Skip to main content\(\newcommand{\powerset}[1]{\mathcal{P}(#1)}
\def\domain{\operatorname{dom}}
\def\codomain{\operatorname{cod}}
\def\funimage{\operatorname{img}}
\def\defeq{\coloneqq}
\def\eqdef{\eqqcolon}
\def\syndef{\Coloneqq}
\def\simp{\Rightarrow}
\newcommand{\edist}{\mathit{edist}}
\newcommand{\rd}{\$d}
\newcommand{\rs}{\$s}
\newcommand{\rt}{\$t}
\newcommand{\lo}{\mathit{lo}}
\newcommand{\hi}{\mathit{hi}}
\newcommand{\pc}{\mathit{pc}}
\newcommand{\shamt}{\mathit{n}}
\newcommand{\offset}{i}
\newcommand{\marke}{\mathit{label}}
\newcommand{\addr}{\mathit{addr}}
\newcommand{\imm}{\mathit{imm}}
\newcommand{\sext}{\mathrm{se}}
\newcommand{\zext}{\mathrm{ze}}
\newcommand{\sei}{\sext(i)}
\newcommand{\zei}{\zext(i)}
\newcommand{\bft}[2]{[#1:#2]}
\newcommand{\ite}[3]{\mathop{\mathit{if}}\,{#1}\mathrel{\mathit{then}}{#2}\mathrel{\mathit{ else }}{#3}}
\newcommand{\cbranch}{\mathit{cbr}}
\newcommand{\Nat} {\mathbb{N}}
\newcommand{\Int} {\mathbb{Z}}
\newcommand{\Bit} {\mathbb{B}}
\newcommand{\bbase} {B}
\newcommand{\bnine} {A}
\newcommand{\bneg} [1] {\overline{#1}}
\newcommand{\band} {\mathrel{\&}}
\newcommand{\bor} {\mathrel{|}}
\newcommand{\bxor} {\mathrel{\hat{\phantom{+}}}}
\newcommand{\badd} {\mathrel{+}}
\newcommand{\blts} {\mathrel{\lt_s}}
\newcommand{\bltu} {\mathrel{\lt_u}}
\newcommand{\baddn} {\mathrel{+_n}}
\newcommand{\bsubn} {\mathrel{-_n}}
\newcommand{\bsll} {\mathrel{\ll}}
\newcommand{\bsrl} {\mathrel{\gg}}
\newcommand{\bsra} {\mathrel{\overset s\gg}}
\newcommand{\bconcat} {\,}
\newcommand{\bseqq} [2] {{#1}_{#2}\dots {#1}_0}
\newcommand{\bseq} [2] {{#1}_{#2-1}\dots {#1}_0}
\newcommand{\bsintp} [1] {[#1]}
\newcommand{\buintp} [1] {\langle #1\rangle}
\newcommand{\buintpb}[2] {\buintp{#1}_{#2}}
\def\Var{\mathit{Var}}
\def\Val{\mathit{Val}}
\def\Addr{\mathit{Addr}}
\def\Expr{\mathit{Expr}}
\def\LExpr{\mathit{LExpr}}
\def\ExprS{\mathit{ExprS}}
\def\Stmt{\mathit{Stmt}}
\def\Prg{\mathit{Prg}}
\def\Conf{\mathit{Conf}}
\def\Ty{\mathit{Ty}}
\def\Assert{\mathit{Assert}}
\newcommand{\TWhile}{\mathtt{While}}
\newcommand{\TIf}{\mathtt{If}}
\newcommand{\TBlock}{\mathtt{Block}}
\newcommand{\TAssign}{\mathtt{Assign}}
\newcommand{\TAbort}{\mathtt{Abort}}
\newcommand{\TVar}{\mathtt{Var}}
\newcommand{\TConst}{\mathtt{Const}}
\newcommand{\TBinary}{\mathtt{Binary}}
\newcommand{\TAddrOf}{\mathtt{Addr}}
