\documentclass{article}
\usepackage{ifthen}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}    \theoremstyle{plain}
\usepackage{stmaryrd}
\usepackage{proof}
\usepackage{array}
\usepackage[usenames,dvipsnames]{color}
\usepackage{graphicx}
\usepackage{mathpartir}

\textwidth=13cm
\hoffset=-1cm
\parskip=0.5ex

\newtheorem{theorem}{Theorem}[section]   %-cd
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{property}[theorem]{Property}

%% Unnumbered variants
\newtheorem*{lemma*}{Lemma}

%%% Logical operators
\newcommand{\True}{\top}              % TRUTH
\newcommand{\False}{\bot}             % FALSEHOOD
\newcommand{\Not}{\lnot}              % NEGATION
\renewcommand{\And}{\mathrel{\land}}  % CONJUNCTION
\newcommand{\Or}{\mathrel{\lor}}      % DISJUNCTION
\newcommand{\Imp}{\mathrel{\supset}}  % IMPLICATION
\newcommand{\Pim}{\mathrel{\subset}}  % REVERSE IMPLICATION
\newcommand{\sE}{\exists}             % EXISTENTIAL QUANTIFIER
\newcommand{\E}[2]{\sE #1.\, #2}
\newcommand{\sA}{\forall}             % UNIVERSAL QUANTIFIER
\newcommand{\A}[2]{\sA #1.\, #2}


%%%%%%%%%%%%%%%%%%%%% TOOLS FOR BUILDING DERIVATIONS %%%%%%%%%%%%%%%%%%%%%

\newcommand{\rname}[1]{\ensuremath{\mathsf{#1}}}
\newcommand{\rn}[1]{\mbox{$\sf \scriptstyle #1$}}

%\def\lowerhalf#1{\hbox{\raise -0.8\baselineskip\hbox{#1}}}
%\def\lowerhalf#1{\hbox{\raise -1.0\baselineskip\hbox{#1}}}
\def\alignRuleX#1#2{\hbox{\raise -#1\baselineskip\hbox{#2}}}
%% Invertible rules
\inferLineSkip=3.5pt
\newcommand{\iRule}[3][]
 {\infer=[\rn{#1}]{#3}{#2}}
% {\lowerhalf{\infer=[\rn{#1}]{#3}{#2}}}
\newcommand{\Rule}[3][]
 {\infer[\!\!\rn{#1}]{#3}{#2}}
% {\lowerhalf{\infer[\!\!\rn{#1}]{#3}{#2}}}
\newcommand{\alignRule}[2][1.0]{\alignRuleX{#1}{#2}}

% COMMAND:     \q, \qq
% SYNOPSIS:    \q
%              \qq
% DESCRIPTION: Various spaces to separate the branches in rules with more
%              than one premiss.

\newcommand{\q}
 {\quad\qquad}
\newcommand{\qq}
 {\quad}


% COMMAND:     \stack
% SYNOPSIS:    \stack[sp]{topExpr}{botExpr}
% DESCRIPTION: Puts the expression "topExpr" on top of "botExpr" with a
%              spacing of "sp" (the default is 1ex).  This is used mainly to
%              put a derivation placeholder on top of a judgement.

\newcommand{\stack}[3][1ex]
 {\begin{array}[b]{@{}c@{}}
     {#2} \\[#1] {#3}
  \end{array}}

\renewenvironment{proof}
 {\par\noindent\textbf{Proof}\ \par\nopagebreak}
 {\hfill$\Box$\par\medskip}

%% Environment to define judgments
\newenvironment{judgment}
 {$$
  \begin{array}{l>{$\em }l<{$}}}
 {\end{array}
  $$\\[-\parsep]}

%% Environment to define grammars
\newenvironment{grammar}
  {$$
   \arraycolsep=0.25em
   \newcommand{\sep}{\hspace{0.5em}|\hspace{0.5em}}
   \begin{array}[t]{@{}>{$\em }r<{$}@{\hspace{0.8em}} rr l>{$\em }l<{$}@{}}}
  {\end{array}
   $$\\[-\parsep]}


%%%%%%%%%%%%%%%%%%%%% Judgements %%%%%%%%%%%%%%%%%%%%%

%% Context shortcuts
\newcommand{\G}{\Gamma}
\newcommand{\D}{\Delta}
\renewcommand{\S}{\Sigma}
\renewcommand{\O}{\Omega}


