doc-src/Exercises/2003/a3/a3.thy

+(*<*)
+theory a3 = Main:
+(*>*)
+
+subsection{* Computing with natural numbers - Magical Methods *}
+
+text{*
+A book about Vedic Mathematics describes three methods to make the calculation of squares of natural numbers easier:
+
+\begin{itemize}
+\item {\em MM1}: Numbers whose predecessors have squares that are known or can easily be calculated. For example:
+\\ Needed: $61^2$  
+\\ Given: $60^2 = 3600$
+\\ Observe: $61^2 = 3600 + 60 + 61 = 3721$
+
+\item {\em MM2}: Numbers greater than, but near 100. For example:
+\\ Needed: $102^2$
+\\ Let $h = 102 - 100 = 2$ , $h^2 = 4$
+\\ Observe: $102^2 = (102+h)$ shifted two places to the left $ + h^2 = 10404$
+ 
+\item {\em MM3}: Numbers ending in $5$. For example:
+\\ Needed: $85^2$
+\\ Observe: $85^2 = (8 * 9)$ appended to $ 25 = 7225$
+\\ Needed: $995^2$
+\\ Observe: $995^2 = (99 * 100)$ appended to $ 25 = 990025 $
+\end{itemize}
+
+
+In this exercise we will show that these methods are not so magical after all!
+
+\begin{itemize}
+\item Based on {\em MM1} define a function @{term "sq"} that calculates the square of a natural number.
+\item Prove the correctness of @{term "sq"} (i.e.\ @{term "sq n = n * n"}).
+\item Formulate and prove the correctness of {\em MM2}.\\ Hints:
+  \begin{itemize}
+  \item Generalise {\em MM2} for an arbitrary constant (instead of $100$).
+  \item Universally quantify all variables other than the induction variable.
+\end{itemize}
+\item Formulate and prove the correctness of {\em MM3}.\\ Hints:
+  \begin{itemize}
+  \item Try to formulate the property `numbers ending in $5$' such that it is easy to get to the rest of the number.
+  \item Proving the binomial formula for $(a+b)^2$ can be of some help.
+  \end{itemize}
+\end{itemize}
+*}
+
+(*<*) end (*>*)