Fix #495.
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Wen Kokke
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4423c37089
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1 changed files with 30 additions and 31 deletions
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@ -1,5 +1,5 @@
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---
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title : "Subtyping: records"
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title : "Subtyping: Records"
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layout : page
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prev : /More/
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permalink : /Subtyping/
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@ -30,7 +30,7 @@ can also have type `B` if `A` is a subtype of `B`.
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⊢<: : ∀{Γ M A B}
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→ Γ ⊢ M ⦂ A
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→ A <: B
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→ A <: B
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-----------
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→ Γ ⊢ M ⦂ B
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@ -58,7 +58,7 @@ written `M # l`, and whose dynamic semantics is defined by the
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following reduction rule, which says that accessing the field `lⱼ`
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returns the value stored at that field.
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{l₁=v₁, ..., lⱼ=vⱼ, ..., lᵢ=vᵢ} # lⱼ —→ vⱼ
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{l₁=v₁, ..., lⱼ=vⱼ, ..., lᵢ=vᵢ} # lⱼ —→ vⱼ
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In this chapter we add records and record types to the simply typed
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lambda calculus (STLC) and prove type safety. It is instructive to see
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@ -138,14 +138,14 @@ Name = String
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A record is traditionally written as follows
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{ l₁ = M₁, ..., lᵢ = Mᵢ }
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so a natural representation is a list of label-term pairs.
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However, we find it more convenient to represent records as a pair of
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vectors (Agda's `Vec` type), one vector of fields and one vector of terms:
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l₁, ..., lᵢ
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M₁, ..., Mᵢ
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This representation has the advantage that the traditional subscript
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notation `lᵢ` corresponds to indexing into a vector.
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@ -244,9 +244,9 @@ If one vector `ns` is a subset of another `ms`, then for any element
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lookup-⊆ : ∀{n m : ℕ}{ns : Vec Name n}{ms : Vec Name m}{i : Fin n}
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→ ns ⊆ ms
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→ Σ[ k ∈ Fin m ] lookup ns i ≡ lookup ms k
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lookup-⊆ {suc n} {m} {x ∷ ns} {ms} {zero} ns⊆ms
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lookup-⊆ {suc n} {m} {x ∷ ns} {ms} {zero} ns⊆ms
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with lookup-∈ (ns⊆ms x (here refl))
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... | ⟨ k , ms[k]=x ⟩ =
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... | ⟨ k , ms[k]=x ⟩ =
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⟨ k , (sym ms[k]=x) ⟩
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lookup-⊆ {suc n} {m} {x ∷ ns} {ms} {suc i} x∷ns⊆ms =
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lookup-⊆ {n} {m} {ns} {ms} {i} (λ x z → x∷ns⊆ms x (there z))
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@ -258,7 +258,7 @@ lookup-⊆ {suc n} {m} {x ∷ ns} {ms} {suc i} x∷ns⊆ms =
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data Type : Set where
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_⇒_ : Type → Type → Type
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`ℕ : Type
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｛_⦂_｝ : {n : ℕ} (ls : Vec Name n) (As : Vec Type n) → .{d : distinct ls} → Type
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｛_⦂_｝ : {n : ℕ} (ls : Vec Name n) (As : Vec Type n) → .{d : distinct ls} → Type
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```
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In addition to function types `A ⇒ B` and natural numbers `ℕ`, we
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@ -366,7 +366,7 @@ Here is the proof of reflexivity, by induction on the size of the type.
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with ty-size-pos {A}
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... | pos rewrite n≤0⇒n≡0 m
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with pos
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... | ()
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... | ()
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<:-refl-aux {suc n}{`ℕ}{m} = <:-nat
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<:-refl-aux {suc n}{A ⇒ B}{m} =
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let A<:A = <:-refl-aux {n}{A}{m+n≤o⇒m≤o _ (≤-pred m) } in
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@ -376,7 +376,7 @@ Here is the proof of reflexivity, by induction on the size of the type.
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where
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G : ls ⦂ As <: ls ⦂ As
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G {i}{j} lij rewrite distinct-lookup-inj (distinct-relevant d) lij =
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let As[i]≤n = ≤-trans (lookup-vec-ty-size {As = As}{i}) (≤-pred m) in
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let As[i]≤n = ≤-trans (lookup-vec-ty-size {As = As}{i}) (≤-pred m) in
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<:-refl-aux {n}{lookup As i}{As[i]≤n}
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```
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The theorem statement uses `n` as an upper bound on the size of the type `A`
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@ -394,7 +394,7 @@ and proceeds by induction on `n`.
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Regarding the latter, we need to show that for any `i` and `j`,
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`lookup ls j ≡ lookup ls i` implies `lookup As j <: lookup As i`.
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Because `lookup` is injective on distinct vectors, we have `i ≡ j`.