\newcommand{\FV}[1]{\mathrm{FV}(#1)}
\newcommand{\m}[2][]{\ensuremath{\mathcal{#2}#1}}
\let\iNoDot\i
\renewcommand{\i}[1]{\ensuremath{\mathit{#1}}}




%%%%%%%%%%%%%%%%%%%%% LOCAL DECLARATIONS %%%%%%%%%%%%%%%%%%%%%

%% Natural deduction
\newcommand{\nd}[3][\S]{#2 \,\vdash_{#1}\, #3}    % Natural deduction
\newcommand{\seq}[3][\S]                          % Sequent calculus
 {#2 \Longrightarrow_{#1} #3}

\newcommand{\tm}[2][\S]{\vdash_{#1}\, #2}         % Valid term


% Local soundness and completeness
\newcommand{\lsound}[3][0.7]{\alignRule[#1]{#2} \qq\Longrightarrow_R\qq #3}
\newcommand{\lcompl}[3][0.7]{#2 \qq\Longrightarrow_E\qq \alignRule[#1]{#3}}

%% Canonical/atomic derivations
\newcommand{\can}[3][\S]{#2 \,\vdash_{#1}\, \mathop{\Uparrow} #3}
\newcommand{\atm}[3][\S]{#2 \,\vdash_{#1}\, \mathop{\downarrow} #3}
\newcommand{\cant}[4][\S]{#2 \,\vdash_{#1} #3 \Uparrow #4}
\newcommand{\atmt}[4][\S]{#2 \,\vdash_{#1} #3 \downarrow #4}


%% Proof terms
\newcommand{\Triv}{\langle\rangle}
\newcommand{\Fst}[1]{\mathsf{fst}\:#1}
\newcommand{\Snd}[1]{\mathsf{snd}\:#1}
\newcommand{\Pair}[2]{\langle{#1},{#2}\rangle}
\newcommand{\Abort}[2][C]{\mathsf{abort}^{#1}\:#2}
\newcommand{\Case}[5]
 {\mathsf{case}\:#1\:%
    \mathsf{of}\:\mathsf{inl}\:#2\Rightarrow #3
    \mid         \mathsf{inr}\:#4\Rightarrow #5}
\newcommand{\Inl}[2][B]{\mathsf{inl}^{#1}\:#2}
\newcommand{\Inr}[2][A]{\mathsf{inr}^{#1}\:#2}
\newcommand{\App}[2]{#1\:#2}
\newcommand{\Lam}[2]{\lambda #1.\,#2}
\newcommand{\Fix}[2]{\mathsf{fix}\:#1.\,#2}
\newcommand{\Let}[3]{\mathsf{let}\:#1=#2 \;\mathsf{in}\: #3}

\newcommand{\NF}[1]{\mathsf{NF}(#1)}

%%% Arithmetic
\newcommand{\nat}{\mathsf{nat}}
\newcommand{\z}{0}
\newcommand{\s}[1]{\mathsf{s}\:#1}
\newcommand{\Rec}[3]{\mathsf{rec}(#1, #2, #3)}


%%% Simplified sequent calculus
\newcommand{\sseq}[3][\S]                          % Sequent calculus
 {#2 \longrightarrow_{#1} #3}

%%% Inversion calculus
\newcommand{\rlseqfboxsep}{2pt}
\newcommand{\rlseqfboxrule}{0.5pt}
\newcommand{\f}[2][Gray]
 {\fboxsep=\rlseqfboxsep%
  \fboxrule=\rlseqfboxrule%
  \fcolorbox{#1}{White}{$#2$}}
\newcommand{\df}[2][Gray]
 {\dashdash=1pt%
  \dashlength=3pt%
  \fboxsep=\rlseqfboxsep%
  \fboxrule=0.5pt%
  {\color{#1}\dbox{\color{Black}$#2$}}}

\newcommand{\Fg}[1]{\f[Green]{#1}}  %% Inversion focus

\newcommand{\iseq}[4][\S]{#2 ; #3 \longrightarrow_{#1} #4}

\newcommand{\rseq}[4][\S]
 {#2 ; #3 \longrightarrow_{#1} \Fg{#4}}
\newcommand{\lseq}[4][\S]
 {#2 ; \Fg{#3} \longrightarrow_{#1} #4}

