works if `V` is any closed term;
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it need not be a value, but we use `V` since we
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@ -637,9 +639,7 @@ reduction₂ =
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∎
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\end{code}
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<!--
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Much of the above, though not all, can be filled in using C-c C-r and C-c C-s.
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-->
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#### Special characters
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@ -861,7 +861,7 @@ or explain why there is no such `A`.
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3. `` ∅ ⊢ λ[ y ∶ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` x · ` y ∶ A ``
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4. `` ∅ , x ∶ A ⊢ λ[ y : 𝔹 ⇒ 𝔹 ] `y · `x : A ``
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For each of the following, give type `A`, `B`, and `C` for which it is derivable,
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For each of the following, give type `A`, `B`, `C`, and `D` for which it is derivable,
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or explain why there are no such types.
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1. `` ∅ ⊢ λ[ y ∶ 𝔹 ⇒ 𝔹 ⇒ 𝔹 ] λ[ x ∶ 𝔹 ] ` y · ` x ∶ A ``
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@ -496,7 +496,7 @@ we show that if `∅ ⊢ V ∶ A` then `Γ ⊢ N [ x := V ] ∶ B`.
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- The remaining cases are similar to application.
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We need a couple of lemmas. A closed term can be weakened to any context, and `just` is injective.
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We need a lemmas stating that a closed term can be weakened to any context.
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\begin{code}
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weaken-closed : ∀ {V A Γ} → ∅ ⊢ V ∶ A → Γ ⊢ V ∶ A
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weaken-closed {V} {A} {Γ} ⊢V = context-lemma Γ~Γ′ ⊢V
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@ -1,7 +1,7 @@
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---
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title : "TypedDB: Typed DeBruijn representation"
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layout : page
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permalink : /Scoped
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permalink : /TypedDB
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---
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## Imports
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@ -13,18 +13,16 @@ module TypedDB where
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\begin{code}
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import Relation.Binary.PropositionalEquality as Eq
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open Eq using (_≡_; refl; sym; trans; cong)
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-- open Eq.≡-Reasoning
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open import Data.Empty using (⊥; ⊥-elim)
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open import Data.Nat using (ℕ; zero; suc; _+_; _∸_)
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open import Data.Product using (_×_; proj₁; proj₂; ∃; ∃-syntax) renaming (_,_ to ⟨_,_⟩)
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open import Data.Sum using (_⊎_; inj₁; inj₂)
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open import Data.Product using (_×_) renaming (_,_ to ⟨_,_⟩)
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open import Data.Unit using (⊤; tt)
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open import Function using (_∘_)
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open import Function.Equivalence using (_⇔_; equivalence)
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open import Relation.Nullary using (¬_; Dec; yes; no)
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open import Relation.Nullary.Decidable using (map)
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open import Relation.Nullary.Negation using (contraposition)
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open import Relation.Nullary.Product using (_×-dec_)
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open import Data.Unit using (⊤; tt)
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open import Data.Empty using (⊥; ⊥-elim)
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open import Function using (_∘_)
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open import Function.Equivalence using (_⇔_; equivalence)
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\end{code}
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@ -77,43 +75,6 @@ data _⊢_ : Env → Type → Set where
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Γ ⊢ B
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\end{code}
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## Writing variables as naturals
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\begin{code}
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postulate
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impossible : ⊥
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_≟Tp_ : ∀ (A B : Type) → Dec (A ≡ B)
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conv : ∀ {Γ} {A} → ℕ → Γ ∋ A
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conv {ε} = ⊥-elim impossible
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conv {Γ , B} (suc n) = S (conv n)
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conv {Γ , A} {B} zero with A ≟Tp B
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... | yes refl = Z
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... | no A≢B = ⊥-elim impossible
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⌈_⌉ : ∀ {Γ} {A} → ℕ → Γ ⊢ A
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⌈ n ⌉ = ⌊ conv n ⌋
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\end{code}
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## Test examples
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\begin{code}
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Ch : Type
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Ch = (o ⇒ o) ⇒ o ⇒ o
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plus : ∀ {Γ} → Γ ⊢ Ch ⇒ Ch ⇒ Ch
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plus = ƛ ƛ ƛ ƛ (⌊ S S S Z ⌋ · ⌊ S Z ⌋ · (⌊ S S Z ⌋ · ⌊ S Z ⌋ · ⌊ Z ⌋))
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two : ∀ {Γ} → Γ ⊢ Ch
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two = ƛ ƛ (⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋))
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four : ∀ {Γ} → Γ ⊢ Ch
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four = ƛ ƛ (⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋))))
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four′ : ∀ {Γ} → Γ ⊢ Ch
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four′ = plus · two · two
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\end{code}
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## Decide whether environments and types are equal
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\begin{code}
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@ -141,6 +102,44 @@ _≟_ : ∀ (Γ Δ : Env) → Dec (Γ ≡ Δ)
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\end{code}
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## Writing variables as naturals
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This turned out to be a lot harder than expected. Probably not a good idea.
