ionchy
/
CIS6700-Spring2025
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Spacing and formatting adjustments

This commit is contained in:
Jonathan Chan 2025-04-22 14:13:06 -04:00
parent 4cdd0c233a
commit 481e81d408
3 changed files with 38 additions and 18 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

26
cbpv.mng
View File

@ -40,6 +40,7 @@
\inputott{rules}
\begin{document}
\maketitle
\section{Introduction}
@ -144,6 +145,8 @@ which expands the set from normal forms
to terms which must reduce to normal forms.
The general recipe for defining these for CBPV is as follows:
\clearpage
\begin{itemize}[rightmargin=\leftmargin]
\item The only strongly neutral values are variables.
\item Strongly neutral computations are eliminators
@ -417,9 +420,10 @@ semantically well formed with respect to a context.
These judgements can be built up inductively,
as demonstrated by the below admissible rules.
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[S]{$[[G ⊧ s]]$}{admissible semantic substitution well-formedness}{Nil,Cons}
}
\endgroup
Then we can define semantic typing in terms of the logical relation,
using semantic well formedness of substitutions to handle the context.
@ -442,10 +446,11 @@ the fundamental theorem of soundness of syntactic typing with respect to semanti
then follows directly.
Normalization holds as a corollary.
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[S]{$[[G ⊧ v : A]]$}{admissible semantic value typing}{Var,Unit,Inl,Inr,Thunk}
\drules[S]{$[[G ⊧ m : B]]$}{admissible semantic computation typing}{Force,Lam,App,Ret,Let,Case}
}
\endgroup
\begin{proof}
By construction using prior lemmas,
@ -573,10 +578,11 @@ Proving them additionally requires showing that small-step reduction preserves n
By mutual induction on the single-step reductions.
\end{proof}
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[sn]{$[[v ∈ sn]]$}{admissible strongly normal values}{Var,Unit,Inl,Inr,Thunk}
\drules[sn]{$[[m ∈ sn]]$}{admissible strongly normal computations}{Lam,Ret,Force,App,Let,Case}
}
\endgroup
\begin{proof}
\Rref{sn-App,sn-Let,sn-Case} are proven by double induction
@ -610,9 +616,10 @@ we first need antisubstitution to be able to undo substitutions.
By induction on the derivation of $[[m{x ↦ v} ∈ sn]]$.
\end{proof}
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[sn]{$[[m ∈ sn]]$}{head expansion}{Force-Thunk,App-Lam,Let-Ret,Case-Inl,Case-Inr}
}
\endgroup
\begin{proof}
\Rref{sn-Force-Thunk} holds directly by induction on the premise;
@ -645,12 +652,15 @@ of the structure of both reductions.
then by cases on the derivation of $[[m ⇝ n1]]$.
\end{proof}
\begingroup
\setlength{\abovedisplayskip}{0pt}
\begin{mathpar}
\mprset{fraction={-~}{-~}{-}}
\fbox{$[[m ∈ sn]]$} \hfill \textit{backward closure in head position} \\
\drule[width=0.45\textwidth]{sn-App-bwd}
\drule[width=0.55\textwidth]{sn-Let-bwd}
\end{mathpar}
\endgroup
\begin{proof}
First, use \rref{sn-App-inv1,sn-App-inv2,sn-Let-inv1,sn-Let-inv2}

BIN
cbpv.pdf (Stored with Git LFS)
View File

Binary file not shown.

26
cbpv.tex
View File

@ -40,6 +40,7 @@
\inputott{rules}
\begin{document}
\maketitle
\section{Introduction}
@ -144,6 +145,8 @@ which expands the set from normal forms
to terms which must reduce to normal forms.
The general recipe for defining these for CBPV is as follows:
\clearpage
\begin{itemize}[rightmargin=\leftmargin]
\item The only strongly neutral values are variables.
\item Strongly neutral computations are eliminators
@ -417,9 +420,10 @@ semantically well formed with respect to a context.
These judgements can be built up inductively,
as demonstrated by the below admissible rules.
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[S]{$ \Gamma \vDash \sigma $}{admissible semantic substitution well-formedness}{Nil,Cons}
}
\endgroup
Then we can define semantic typing in terms of the logical relation,
using semantic well formedness of substitutions to handle the context.
@ -442,10 +446,11 @@ the fundamental theorem of soundness of syntactic typing with respect to semanti
then follows directly.
Normalization holds as a corollary.
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[S]{$ \Gamma \vDash \ottnt{v} : \ottnt{A} $}{admissible semantic value typing}{Var,Unit,Inl,Inr,Thunk}
\drules[S]{$ \Gamma \vDash \ottnt{m} : \ottnt{B} $}{admissible semantic computation typing}{Force,Lam,App,Ret,Let,Case}
}
\endgroup
\begin{proof}
By construction using prior lemmas,
@ -573,10 +578,11 @@ Proving them additionally requires showing that small-step reduction preserves n
By mutual induction on the single-step reductions.
\end{proof}
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[sn]{$ \ottnt{v} \in \kw{sn} $}{admissible strongly normal values}{Var,Unit,Inl,Inr,Thunk}
\drules[sn]{$ \ottnt{m} \in \kw{sn} $}{admissible strongly normal computations}{Lam,Ret,Force,App,Let,Case}
}
\endgroup
\begin{proof}
\Rref{sn-App,sn-Let,sn-Case} are proven by double induction
@ -610,9 +616,10 @@ we first need antisubstitution to be able to undo substitutions.
By induction on the derivation of $ \ottnt{m} [ \ottmv{x} \mapsto \ottnt{v} ] \in \kw{sn} $.
\end{proof}
{\mprset{fraction={-~}{-~}{-}}
\begingroup
\mprset{fraction={-~}{-~}{-}}
\drules[sn]{$ \ottnt{m} \in \kw{sn} $}{head expansion}{Force-Thunk,App-Lam,Let-Ret,Case-Inl,Case-Inr}
}
\endgroup
\begin{proof}
\Rref{sn-Force-Thunk} holds directly by induction on the premise;
@ -645,12 +652,15 @@ of the structure of both reductions.
then by cases on the derivation of $ \ottnt{m} \rightsquigarrow \ottnt{n_{{\mathrm{1}}}} $.
\end{proof}
\begingroup
\setlength{\abovedisplayskip}{0pt}
\begin{mathpar}
\mprset{fraction={-~}{-~}{-}}
\fbox{$ \ottnt{m} \in \kw{sn} $} \hfill \textit{backward closure in head position} \\
\drule[width=0.45\textwidth]{sn-App-bwd}
\drule[width=0.55\textwidth]{sn-Let-bwd}
\end{mathpar}
\endgroup
\begin{proof}
First, use \rref{sn-App-inv1,sn-App-inv2,sn-Let-inv1,sn-Let-inv2}