\newcommand{\coloredbox}[3][White]{
  \bgroup%
  \fboxsep=2.5pt%
  \fcolorbox{#1}{#2}{$#3$}%
  \egroup}

\definecolor{PaleGreen}{RGB}{192,251,192}
\newcommand{\stable}[1]{\coloredbox{PaleGreen}{#1}}
\definecolor{LightSalmon}{RGB}{255,208,188}
\newcommand{\chosen}[1]{\coloredbox{LightSalmon}{#1}}

\newcommand{\absinvseq}[3][\S]
 {#2 \xrightarrow{\scriptscriptstyle\mathrm{ABS}}_{#1} #3}
\newcommand{\invseq}[3][\S]
 {#2 \xrightarrow{\scriptscriptstyle\mathrm{INV}}_{#1} #3}
\newcommand{\choiceseq}[3][\S]
 {#2 \xrightarrow{\scriptscriptstyle\mathrm{CHOOSE}}_{#1} #3}

\newcommand{\sidebox}[3][0pt]
 {\fboxsep=0pt%
  \fboxrule=#1%
  \framebox[#2]{#3}}

%%% Contraction free sequent calculus
\newcommand{\size}[1]{|#1|}
\newcommand{\mset}[1]{\Lbag #1 \Rbag}


%%% Chaining
\newcommand{\F}[1]{\f[Red]{#1}}
\newcommand{\Fd}[1]{\df[Red]{#1}}
\newcommand{\chseq}[3][\S]
 {#2 \xrightarrow{\scriptscriptstyle\mathrm{CH}}_{#1} #3}

%%% Backward logic programming
% Goal sequencing
\newcommand{\gsseq}[4][\S]{\sseq[#1]{#2}{#3 + #4}}
% Backtracking
\newcommand{\Bsseq}[2]{#1 {\color{blue}\;|\; #2}}
\newcommand{\bck}[1][\G']{#1 \triangleright}
\newcommand{\okchoice}[2][\G]{#2 \;\mathsf{ok}_{#1}}
\newcommand{\porSep}[1][-1.8ex]
{\qq\raisebox{#1}[0pt][0pt]{\rule{0.4pt}{4.5ex}}\qq}


%%% Linear Logic
\newcommand{\sTop}{\top}               % ADDITIVE TRUTH
\newcommand{\Top}
 {\sTop}
\newcommand{\sZero}{{\bf 0}}           % ADDITIVE FALSEHOOD
\newcommand{\Zero}
 {\sZero}
\newcommand{\sWith}                    % ADDITIVE CONJUCTION
% {\&}
 {\binampersand}
\newcommand{\With}
 {\mathop{\sWith}}
\newcommand{\sPlus}{\oplus}            % ADDITIVE DISJUNCTION
\newcommand{\Plus}
 {\mathop{\sPlus}}
\newcommand{\sWave}{\leadsto}          % ADDITIVE IMPLICATION
\newcommand{\Wave}
 {\mathop{\sWave}}
\newcommand{\sOne}{{\bf 1}}            % MULTIPLICATIVE TRUTH
\newcommand{\One}
 {\sOne}
\newcommand{\sBot}{\bot}               % MULTIPLICATIVE FALSEHOOD
\newcommand{\Bot}
 {\sBot}
\newcommand{\sTimes}{\otimes}          % MULTIPLICATIVE CONJUCTION
\newcommand{\Times}
 {\mathop{\sTimes}}
\newcommand{\sPar}                     % MULTIPLICATIVE DISJUNCTION
% {\#}
 {\bindnasrepma}
\newcommand{\Par}
 {\mathop{\sPar}}
\newcommand{\sLolli}                   % MULTIPLICATIVE IMPLICATION
 {\mbox{$-\!\circ$}}
\newcommand{\Lolli}
 {\mathop{\sLolli}}
\newcommand{\sNil}{{^\bot}}            % LINEAR NEGATION
\newcommand{\Nil}
 {\sNil}
\newcommand{\sBang}{!}                 % EXPONENTIAL "OF COURSE"
\newcommand{\Bang}
 {\mathopen{\sBang}}
\newcommand{\sWhyNot}{?}               % EXPONENTIAL "WHY NOT"
\newcommand{\WhyNot}
 {\mathopen{\sWhyNot}}

