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zi@FSdKoH6ttF$GAz5T1Nxr(NKlnpbk2*pVadGV)Ums;)B66#Tv0@KXrlAlb18q`;mt%G7y8FwrN)p9z5*$E+Ga-esmc!6qx6G<~Cv<1$!UyAj zkt|Gge<&0ZoYTQhlyW9;L z=ts}-GC806-n?p3>f6+Cp6pw?#$iLNGl~z%Q(3izuDXh5PYLUF3n2~@f(lMZatn*O zcGj0%B(LXDVV+}$FOUmyS}i7|9GCV>Y;9SdR$TTN0Yz4udVFX3G`co5!1u!Uk?a== arO|z<^cZSDAQWk0jKiRzaJa3r9rSO0@|EHM diff --git a/prover/coq_rules/coq_rules.tex b/prover/coq_rules/coq_rules.tex index 7d3f06841..c5cfa3ac3 100644 --- a/prover/coq_rules/coq_rules.tex +++ b/prover/coq_rules/coq_rules.tex @@ -343,8 +343,8 @@ \subsection{Conversion Rules} \end{center} In addition to the above convertibility rules, we also allow identifying a term $t$ of type $\cfa{x}{t}{U}$ with its \emph{$\eta$-expansion} $\clm{x}{T}{(t x)}$ -for $x$ fresh in $t$. Note \emph{$\eta$-reduction} is deliberately not defined -(TODO: show example?). +for $x$ fresh in $t$. Note \emph{$\eta$-reduction} is deliberately not defined. +% TODO: show example? \begin{notation} We write $E[\Gamma] \vdash t \red u$ for the contextual closure of the rules @@ -365,7 +365,22 @@ \subsection{Conversion Rules} \[ E[\Gamma] \vdash t_1 \red \ldots \red u_1 \text{ and } E[\Gamma] \vdash t_2 \red \ldots \red u_2 \] and either $u_1$ and $u_2$ are identical up irrelevant subterms, or they are convertible up to $\eta$-exxpansion. We denote this as -$E[\Gamma] \vdash t_1 =_{\beta\delta\iota\zeta\eta} t_2$ +$E[\Gamma] \vdash t_1 \conveq t_2$ \end{definition} +\subsection{Subtyping Rules} +The \emph{subtyping} relation is inductively defined as follows: +% TODO: display as proof rules? +\begin{itemize} +\item if $E[\Gamma] \vdash t \conveq u$, then $E[\Gamma] \vdash t \subty u$ +\item if $i \leq j$, then $E[\Gamma] \vdash \Type(i) \subty \Type(j)$ +\item $E[\Gamma] \vdash \Set \subty \Type(i)$ for all $i$ +\item $E[\Gamma] \vdash \Prop \subty \Set$ +\item if $E[\Gamma] \vdash T \conveq U$ and + $E[\Gamma \dcln (\lasm{x}{T})] \vdash T' \subty U'$, then + $E[\Gamma] \vdash \cfa{x}{T}{T'} \subty \cfa{x}{U}{U'}$ +\end{itemize} + +\subsection{Conversion/Subtyping: Polymorphic Universes} + \end{document} diff --git a/prover/coq_rules/newcommands.tex b/prover/coq_rules/newcommands.tex index 46d901cfb..2afec7ea3 100644 --- a/prover/coq_rules/newcommands.tex +++ b/prover/coq_rules/newcommands.tex @@ -35,3 +35,5 @@ \newcommand{\sd}[1]{{{\color{JungleGreen} #1}}} \newcommand{\red}{\triangleright} +\newcommand{\conveq}{=_{\beta\delta\iota\zeta\eta}} +\newcommand{\subty}{\leq_{\beta\delta\iota\zeta\eta}}