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\begin{code}
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postulate
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impossible : ⊥
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conv : ∀ {Γ} {A} → ℕ → Γ ∋ A
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conv {ε} = ⊥-elim impossible
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conv {Γ , B} (suc n) = S (conv n)
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conv {Γ , A} {B} zero with A ≟T B
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... | yes refl = Z
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... | no A≢B = ⊥-elim impossible
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var : ∀ {Γ} {A} → ℕ → Γ ⊢ A
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var n = ⌊ conv n ⌋
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\end{code}
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## Test examples
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\begin{code}
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Ch : Type
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Ch = (o ⇒ o) ⇒ o ⇒ o
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plus : ∀ {Γ} → Γ ⊢ Ch ⇒ Ch ⇒ Ch
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plus = ƛ ƛ ƛ ƛ ⌊ S S S Z ⌋ · ⌊ S Z ⌋ · (⌊ S S Z ⌋ · ⌊ S Z ⌋ · ⌊ Z ⌋)
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two : ∀ {Γ} → Γ ⊢ Ch
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two = ƛ ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋)
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four : ∀ {Γ} → Γ ⊢ Ch
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four = ƛ ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋)))
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four′ : ∀ {Γ} → Γ ⊢ Ch
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four′ = plus · two · two
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\end{code}
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# Denotational semantics
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\begin{code}
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@ -202,22 +201,22 @@ substitute N M = subst (ext⊢ ⌊_⌋ M) N
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## Normal
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\begin{code}
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data Normal : ∀ {Γ} {A} → Γ ⊢ A → Set
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data Normal : ∀ {Γ} {A} → Γ ⊢ A → Set
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data Neutral : ∀ {Γ} {A} → Γ ⊢ A → Set
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data Normal where
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Fun : ∀ {Γ} {A B} {N : Γ , A ⊢ B} → Normal N → Normal (ƛ N)
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Neu : ∀ {Γ} {A} {M : Γ ⊢ A} → Neutral M → Normal M
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ƛ_ : ∀ {Γ} {A B} {N : Γ , A ⊢ B} → Normal N → Normal (ƛ N)
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⌈_⌉ : ∀ {Γ} {A} {M : Γ ⊢ A} → Neutral M → Normal M
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data Neutral where
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Var : ∀ {Γ} {A} → {n : Γ ∋ A} → Neutral ⌊ n ⌋
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App : ∀ {Γ} {A B} → {L : Γ ⊢ A ⇒ B} {M : Γ ⊢ A} → Neutral L → Normal M → Neutral (L · M)
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⌊_⌋ : ∀ {Γ} {A} → (n : Γ ∋ A) → Neutral ⌊ n ⌋
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_·_ : ∀ {Γ} {A B} → {L : Γ ⊢ A ⇒ B} {M : Γ ⊢ A} → Neutral L → Normal M → Neutral (L · M)
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\end{code}
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## Reduction step
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\begin{code}
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infix 2 _⟶_
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infix 2 _⟶_
