68 lines
1.6 KiB
Markdown
68 lines
1.6 KiB
Markdown
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---
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title: quantity polymorphism (draft)
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date: 2024-04-21
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tags: [quox, computer]
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...
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im getting sick of writing `elim` and `elimω`, and `fold` and `foldω` and—
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so let's call the quantity context, uh, $Ξ$. it's just a list of bound vars.
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quantity arguments have to be full expressions though because arithmetic shows
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up in the typing rules
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## syntax
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:::texdefs
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$$ \newcommand\HL{\textcolor{blueviolet}} $$
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:::
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$$
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\begin{aligned}
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D &::=
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\cdots \mid \mathtt{def} \; \mathsf{a}_{\HL{\overline α}} : A = s
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& \text{declarations} \\
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e &::=
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\cdots \mid \mathsf{a}_{\HL{\overline π}}^{ℓ}
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& \text{eliminations} \\
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π, ρ &::=
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𝔮 \HL{\mid α \mid π + ρ \mid π \cdot ρ}
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& \HL{\text{quantity expressions}} \\[1em]
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\HL{α, β} && \HL{\text{quantity variables}} \\
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𝔮 &::=
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0 \mid 1 \mid ω
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& \text{quantity constants} \\
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\HL{Ξ} &\mathrel{\HL{::=}}
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\HL{\cdot \mid Ξ, α}
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& \HL{\text{quantity contexts}}
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\end{aligned}
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$$
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- a _quantity value_ $π̃$ in a context $Ξ$ is a polynomial with variables from
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$Ξ$ and coefficients from $\{0,1,ω\}$.
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## typing
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:::texdefs
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$$
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\newcommand\FracS[2]{\frac{\;#1\;}{\;#2\;}}
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\newcommand\Many[1]{\begin{gathered}#1\end{gathered}}
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\newcommand\Rule[3][]{\FracS{\Many{#2}}{\Many{#3}}\;\text{\small[#1]}}
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\newenvironment{Rules}
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{\begin{gathered}\newcommand\nl{\\[1em]}}
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{\end{gathered}}
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$$
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:::
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:::rulebox
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$$
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\newcommand\Syn\Rightarrow \newcommand\Chk\Leftarrow
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\begin{gathered}
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\HL{Ξ \mid} Ψ \mid Γ \vdash σ \cdot s \Chk A \rhd Σ \\
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\HL{Ξ \mid} Ψ \mid Γ \vdash σ \cdot e \Syn A \rhd Σ
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\end{gathered}
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$$
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:::
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