\newcommand{\TIndir}{\mathtt{Indir}}
\newcommand{\TSeq}{\mathtt{Seq}}
\newcommand{\TTerm}{\mathtt{Term}}
\newcommand{\TCrash}{\mathtt{Crash}}
\newcommand{\AIf}[3]{\TIf\left[#1,#2,#3\right]}
\newcommand{\AWhile}[2]{\TWhile\left[#1,#2\right]}
\newcommand{\ABlock}[2]{\TBlock_{#1}\left[#2\right]}
\newcommand{\AAssign}[2]{\TAssign\left[#1,#2\right]}
\newcommand{\AAbort}{\TAbort}
\newcommand{\ASeq}[2]{\TSeq\left[#1,#2\right]}
\newcommand{\ATerm}{\TTerm}
\newcommand{\ACrash}{\TCrash}
\newcommand{\AVar}[1]{\mathtt{Var}_{#1}}
\newcommand{\ALVal}[1]{\mathtt{LVal}\left[#1\right]}
\newcommand{\AConst}[1]{\TConst_{#1}}
\newcommand{\ABinary}[3]{\TBinary_{#1}\left[#2,#3\right]}
\newcommand{\AAddrOf}[1]{\TAddrOf\left[#1\right]}
\newcommand{\AIndir}[1]{\TIndir\left[#1\right]}
\newcommand{\CIf}[3]{\mathtt{if}\,(#1)\,#2\,\mathtt{else}\,#3}
\newcommand{\CWhile}[2]{\mathtt{while}\,(#1)\,#2}
\newcommand{\CWhileInv}[3]{\mathtt{while}\,(#1)\,\mathtt{\_Inv}(#3)\,#2}
\newcommand{\CBlock}[1]{\{#1\}}
\newcommand{\CAssign}[2]{#1=#2;}
\newcommand{\CAbort}{\mathtt{abort();}}
\newcommand{\CBinary}[3]{#2\mathop{#1}#3}
\newcommand{\CAddrOf}[1]{\}
\newcommand{\CIndir}[1]{*#1}
\newcommand{\CSeq}[2]{{#1}\ {#2}}
\newcommand{\CTerm}{\varepsilon}
\newcommand{\CCrash}{\Lightning}
\newcommand{\CConcat}[2]{#1\mathop{+\!+}#2}
\newcommand{\CAssert}[1]{\mathtt{assert}(#1)}
\newcommand{\CAssume}[1]{\mathtt{assume}(#1)}
\newcommand{\CPtr}[1]{#1*}
\newcommand{\CAssnVV}[2]{\CAssign{\CVar #1}{\CVar #2}}
\newcommand{\CAssnVC}[2]{\CAssign{\CVar #1}{\CConst #2}}
\newcommand{\CSeqThree}[3]{\CSeq{#1}{\CSeq{#2}{#3}}}
\newcommand{\CSeqFour}[4]{\CSeq{#1}{\CSeqThree{#2}{#3}{#4}}}
\newcommand{\COp}[1]{\mathbin{\mathtt{#1}}}
\newcommand{\CVar}[1]{\mathtt{#1}}
\newcommand{\CConst}[1]{\mathtt{#1}}
\newcommand{\CPrg}[1]{\texttt{#1}}
\newcommand{\CInt}{\mathtt{int}}
\newcommand{\CChar}{\mathtt{char}}
\newcommand{\CVoid}{\mathtt{void}}
\newcommand{\tptr}[1]{#1\texttt{*}}
\def\pto{\rightharpoonup}
\def\steprel{\rightarrow}
\def\terminates{\mathrel{\downarrow}}
\newcommand\config[2]{\langle#1\mid #2\rangle}
\newcommand\configt[1]{\config{\CTerm}{#1}}
\def\castrel{\leftrightarrow}
\newcommand{\stmtjudge}[2]{{#1}\mathrel{\vdash}{#2}}
\newcommand{\stmtGamma}[1]{\stmtjudge{\Gamma}{#1}}
\newcommand{\typejudge}[3]{{#1}\mathrel{\vdash}{#2}\mathrel{:}{#3}}