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We then use the induction hypothesis to show that
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We then use the induction hypothesis to show that
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`lookup As i <: lookup As i`, noting that the size of
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`lookup As i` is less-than-or-equal to the size of `As`, which
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is one smaller than the size of `｛ ls ⦂ As ｝`.
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@ -421,7 +421,7 @@ The following corollary packages up reflexivity for ease of use.
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where
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As<:Cs : ls ⦂ As <: ns ⦂ Cs
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As<:Cs {i}{j} ls[j]=ns[i]
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with lookup-⊆ {i = i} ns⊆ms
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with lookup-⊆ {i = i} ns⊆ms
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... | ⟨ k , ns[i]=ms[k] ⟩ =
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let As[j]<:Bs[k] = As<:Bs {k}{j} (trans ls[j]=ns[i] ns[i]=ms[k]) in
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let Bs[k]<:Cs[i] = Bs<:Cs {i}{k} (sym ns[i]=ms[k]) in
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@ -450,7 +450,7 @@ The proof is by induction on the derivations of `A <: B` and `B <: C`.
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if we can prove that `lookup ls j ≡ lookup ms k`.
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We already have `lookup ls j ≡ lookup ns i` and we
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obtain `lookup ns i ≡ lookup ms k` by use of the lemma `lookup-⊆`,
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noting that `ns ⊆ ms`.
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noting that `ns ⊆ ms`.
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We can obtain the later, that `lookup Bs k <: lookup Cs i`,
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from `｛ ms ⦂ Bs ｝ <: ｛ ns ⦂ Cs ｝`.
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It suffices to show that `lookup ms k ≡ lookup ns i`,
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@ -527,7 +527,7 @@ The typing judgment takes the form `Γ ⊢ M ⦂ A` and relies on an
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auxiliary judgment `Γ ⊢* Ms ⦂ As` for typing a vector of terms.
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```
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data _⊢*_⦂_ : Context → ∀ {n} → Vec Term n → Vec Type n → Set
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data _⊢*_⦂_ : Context → ∀ {n} → Vec Term n → Vec Type n → Set
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data _⊢_⦂_ : Context → Term → Type → Set where
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@ -607,7 +607,7 @@ chapter. Here we discuss the three new rules.
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* Rule `⊢<:`: (Subsumption) If a term `M` has type `A`, and `A <: B`,
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then term `M` also has type `B`.
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## Renaming and Substitution
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@ -712,7 +712,7 @@ data Value : Term → Set where
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→ Value (`suc V)
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V-rcd : ∀{n}{ls : Vec Name n}{Ms : Vec Term n}
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→ Value ｛ ls := Ms ｝
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→ Value ｛ ls := Ms ｝
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```
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## Reduction
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@ -827,10 +827,10 @@ canonical ⊢zero V-zero = C-zero
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canonical (⊢suc ⊢V) (V-suc VV) = C-suc (canonical ⊢V VV)
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canonical (⊢case ⊢L ⊢M ⊢N) ()
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canonical (⊢μ ⊢M) ()
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canonical (⊢rcd ⊢Ms d) VV = C-rcd {dls = d} ⊢Ms d <:-refl
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canonical (⊢rcd ⊢Ms d) VV = C-rcd {dls = d} ⊢Ms d <:-refl
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canonical (⊢<: ⊢V <:-nat) VV = canonical ⊢V VV
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canonical (⊢<: ⊢V (<:-fun {A}{B}{C}{D} C<:A B<:D)) VV
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with canonical ⊢V VV
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canonical (⊢<: ⊢V (<:-fun {A}{B}{C}{D} C<:A B<:D)) VV
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with canonical ⊢V VV
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... | C-ƛ {N}{A′}{B′}{A}{B} ⊢N AB′<:AB = C-ƛ ⊢N (<:-trans AB′<:AB (<:-fun C<:A B<:D))
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canonical (⊢<: ⊢V (<:-rcd {ks = ks}{ls = ls}{d2 = dls} ls⊆ks ls⦂Ss<:ks⦂Ts)) VV
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with canonical ⊢V VV
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@ -915,8 +915,8 @@ progress (⊢· ⊢L ⊢M)
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... | done VL
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with progress ⊢M
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... | step M—→M′ = step (ξ-·₂ VL M—→M′)
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... | done VM
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with canonical ⊢L VL
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... | done VM
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with canonical ⊢L VL
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... | C-ƛ ⊢N _ = step (β-ƛ VM)
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progress ⊢zero = done V-zero
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progress (⊢suc ⊢M) with progress ⊢M
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@ -950,7 +950,7 @@ progress (⊢<: {A = A}{B} ⊢M A<:B) = progress ⊢M
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* Case `⊢rcd`: we immediately characterize the record as a value.
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* Case `⊢<:`: we invoke the induction hypothesis on sub-derivation of `Γ ⊢ M ⦂ A`.
