From 74f33591f06d45827eb3d2ee543579fb1c7bae0b Mon Sep 17 00:00:00 2001 From: Lucas Pena Date: Fri, 24 Apr 2020 15:43:22 -0500 Subject: [PATCH] coq_rules: more on conversion rules --- prover/coq_rules/coq_rules.pdf | Bin 193706 -> 196476 bytes prover/coq_rules/coq_rules.tex | 30 ++++++++++++++++++++++++++---- prover/coq_rules/newcommands.tex | 2 +- prover/ext/k | 2 +- 4 files changed, 28 insertions(+), 6 deletions(-) diff --git a/prover/coq_rules/coq_rules.pdf b/prover/coq_rules/coq_rules.pdf index b4d0dd3d7d410fdfd00925b3ac200d9b34c8004a..4d3b943a7f703ffa34587f8e92844e0f823eaff2 100644 GIT binary patch delta 64060 zcmV(#K;*xw=?nb+3y_F^g}WFG1~c7XcYob8OT2ow!mGc0g-?9`>GG@dS46CcLX-%; zy1ZUdf{`G4C8b8hK+|^hCi`}`J6khMvzLdnHI>_(u!9*wvLB9{y!Nd> zol%v+*zot?XM(6~pWD$_LLfu!cnona!*uVea_1kT{Du9ZGKwj-4@_im{)&l}Mn+1v zegfmk6+D8d(qXu(x~cL_Gxs=;9+O)E9DneovCTBAkGHpZRsLRtZn+aUd}(e9-{Y>b zPnzs-?Qc{;_He6h3VoT^l6G2xtMQY0U6*GhtDD@WlO9&EYrXGy+c&B#i+PV9uETNuA8ug*8es(&i> z&3RQ^-{$-CUA}*J%-G}q-lzz3!Q 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+$\prftree{E[\Gamma] \vdash ((\clm{x}{T}{t}) u) \red_{\beta} \sbs{t}{x}{u}}$& We say that $\sbs{t}{x}{u}$ is the \emph{$\beta$-contraction} of $((\clm{x}{T}{t}) u)$, and that $((\clm{x}{T}{t}) u)$ is the \emph{$\beta$-expansion} of $\sbs{t}{x}{u}$ \\ @@ -323,20 +323,20 @@ \subsection{Conversion Rules} $\prftree {\wf(E)[\Gamma]} {(\ldef{x}{t}{T}) \in \Gamma} -{E[\Gamma] \vdash x \red{\Delta} t}$& +{E[\Gamma] \vdash x \red_{\Delta} t}$& Reducing variable defined in local context \\ \prule{Delta-Local}& $\prftree {\wf(E)[\Gamma]} {(\gldef{c}{t}{T}) \in E} -{E[\Gamma] \vdash c \red{\delta} t}$& +{E[\Gamma] \vdash c \red_{\delta} t}$& Reducing constant defined in global context \\ \prule{Zeta}& $\prftree {\wf(E)[\Gamma]} {E[\Gamma] \vdash \lasm{u}{U}} {E[\Gamma \dcln (\ldef{x}{u}{U})] \vdash \lasm{t}{T}} -{E[\Gamma] \vdash \clt{x}{u}{U}{t} \red{\zeta} \sbs{t}{x}{x}}$& +{E[\Gamma] \vdash \clt{x}{u}{U}{t} \red_{\zeta} \sbs{t}{x}{x}}$& Remove local definitions occurring in terms \\ \end{longtable} \egroup @@ -346,4 +346,26 @@ \subsection{Conversion Rules} 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 +defined above. That is, $t$ reduces to $u$ with global environment $E$ and local +context $\Gamma$ with one of the previous reductions $\beta, \Delta, \delta, +\iota,$ or $\zeta$. +\end{notation} + +\begin{definition} +Two terms are called \emph{irrelevant} if they share a common type of a strict +proposition $\gasm{A}{\SProp}$. Irrelevant terms can be identified. +\end{definition} + +\begin{definition} +Two terms $t_1, t_2$ are called \emph{$\beta\delta\iota\zeta\eta$-convertible}, or +\emph{convertible}, or \emph{equivalent} in global environment $E$ and local +context $\Gamma$ iff there exists $t_1, t_2$ such that +\[ E[\Gamma] \vdash t_1 \red \ldots \red u_1 \text{ and } E[\Gamma] \vdash t_2 \red \ldots \red u_2 \] +and either $u_1$ and $u_2$ are identical up irrelevant subterms, or they are +convertible up to $\eta$-exxpansion. We denote this as +$E[\Gamma] \vdash t_1 =_{\beta\delta\iota\zeta\eta} t_2$ +\end{definition} + \end{document} diff --git a/prover/coq_rules/newcommands.tex b/prover/coq_rules/newcommands.tex index 08f522d46..46d901cfb 100644 --- a/prover/coq_rules/newcommands.tex +++ b/prover/coq_rules/newcommands.tex @@ -34,4 +34,4 @@ \newcommand{\prule}[1]{\textsc{(#1)}} \newcommand{\sd}[1]{{{\color{JungleGreen} #1}}} -\newcommand{\red}[1]{\triangleright_{#1}} +\newcommand{\red}{\triangleright} diff --git a/prover/ext/k b/prover/ext/k index 0d2819ac8..8ebd956aa 160000 --- a/prover/ext/k +++ b/prover/ext/k @@ -1 +1 @@ -Subproject commit 0d2819ac873a6c9a448de4da3060b1ba1e682bb2 +Subproject commit 8ebd956aa1e344ed1d2e0ce15b22b251435839c3