\newcommand{\sSmallLolli}              % SCRIPTSIZE MULTIPLICATIVE IMPLICATION
 {\mbox{$\scriptstyle -\!\circ$}}
\newcommand{\SmallLolli}
 {\mathop{\sSmallLolli}}
\newcommand{\btimes}
 {\mbox{$\sBang\sTimes$}}
\newcommand{\BTimes}
 {\mathop{\btimes}}




\newcommand{\NUM}{xx}         % Homework number
\newcommand{\NAME}{Anonymous} % Student name
\title{\textbf{15-317 Fall 2015\\Homework \NUM}}
\author{by \NAME}







\newcommand{\TM}[2]{\textcolor{blue}{#1} : #2}
\begin{document}
\maketitle

% Proof terms
\begin{figure}[p]
\newcommand{\sep}{\;\;|\;\;}
\begin{center}
\newcommand{\rsp}{4ex}
\newcommand{\rspl}{-.5ex}
\newcommand{\hhfill}{\hfill}
\fboxsep=0pt

$A \;\;::=\;\; a \sep \True \sep A \And B \sep \False \sep A \Or B
  \sep A \Imp B \sep \A{x}{A} \sep \E{x}{A}$

\bigskip

\fbox{%
$\begin{array}{@{\hfill}c@{\hfill}}
\makebox[\textwidth]{}\\[-1ex]
  \Rule[id]{}
   {\nd[]{\G, \TM{x}{A}}{\TM{x}{A}}}
\\[\rsp]
  \text{\em (No \rname{\True_E})}
\hfill
  \Rule[\True_I]{}
   {\nd[]{\G}{\TM{\Triv}{\True}}}
\\[\rsp]
  \Rule[\And_{E1}]
   {\nd[]{\G}{\TM{M}{A \And B}}}
   {\nd[]{\G}{\TM{\Fst{M}}{A}}}
\hfill
  \Rule[\And_{E2}]
   {\nd[]{\G}{\TM{M}{A \And B}}}
   {\nd[]{\G}{\TM{\Snd{M}}{B}}}
\hfill
  \Rule[\And_I]
   {\nd[]{\G}{\TM{M}{A}}
   &\nd[]{\G}{\TM{N}{B}}}
   {\nd[]{\G}{\TM{\Pair{M}{N}}{A \And B}}}
\\[\rsp]
  \Rule[\False_E]
   {\nd[]{\G}{\TM{M}{\False}}}
   {\nd[]{\G}{\TM{\Abort{M}}{C}}}
\hfill
  \text{\em (No \rname{\False_I})}
\\[\rsp]
  \Rule[\Or_E]
   {\nd[]{\G}{\TM{M}{A \Or B}}
   &\nd[]{\G, \TM{x}{A}}{\TM{N}{C}}
   &\nd[]{\G, \TM{y}{B}}{\TM{O}{C}}}
   {\nd[]{\G}{\TM{\Case{M}{x}{N}{y}{O}}{C}}}
\\[\rsp]
  \Rule[\Or_{I1}]
   {\nd[]{\G}{\TM{M}{A}}}
   {\nd[]{\G}{\TM{\Inl{M}}{A \Or B}}}
\hfill
  \Rule[\Or_{I2}]
   {\nd[]{\G}{\TM{M}{B}}}
   {\nd[]{\G}{\TM{\Inr{M}}{A \Or B}}}
\\[\rsp]
  \Rule[\Imp_E]
   {\nd[]{\G}{\TM{M}{A \Imp B}}
   &\nd[]{\G}{\TM{N}{A}}}
   {\nd[]{\G}{\TM{\App{M}{N}}{B}}}
\hfill
  \Rule[\Imp_I]
   {\nd[]{\G, \TM{x}{A}}{\TM{M}{B}}}
   {\nd[]{\G}{\TM{\Lam{x}{M}}{A \Imp B}}}
\\[\rsp]
  \Rule[\sA_E]
   {\nd{\G}{\TM{M}{\A{x}{A}}}
   &\tm{t}}
   {\nd{\G}{\TM{\App{M}{t}}{[t/x]A}}}
\hfill
  \Rule[\sA_I]
   {\nd[\S,x]{\G}{\TM{M}{A}}}
   {\nd{\G}{\TM{\Lam{x}{M}}{\A{x}{A}}}}
\\[\rsp]
  \Rule[\sE_E]
   {\nd{\G}{\TM{M}{\E{x}{A}}}
   &\nd[\S,x]{\G,\TM{y}{A}}{\TM{N}{C}}}
   {\nd{\G}{\TM{\Let{\Pair{x}{y}}{M}{N}}{C}}}
\hfill
  \Rule[\sE_I]
   {\tm{t}
   &\nd{\G}{\TM{M}{[t/x]A}}}
   {\nd{\G}{\TM{\Pair{t}{M}}{\E{x}{A}}}}
\\[-1ex]\relax
\end{array}$}
\end{center}
%\caption{Natural Deduction with Proof Terms}
\end{figure}