\newcommand{\typeGamma}[2]{\typejudge{\Gamma}{#1}{#2}}
\newcommand{\codeE}[5]{\mathit{#1}\enspace #2\enspace #3\enspace #4 = #5}
\newcommand{\codeR}[3]{\codeE{codeR}{\mathit{offs}}{#1}{#2}{#3}}
\newcommand{\codeL}[3]{\codeE{codeL}{\mathit{offs}}{#1}{#2}{#3}}
\newcommand{\codeSO}[3]{\mathit{codeS}\enspace #1\enspace #2 = #3}
\newcommand{\codeS}[2]{\mathit{codeS}\enspace \mathit{offs}\enspace #1 = #2}
\newcommand{\codePO}[3]{\mathit{codeP}\enspace #1\enspace #2 = #3}
\newcommand{\codeP}[2]{\mathit{codeP}\enspace \mathit{offs}\enspace #1 = #2}
\newcommand{\mreg}[1]{\mathtt{\$#1}}
\newcommand{\lcc}{\mathbin{::}}
\def\rsl{r_1\lcc rs}
\def\rsls{r_1\lcc r_2\lcc rs}
\def\rlist{r\lcc{rs}}
\ifdefined\isLatex
\def\adra{\bigtriangleup}
\def\adrb{\pentagon}
\def\adrc{\diamond}
\def\adrd{\bigtriangledown}
\def\adre{\bigstar}
\def\adrf{\male}
\def\adrg{\female}
\let\oldLightning\Lightning
\renewcommand{\Lightning}{\text{\oldLightning}}
\newcommand{\denot}[1]{\llbracket {#1} \rrbracket}
\else
\def\adra{\unicode{x25b3}}
\def\adrb{\unicode{x2b20}}
\def\adrc{\unicode{x25c7}}
\def\adrd{\unicode{x25bd}}
\def\adre{\unicode{x2605}}
\def\adrf{\unicode{x2642}}
\def\adrg{\unicode{x2640}}
\def\Lightning{\unicode{x21af}}
\newcommand{\denot}[1]{[\![{#1}]\!]}
\fi
\newcommand{\denotL}[1]{L\denot{#1}}
\newcommand{\denotR}[1]{R\denot{#1}}
\newcommand{\indetval}{\mathord{?}}
\def\adra{\bigtriangleup}
\def\adrb{\bigcirc}
\def\adrc{\lozenge}
\def\adrd{\bigtriangledown}
\def\adre{\bigstar}
\def\adrf{\dagger}
\def\adrg{\ddagger}
\newcommand{\cleave}{\mathbin{\blacksquare}}
\def\models{\vDash}
\def\limp{\Rightarrow}
\def\liff{\Leftrightarrow}
\def\lneg{\neg}
\def\bigstep{\mathrel{\Downarrow}}
\def\ltrue{\mathit{true}}
\def\lfalse{\mathit{false}}
\def\exprdef{\operatorname{def}}
\newcommand{\hpc}{\operatorname{pc}}
\newcommand{\hvc}{\operatorname{vc}}
\newcommand{\amodels}[1]{M\denot{#1}}
\newcommand{\hoare}[3]{\{#1\}\mathrel{#2}\{#3\}}
\newcommand{\hoaret}[3]{[#1]\mathrel{#2}[#3]}
\newcommand{\passingt}[1]{{#1}_\checkmark}
\newcommand{\failingt}[1]{{#1}_\times}
\newcommand{\passing}{\passingt\sigma}
\newcommand{\failing}{\failingt\sigma}
\newcommand{\slabel}[1]{\mathit{lab}}
\newcommand{\subtyperel} {\mathrel{\le}}
\newcommand{\prgltr}{s}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Section D.2 Exercise Sheet 02 - Arithmetic and MIPS (Sample Solution)
