96 lines
2.3 KiB
Markdown
96 lines
2.3 KiB
Markdown
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---
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title: records
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date: 2024-06-02
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tags: ['quox (language)']
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...
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n-ary products to replace pairs _and_ boxes, since the nested `case`
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expressions for nested pairs are actually really annoying.
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# syntax
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$$
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\newcommand\CEQ{\operatorname{::=}}
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\begin{aligned}
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A, B, s, t &\CEQ \dotsb
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\mid ꞌ[Δ]
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\mid [𝔇]
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& \text{terms} \\
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e, f &\operatorname{::=} \dotsb
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\mid e\mathord·ℓ
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& \text{eliminations} \\
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𝔅 &\operatorname{::=} \dotsb
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\mid [ \overline{ℓ=x} ] ⇒ s
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& \text{case bodies} \\[1em]
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ℓ &\operatorname{::=} x \mid n & \text{labels ($n ∈ ℕ$)} \\[1em]
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Δ &\operatorname{::=} \overline{π· ℓ=x : A} & \text{telescopes} \\
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𝔇 &\operatorname{::=} \overline{ℓ=s} & \text{list of bindings} \\
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ϕ &\operatorname{::=} \overline{x=e} & \text{substitutions} \\
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\end{aligned}
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$$
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- this __replaces__ the box syntax. pair syntax __still exists__ but
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expands to this. this means that
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$A \times B \times C$ and $A \times (B \times C)$ are no longer equivalent.
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i guess that's fine but it is a change.
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- overlined things are comma separated.
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- in $Δ$, $π$ is optional and defaults to $1$ like everywhere else.
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- the pair syntax
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$(x_0, \dotsc, x_n) : (x_0 : A_0) \times \dotsb \times A_n$
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is now an abbreviation for
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$ꞌ[0=x_0, \dotsc, n=x_n] : [0=x_0: A_0, \dotsc, n: A_n]$.
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- instead of $\operatorname{\mathsf{fst}} e$ and $\operatorname{\mathsf{snd}} e$, it's now $e\mathord.0$
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and $e\mathord.1$ (and $e\mathord.2$, …).
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- "`'[0. 0 : A]`" looks bad. i know.
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- a telescope $Δ$ can be used as a context by ignoring the outer labels $ℓ$.
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# typing
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## telescope checking
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:::rulebox
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$$Γ ⊢ Δ$$
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:::
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$$
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\frac{}{Γ ⊢ •}
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\qquad
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\frac{
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Γ ⊢ Δ \qquad
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Γ, Δ ⊢_0 A ⇐ ★
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}{
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Γ ⊢ Δ, π·ℓ = x : A
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}
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$$
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## binding list checking
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:::rulebox
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$$ Γ ⊢ σ·𝔇 ⇐ Δ ⊳ Σ $$
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:::
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$$ \frac{}{Γ ⊢ σ·• ⇐ • ⊳ 𝟎} $$
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$$
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\frac{
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Γ ⊢ σ·𝔇 ⇐ Δ ⊳ Σ_𝔇 \qquad
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Γ ⊢ ⌊σπ⌋ · s ⇐ A[𝔇:Δ] ⊳ Σ_s
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}{
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Γ ⊢ σ· (𝔇, ℓ = s) ⇐ (Δ, π·ℓ = x : A) ⊳ Σ_𝔇 + πΣ_s
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}
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$$
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:::rulebox
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$$ 𝔇 : Δ = ϕ $$
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:::
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produce a substitution from a binding list and a telescope by distributing the
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type annotations
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$$
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\begin{aligned}
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• : • &= • \\
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(𝔇, x = s) : (Δ, π·ℓ=x : A) &= (𝔇 : Δ), (x = s ∷ A)
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\end{aligned}
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$$
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