\clearpage

\begin{figure}[p]
\newcommand{\sep}{\;\;|\;\;}
\begin{center}
\newcommand{\rsp}{4ex}
\newcommand{\rspl}{-.5ex}
\newcommand{\hhfill}{\hfill}
\fboxsep=0pt

$A \;\;::=\;\; a \sep \True \sep A \And B \sep \False \sep A \Or B
  \sep A \Imp B \sep \A{x}{A} \sep \E{x}{A}$

\bigskip

\fbox{%
\renewcommand{\TM}[2]{#2}
$\begin{array}{@{\hfill}c@{\hfill}}
\makebox[\textwidth]{}\\[-1ex]
  \Rule[id]{}
   {\atm[]{\G, \TM{x}{A}}{\TM{x}{A}}}
\\[\rsp]
  \text{\em (No \rname{\True_E})}
\hfill
  \Rule[\And_{E1}]
   {\atm[]{\G}{\TM{M}{A \And B}}}
   {\atm[]{\G}{\TM{\Fst{M}}{A}}}
\hfill
  \Rule[\And_{E2}]
   {\atm[]{\G}{\TM{M}{A \And B}}}
   {\atm[]{\G}{\TM{\Snd{M}}{B}}}
\\[\rsp]
  \Rule[\False_E]
   {\atm[]{\G}{\TM{M}{\False}}}
   {\can[]{\G}{\TM{\Abort{M}}{C}}}
\hfill
  \Rule[\Or_E]
   {\atm[]{\G}{\TM{M}{A \Or B}}
   &\can[]{\G, \TM{x}{A}}{\TM{N}{C}}
   &\can[]{\G, \TM{y}{B}}{\TM{O}{C}}}
   {\can[]{\G}{\TM{\Case{M}{x}{N}{y}{O}}{C}}}
\\[\rsp]
  \Rule[\Imp_E]
   {\atm[]{\G}{\TM{M}{A \Imp B}}
   &\can[]{\G}{\TM{N}{A}}}
   {\atm[]{\G}{\TM{\App{M}{N}}{B}}}
\\[\rsp]
  \Rule[\sA_E]
   {\atm{\G}{\TM{M}{\A{x}{A}}}
   &\tm{t}}
   {\atm{\G}{\TM{\App{M}{t}}{[t/x]A}}}
\hfill
  \Rule[\sE_E]
   {\atm{\G}{\TM{M}{\E{x}{A}}}
   &\can[\S,x]{\G,\TM{y}{A}}{\TM{N}{C}}}
   {\can{\G}{\TM{\Let{\Pair{x}{y}}{M}{N}}{C}}}
\\[\rsp]
  \Rule[\downarrow\Uparrow]
   {\atm{\G}{\TM{M}{a}}}
   {\can{\G}{\TM{M}{a}}}
\\[\rsp]
  \Rule[\True_I]{}
   {\can[]{\G}{\TM{\Triv}{\True}}}
\hfill
  \Rule[\And_I]
   {\can[]{\G}{\TM{M}{A}}
   &\can[]{\G}{\TM{N}{B}}}
   {\can[]{\G}{\TM{\Pair{M}{N}}{A \And B}}}
\\[\rsp]
  \text{\em (No \rname{\False_I})}
\hfill
  \Rule[\Or_{I1}]
   {\can[]{\G}{\TM{M}{A}}}
   {\can[]{\G}{\TM{\Inl{M}}{A \Or B}}}
\hfill
  \Rule[\Or_{I2}]
   {\can[]{\G}{\TM{M}{B}}}
   {\can[]{\G}{\TM{\Inr{M}}{A \Or B}}}
\\[\rsp]
  \Rule[\Imp_I]
   {\can[]{\G, \TM{x}{A}}{\TM{M}{B}}}
   {\can[]{\G}{\TM{\Lam{x}{M}}{A \Imp B}}}
\\[\rsp]
  \Rule[\sA_I]
   {\can[\S,x]{\G}{\TM{M}{A}}}
   {\can{\G}{\TM{\Lam{x}{M}}{\A{x}{A}}}}
\hfill
  \Rule[\sE_I]
   {\tm{t}
   &\can{\G}{\TM{M}{[t/x]A}}}
   {\can{\G}{\TM{\Pair{t}{M}}{\E{x}{A}}}}
\\[-1ex]\relax
\end{array}$}
\end{center}
%\caption{Natural Deduction with Proof Terms}
\end{figure}


