Révision 253 CSL17/tech-report/preliminaries.tex

preliminaries.tex (revision 253)
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%\anupam{Should add $+(;x,y)$ as a basic function (check satisfies polychecking lemma!), since otherwise not definable with safe input. Then define unary successor $\succ{} (;x)$ as $+(;x,\succ 1 0)$. }


  		





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We denote $|x|=\ulcorner \log(x+1) \urcorner.$


  		





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\begin{example}


  		





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	[Some simple functions]


  		





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	\label{ex:simple-bc-fns}


  		





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	%\item $\bit(l;x)$ returns the $\mode l$th bit of $x$.


  		





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	\item We can define the function $\bit(l;x)$ which returns the $|l|$th least significant bit of $x$. For instance $\bit(11;1101)=0$.


  		





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	For that let us first define the function $\shorten(l;x)$ which returns the $|x|- |l|+1$ prefix of $x$, as follows:


  		





  
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	For that let us first define the function $\shorten(l;x)$ which returns the $|x|- |l|$ prefix of $x$, as follows:


  		





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	\begin{eqnarray*}


  		





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	\shorten(0;x) &=&x\\


  		





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	\shorten(\succ{i}l;x) &=&p(;\shorten(l;x))


  		












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