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## Preservation <a name="subtyping-preservation"></a>
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@ -1047,9 +1047,9 @@ subst-pres : ∀ {Γ Δ σ N A}
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-----------------
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→ Δ ⊢ subst σ N ⦂ A
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subst-pres σ⦂ (⊢` eq) = σ⦂ eq
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subst-pres {σ = σ} σ⦂ (⊢ƛ ⊢N) = ⊢ƛ (subst-pres{σ = exts σ}(exts-pres {σ = σ} σ⦂) ⊢N)
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subst-pres {σ = σ} σ⦂ (⊢· ⊢L ⊢M) = ⊢· (subst-pres{σ = σ} σ⦂ ⊢L) (subst-pres{σ = σ} σ⦂ ⊢M)
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subst-pres {σ = σ} σ⦂ (⊢μ ⊢M) = ⊢μ (subst-pres{σ = exts σ} (exts-pres{σ = σ} σ⦂) ⊢M)
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subst-pres {σ = σ} σ⦂ (⊢ƛ ⊢N) = ⊢ƛ (subst-pres{σ = exts σ}(exts-pres {σ = σ} σ⦂) ⊢N)
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subst-pres {σ = σ} σ⦂ (⊢· ⊢L ⊢M) = ⊢· (subst-pres{σ = σ} σ⦂ ⊢L) (subst-pres{σ = σ} σ⦂ ⊢M)
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subst-pres {σ = σ} σ⦂ (⊢μ ⊢M) = ⊢μ (subst-pres{σ = exts σ} (exts-pres{σ = σ} σ⦂) ⊢M)
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subst-pres σ⦂ (⊢rcd ⊢Ms dls) = ⊢rcd (subst-vec-pres σ⦂ ⊢Ms ) dls
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subst-pres {σ = σ} σ⦂ (⊢# {d = d} ⊢M lif liA) =
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⊢# {d = d} (subst-pres {σ = σ} σ⦂ ⊢M) lif liA
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@ -1097,7 +1097,7 @@ The proof is by induction on the typing derivation.
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* Case `⊢-*-∷`: So we have `∅ ⊢ M ⦂ B` and `∅ ⊢* Ms ⦂ As`. We proceed by cases on `i`.
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* If it is `0`, then lookup yields term `M` and `A ≡ B`, so we conclude that `∅ ⊢ M ⦂ A`.
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* If it is `suc i`, then we conclude by the induction hypothesis.
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@ -1144,7 +1144,7 @@ with cases on `M —→ N`.
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* Case `⊢#` and `ξ-#`: So `∅ ⊢ M ⦂ ｛ ls ⦂ As ｝`, `lookup ls i ≡ l`, `lookup As i ≡ A`,
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and `M —→ M′`. We apply the induction hypothesis to obtain `∅ ⊢ M′ ⦂ ｛ ls ⦂ As ｝`
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and then conclude by rule `⊢#`.
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* Case `⊢#` and `β-#`. We have `∅ ⊢ ｛ ks := Ms ｝ ⦂ ｛ ls ⦂ As ｝`, `lookup ls i ≡ l`,
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`lookup As i ≡ A`, `lookup ks j ≡ l`, and `｛ ks := Ms ｝ # l —→ lookup Ms j`.
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By the canonical forms lemma, we have `∅ ⊢* Ms ⦂ Bs`, `ls ⊆ ks` and `ks ⦂ Bs <: ls ⦂ As`.
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@ -1230,7 +1230,7 @@ The non-algorithmic subtyping rules for variants are
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∀i∈1..u, ∃j∈1..u, kⱼ = lᵢ and Aⱼ = Bᵢ
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----------------------------------------------
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〈k₁=A₁, ..., kᵤ=Aᵤ〉 <: 〈l₁=B₁, ..., lᵤ=Bᵤ〉
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Come up with an algorithmic subtyping rule for variant types.
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#### Exercise `<:-alternative` (stretch)
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@ -1260,7 +1260,7 @@ of the form:
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If S <: { lᵢ : Tᵢ | i ∈ 1..n }, then S ≡ { kⱼ : Sⱼ | j ∈ 1..m }
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and { lᵢ | i ∈ 1..n } ⊆ { kⱼ | j ∈ 1..m }
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and Sⱼ <: Tᵢ for every i and j such that lᵢ = kⱼ.
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and Sⱼ <: Tᵢ for every i and j such that lᵢ = kⱼ.
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## References
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@ -1274,6 +1274,5 @@ of the form:
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* Barbara H. Liskov and Jeannette M. Wing. A Behavioral Notion of
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Subtyping. In ACM Trans. Program. Lang. Syst. Volume 16, 1994.
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* Types and Programming Languages. Benjamin C. Pierce. The MIT Press. 2002.
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* Types and Programming Languages. Benjamin C. Pierce. The MIT Press. 2002.
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