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alg-lin.tex
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alg-lin.tex
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\documentclass[reqno]{amsart}
\usepackage{amssymb}
\usepackage{hyperref}
\usepackage{mathtools}
\usepackage[all]{xy}
\usepackage{verbatim}
\usepackage{ifthen}
\usepackage{xargs}
\usepackage{bussproofs}
\usepackage{turnstile}
\usepackage{etex}
\usepackage{todonotes}
\hypersetup{colorlinks=true,linkcolor=blue}
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\theoremstyle{definition}
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\begin{document}
\title{Algebraic Linear Type Theories}
\author{Valery Isaev}
\begin{abstract}
\end{abstract}
\maketitle
\makeatletter
\providecommand\@dotsep{5}
\makeatother
\listoftodos\relax
\section{Introduction}
\section{Linear non-dependent type theories}
Let $\mathbb{L}_0$ be a partial Horn theory with sorts $\{ ty \} \cup \{ (tm,n)\ |\ n \in \mathbb{N} \}$,
function symbols $ctx_{n,i} : tm_n \to ty$, $1 \leq i \leq n$, and $ty : tm \to ty$, and no axioms.
Let $\mathbb{L}_1$ be a theory under $\mathbb{L}_0$ with the following additional function symbols:
\begin{align*}
v & : ty \to (tm,1) \\
subst_{n_1, \ldots n_k} & : (tm,k) \times (tm,n_i)^k \to (tm, n_1 + \ldots + n_k)
\end{align*}
As in the case of algebraic dependent type theories we can omit the first argument of $v$.
We will write $a : A$ for formula $ty(a) = A$ and $A_1, \ldots A_n \vdash a : A$ for formula $ty(a) = A \land ctx_{n,i}(a) = A_i$.
Theory $\mathbb{L}_1$ has the following axioms:
\medskip
\begin{center}
\AxiomC{}
\UnaryInfC{$A \vdash v : A$}
\DisplayProof
\qquad
\AxiomC{$A_1, \ldots A_k \vdash b : B$}
\AxiomC{$\Delta_i \vdash a_i : A_i$}
\BinaryInfC{$\Delta_1, \ldots \Delta_k \vdash subst_{n_1, \ldots n_k}(b, a_1, \ldots a_k) : B$}
\DisplayProof
\end{center}
\medskip
\begin{align*}
subst_{n_1, \ldots n_k}(b, a_1, \ldots a_k)\!\downarrow\ & \sststile{}{b, a_1, \ldots a_k} ctx_{k,1}(b) = ty(a_1) \land \ldots \land ctx_{k,k}(b) = ty(a_k) \\
& \sststile{}{a} subst_n(v, a) = a \\
& \sststile{}{b, A} subst_{1, \ldots 1}(b, v(A), \ldots v(A)) = b
\end{align*}
\medskip
\begin{center}
\AxiomC{$B_1, \ldots B_m \vdash c : C$}
\AxiomC{$A^i_1, \ldots A^i_{k_i} \vdash b_i : B_i$}
\AxiomC{$\Delta^i_{j_i} \vdash a^i_{j_i} : A^i_{j_i}$}
\TrinaryInfC{$subst_{n^1_1, \ldots n^1_{k_1}, \ldots n^m_1, \ldots n^m_{k_m}}(c', a^1_1, \ldots a^1_{k_1}, \ldots a^m_1, \ldots a^m_{k_m}) =
subst_{N_1, \ldots N_m}(c, b_1', \ldots b_m')$}
\DisplayProof
\end{center}
where $c' = subst_{k_1, \ldots k_m}(c, b_1, \ldots b_m)$, $b_i' = subst_{n^i_1, \ldots n^i_{k_i}}(b_i, a^i_1, \ldots a^i_{k_i})$ and $N_i = n^i_1 + \ldots + n^i_{k_i}$.
The category of models of $\mathbb{L}_1$ is isomorphic to the category of small multicategories.
\bibliographystyle{amsplain}
\bibliography{ref}
\end{document}