\clearpage

\begin{figure}[p]
\newcommand{\sep}{\;\;|\;\;}
\begin{center}
\newcommand{\rsp}{4ex}
\newcommand{\rspl}{-.5ex}
\newcommand{\hhfill}{\hfill}
\fboxsep=0pt


%\renewcommand{\seq}[3][\S]{#2 \Longrightarrow #3}
$A \;\;::=\;\; a \sep \True \sep A \And B \sep \False \sep A \Or B \sep A \Imp B
 \sep \A{x}{A} \sep \E{x}{A}
$

\bigskip

\fbox{%
\renewcommand{\TM}[2]{#2}
$\begin{array}{@{\hfill}c@{\hfill}}
\makebox[\textwidth]{}\\[-1ex]
  \Rule[init]{}
   {\seq{\G,a}{a}}
\\[\rsp]
  \Rule[\And_{L1}]
   {\seq{\G, A \And B, A}{C}}
   {\seq{\G, A \And B}{C}}
\hfill
  \Rule[\And_{L2}]
   {\seq{\G, A \And B, B}{C}}
   {\seq{\G, A \And B}{C}}
\hfill
  \Rule[\And_R]
   {\seq{\G}{A}
   &\seq{\G}{B}}
   {\seq{\G}{A \And B}}
\\[\rsp]
\text{\em (no \rname{\True_L})}
\hfill
  \Rule[\True_R]{}
   {\seq{\G}{\True}}
\\[\rsp]
  \Rule[\Or_L]
   {\seq{\G, A \Or B, A}{C}
   &\seq{\G, A \Or B, B}{C}}
   {\seq{\G, A \Or B}{C}}
% \hfill\hfill\hfill\hfill
% \\[\rsp]
% \hfill\hfill\hfill\hfill
\hfill
  \Rule[\Or_{R1}]
   {\seq{\G}{A}}
   {\seq{\G}{A \Or B}}
\hfill
  \Rule[\Or_{R2}]
   {\seq{\G}{B}}
   {\seq{\G}{A \Or B}}
\\[\rsp]
  \Rule[\False_L]{}
   {\seq{\G,\False}{C}}
\hfill
\text{\em (no \rname{\False_R})}
\\[\rsp]
  \Rule[\Imp_L]
   {\seq{\G, A \Imp B}{A}
   &\seq{\G, A \Imp B, B}{C}}
   {\seq{\G, A \Imp B}{C}}
\hfill
  \Rule[\Imp_R]
   {\seq{\G, A}{B}}
   {\seq{\G}{A \Imp B}}
\\[\rsp]
  \Rule[\sA_L]
   {\tm{t}
   &\seq{\G,\A{x}{A},[t/x]A}{C}}
   {\seq{\G,\A{x}{A}}{C}}
\hfill
  \Rule[\sA_R]
   {\seq[\S,x]{\G}{A}}
   {\seq{\G}{\A{x}{A}}}
\\[\rsp]
  \Rule[\sE_L]
   {\seq[\S,x]{\G,\E{x}{A}, A}{C}}
   {\seq{\G,\E{x}{A}}{C}}
\hfill
  \Rule[\sE_R]
   {\tm{t}
   &\seq{\G}{[t/x]A}}
   {\seq{\G}{\E{x}{A}}}
\\[-1ex]\relax
\end{array}$}
\end{center}
%\caption{Sequent Calculus}
\end{figure}



\end{document}




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