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simplyfying some inference rules by removing unnecessary contexts #823

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simplyfying some inference rules by removing unnecessary contexts #823

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36 changes: 18 additions & 18 deletions formal.tex
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Expand Up @@ -612,8 +612,8 @@ \subsection{Type universes}
%
\begin{mathpar}
\inferrule*[right=\UU-\textsc{intro}]
{\wfctx \Gamma }
{\oftp\Gamma{\UU_i}{\UU_{i+1}}}
{ }
{\oftp{}{\UU_i}{\UU_{i+1}}}
\and
\inferrule*[right=\UU-\textsc{cumul}]
{\oftp\Gamma{A}{\UU_i}}
Expand Down Expand Up @@ -792,8 +792,8 @@ \subsection{The empty type \texorpdfstring{$\emptyt$}{0}}

\begin{mathparpagebreakable}
\inferrule*[right=$\emptyt$-\rform]
{\wfctx\Gamma}
{\oftp\Gamma\emptyt{\UU_i}}
{ }
{\oftp{}\emptyt{\UU_i}}
\and
\inferrule*[right=$\emptyt$-\relim]
{\oftp{\Gamma,\tmtp x\emptyt}{C}{\UU_i} \\ \oftp\Gamma{a}{\emptyt}}
Expand All @@ -811,12 +811,12 @@ \subsection{The unit type \texorpdfstring{$\unit$}{1}}

\begin{mathparpagebreakable}
\inferrule*[right=$\unit$-\rform]
{\wfctx\Gamma}
{\oftp\Gamma\unit{\UU_i}}
{ }
{\oftp{}\unit{\UU_i}}
\and
\inferrule*[right=$\unit$-\rintro]
{\wfctx\Gamma}
{\oftp\Gamma{\ttt}{\unit}}
{ }
{\oftp{}{\ttt}{\unit}}
\and
\inferrule*[right=$\unit$-\relim]
{\oftp{\Gamma,\tmtp x\unit}{C}{\UU_i} \\
Expand Down Expand Up @@ -851,12 +851,12 @@ \subsection{The natural number type}
\oftp{\Gamma,\tmtp{x}\N,\tmtp y C}{c_s}{C[\suc(x)/x]}}
%
\inferrule*[right=$\N$-\rform]
{\wfctx\Gamma}
{\oftp\Gamma{\N}{\UU_i}}
{ }
{\oftp{}{\N}{\UU_i}}
\and
\inferrule*[right=$\N$-\rintro${}_1$]
{\wfctx\Gamma}
{\oftp\Gamma{0}{\N}}
{ }
{\oftp{}{0}{\N}}
\and
\inferrule*[right=$\N$-\rintro${}_2$]
{\oftp\Gamma{n}{\N}}
Expand Down Expand Up @@ -1007,16 +1007,16 @@ \subsection{The circle}

\begin{mathparpagebreakable}
\inferrule*[right=$\Sn^1$-\rform]
{\wfctx\Gamma}
{\oftp\Gamma{\Sn^1}{\UU_i}}
{ }
{\oftp{}{\Sn^1}{\UU_i}}
\and
\inferrule*[right=$\Sn^1$-\rintro${}_1$]
{\wfctx\Gamma}
{\oftp\Gamma{\base}{\Sn^1}}
{ }
{\oftp{}{\base}{\Sn^1}}
\and
\inferrule*[right=$\Sn^1$-\rintro${}_2$]
{\wfctx\Gamma}
{\oftp\Gamma{\lloop}{\id[\Sn^1]{\base}{\base}}}
{ }
{\oftp{}{\lloop}{\id[\Sn^1]{\base}{\base}}}
\and
\inferrule*[right=$\Sn^1$-\relim]
{\oftp{\Gamma,\tmtp x{\Sn^1}}{C}{\UU_i} \\
Expand Down