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data _⟶_ : ∀ {Γ} {A} → Γ ⊢ A → Γ ⊢ A → Set where
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@ -247,43 +246,28 @@ data _⟶_ : ∀ {Γ} {A} → Γ ⊢ A → Γ ⊢ A → Set where
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\begin{code}
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infix 2 _⟶*_
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data _⟶*_ : ∀ {Γ} {A} → Γ ⊢ A → Γ ⊢ A → Set where
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reflexive : ∀ {Γ} {A} {M : Γ ⊢ A} →
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--------
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M ⟶* M
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inclusion : ∀ {Γ} {A} {M N : Γ ⊢ A} →
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M ⟶ N →
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---------
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M ⟶* N
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transitive : ∀ {Γ} {A} {L M N : Γ ⊢ A} →
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L ⟶* M →
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M ⟶* N →
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----------
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L ⟶* N
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\end{code}
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## Displaying reduction sequences
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\begin{code}
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infix 1 begin_
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infixr 2 _⟶⟨_⟩_
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infix 3 _∎
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data _⟶*_ : ∀ {Γ} {A} → Γ ⊢ A → Γ ⊢ A → Set where
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_∎ : ∀ {Γ} {A} (M : Γ ⊢ A) →
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-------------
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M ⟶* M
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_⟶⟨_⟩_ : ∀ {Γ} {A} (L : Γ ⊢ A) {M N : Γ ⊢ A} →
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L ⟶ M →
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M ⟶* N →
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---------
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L ⟶* N
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begin_ : ∀ {Γ} {A} {M N : Γ ⊢ A} → (M ⟶* N) → (M ⟶* N)
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begin M⟶*N = M⟶*N
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_⟶⟨_⟩_ : ∀ {Γ} {A} (L : Γ ⊢ A) {M N : Γ ⊢ A} → (L ⟶ M) → (M ⟶* N) → (L ⟶* N)
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L ⟶⟨ L⟶M ⟩ M⟶*N = transitive (inclusion L⟶M) M⟶*N
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_∎ : ∀ {Γ} {A} (M : Γ ⊢ A) → M ⟶* M
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M ∎ = reflexive
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\end{code}
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## Example reduction sequence
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## Example reduction sequences
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\begin{code}
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id : ∀ (A : Type) → ε ⊢ A ⇒ A
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@ -292,65 +276,67 @@ id A = ƛ ⌊ Z ⌋
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_ : id (o ⇒ o) · id o ⟶* id o
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_ =
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begin
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id (o ⇒ o) · id o
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⟶⟨ β (Fun (Neu Var)) ⟩
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id o
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(ƛ ⌊ Z ⌋) · (ƛ ⌊ Z ⌋)
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⟶⟨ β (ƛ ⌈ ⌊ Z ⌋ ⌉) ⟩
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ƛ ⌊ Z ⌋
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∎
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Nmplus : Normal (plus {ε})
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Nmplus = Fun (Fun (Fun (Fun (Neu (App (App Var (Neu Var)) (Neu (App (App Var (Neu Var)) (Neu Var))))))))
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Nmtwo : Normal (two {ε})
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Nmtwo = Fun (Fun (Neu (App Var (Neu (App Var (Neu Var))))))
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_ : four′ {ε} ⟶* four {ε}
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_ =
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begin
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plus · two · two
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⟶⟨ ξ₁ (β Nmtwo) ⟩
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(ƛ ƛ ƛ (two · ⌊ S Z ⌋ · (⌊ S S Z ⌋ · ⌊ S Z ⌋ · ⌊ Z ⌋))) · two
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⟶⟨ β Nmtwo ⟩
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(ƛ ƛ (two · ⌊ S Z ⌋ · (two · ⌊ S Z ⌋ · ⌊ Z ⌋)))
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⟶⟨ ζ (ζ (ξ₁ (β (Neu Var)))) ⟩
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(ƛ ƛ ((ƛ (⌊ S S Z ⌋ · (⌊ S S Z ⌋ · ⌊ Z ⌋))) · (two · ⌊ S Z ⌋ · ⌊ Z ⌋)))
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⟶⟨ ζ (ζ (ξ₂ (Fun (Neu (App Var (Neu (App Var (Neu Var)))))) (ξ₁ (β (Neu Var))))) ⟩
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(ƛ ƛ ((ƛ (⌊ S S Z ⌋ · (⌊ S S Z ⌋ · ⌊ Z ⌋))) · ((ƛ (⌊ S S Z ⌋ · (⌊ S S Z ⌋ · ⌊ Z ⌋))) · ⌊ Z ⌋)))
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⟶⟨ ζ (ζ (ξ₂ (Fun (Neu (App Var (Neu (App Var (Neu Var)))))) (β (Neu Var)))) ⟩
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(ƛ ƛ ((ƛ (⌊ S S Z ⌋ · (⌊ S S Z ⌋ · ⌊ Z ⌋))) · (⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋))))
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⟶⟨ ζ (ζ (β (Neu (App Var (Neu (App Var (Neu Var))))))) ⟩
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(ƛ ƛ (⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋)))))
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⟶⟨ ξ₁ (β (ƛ (ƛ ⌈ ⌊ S Z ⌋ · ⌈ ⌊ S Z ⌋ · ⌈ ⌊ Z ⌋ ⌉ ⌉ ⌉))) ⟩
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(ƛ ƛ ƛ two · ⌊ S Z ⌋ · (⌊ S (S Z) ⌋ · ⌊ S Z ⌋ · ⌊ Z ⌋)) · two
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⟶⟨ ξ₁ (ζ (ζ (ζ (ξ₁ (β ⌈ ⌊ S Z ⌋ ⌉))))) ⟩
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(ƛ ƛ ƛ (ƛ ⌊ S (S Z) ⌋ · (⌊ S (S Z) ⌋ · ⌊ Z ⌋)) · (⌊ S (S Z) ⌋ · ⌊ S Z ⌋ · ⌊ Z ⌋)) · two
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⟶⟨ ξ₁ (ζ (ζ (ζ (β ⌈ (⌊ S (S Z) ⌋ · ⌈ ⌊ S Z ⌋ ⌉) · ⌈ ⌊ Z ⌋ ⌉ ⌉)))) ⟩
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(ƛ ƛ ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S (S Z) ⌋ · ⌊ S Z ⌋ · ⌊ Z ⌋))) · two
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⟶⟨ β (ƛ (ƛ ⌈ ⌊ S Z ⌋ · ⌈ ⌊ S Z ⌋ · ⌈ ⌊ Z ⌋ ⌉ ⌉ ⌉)) ⟩
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ƛ ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · ((ƛ (ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋))) · ⌊ S Z ⌋ · ⌊ Z ⌋))
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⟶⟨ ζ (ζ (ξ₂ ⌈ ⌊ S Z ⌋ ⌉ (ξ₂ ⌈ ⌊ S Z ⌋ ⌉ (ξ₁ (β ⌈ ⌊ S Z ⌋ ⌉))))) ⟩
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ƛ ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · ((ƛ ⌊ S (S Z) ⌋ · (⌊ S (S Z) ⌋ · ⌊ Z ⌋)) · ⌊ Z ⌋))
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⟶⟨ ζ (ζ (ξ₂ ⌈ ⌊ S Z ⌋ ⌉ (ξ₂ ⌈ ⌊ S Z ⌋ ⌉ (β ⌈ ⌊ Z ⌋ ⌉)))) ⟩
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ƛ ƛ ⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · (⌊ S Z ⌋ · ⌊ Z ⌋)))
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∎
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\end{code}
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## Progress
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\begin{code}
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progress : ∀ {Γ} {A} → (M : Γ ⊢ A) → (∃[ N ] (M ⟶ N)) ⊎ Normal M
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progress ⌊ x ⌋ = inj₂ (Neu Var)
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data Progress {Γ A} (M : Γ ⊢ A) : Set where
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step : ∀ (N : Γ ⊢ A) → M ⟶ N → Progress M
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done : Normal M → Progress M
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progress : ∀ {Γ} {A} → (M : Γ ⊢ A) → Progress M
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progress ⌊ x ⌋ = done ⌈ ⌊ x ⌋ ⌉
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progress (ƛ N) with progress N
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progress (ƛ N) | inj₁ ⟨ N′ , r ⟩ = inj₁ ⟨ ƛ N′ , ζ r ⟩
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progress (ƛ V) | inj₂ NmV = inj₂ (Fun NmV)
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progress (ƛ N) | step N′ r = step (ƛ N′) (ζ r)
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progress (ƛ V) | done NmV = done (ƛ NmV)
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progress (L · M) with progress L
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progress (L · M) | inj₁ ⟨ L′ , r ⟩ = inj₁ ⟨ L′ · M , ξ₁ r ⟩
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progress (V · M) | inj₂ NmV with progress M
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progress (V · M) | inj₂ NmV | inj₁ ⟨ M′ , r ⟩ = inj₁ ⟨ V · M′ , ξ₂ NmV r ⟩
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progress (V · W) | inj₂ (Neu NeV) | inj₂ NmW = inj₂ (Neu (App NeV NmW))
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progress ((ƛ V) · W) | inj₂ (Fun NmV) | inj₂ NmW = inj₁ ⟨ substitute V W , β NmW ⟩
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progress (L · M) | step L′ r = step (L′ · M) (ξ₁ r)
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progress (V · M) | done NmV with progress M
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progress (V · M) | done NmV | step M′ r = step (V · M′) (ξ₂ NmV r)
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progress (V · W) | done ⌈ NeV ⌉ | done NmW = done ⌈ NeV · NmW ⌉
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progress ((ƛ V) · W) | done (ƛ NmV) | done NmW = step (substitute V W) (β NmW)
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\end{code}
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|
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## Normalise
|
||||
|
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\begin{code}
|
||||
{-
|
||||
ex₃ : progress (app (app plus one) one) ≡
|
||||
inj₁ ⟨ (app (abs (abs (abs (app (app Γone (var (S Z))) (app (app (var (S (S Z))) (var (S Z))) (var Z)))))) Γone) , ξ₁ (β Fun) ⟩
|
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ex₃ = refl
|
||||
data Normalise {Γ A} (M : Γ ⊢ A) : Set where
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out-of-gas : Normalise M
|
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normal : ∀ (N : Γ ⊢ A) → Normal N → M ⟶* N → Normalise M
|
||||
|
||||
ex₄ : progress (app (abs (abs (abs (app (app Γone (var (S Z))) (app (app (var (S (S Z))) (var (S Z))) (var Z)))))) Γone) ≡
|
||||
inj₁ ⟨ (abs (abs (app (app Γone (var (S Z))) (app (app Γone (var (S Z))) (var Z))))) , β Fun ⟩
|
||||
ex₄ = refl
|
||||
|
||||
ex₅ : progress (abs (abs (app (app Γone (var (S Z))) (app (app Γone (var (S Z))) (var Z))))) ≡ inj₂ Fun
|
||||
ex₅ = refl
|
||||
|
||||
-}
|
||||
normalise : ∀ {Γ} {A} → ℕ → (M : Γ ⊢ A) → Normalise M
|
||||
normalise zero L = out-of-gas
|
||||
normalise (suc n) L with progress L
|
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... | done NmL = normal L NmL (L ∎)
|
||||
... | step M L⟶M with normalise n M
|
||||
... | out-of-gas = out-of-gas
|
||||
... | normal N NmN M⟶*N = normal N NmN (L ⟶⟨ L⟶M ⟩ M⟶*N)
|
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\end{code}
|
||||
|
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|
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|
|
|
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Loading…
Reference in a new issue