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\title{Irrational numbers from THE BOOK}
\author{Lawrence C. Paulson}
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\begin{abstract}
An elementary proof is formalised: that $\exp r$ is irrational for every nonzero rational number~$r$. The mathematical development comes from the well-known volume \emph{Proofs from THE BOOK}~\cite[pp.\thinspace51--2]{aigner-proofs}, by Aigner and Ziegler, who credit the idea to Hermite. The development illustrates a number of basic Isabelle techniques: the manipulation of summations, the calculation of quite complicated derivatives and the estimation of integrals.
We also see how to import another AFP entry (Stirling's formula)~\cite{Stirling_Formula-AFP}.

As for the theorem itself, note that a much stronger and more general result (the Hermite--Lindemann--Weierstra\ss{} transcendence theorem) is already available in the AFP~\cite{Hermite_Lindemann-AFP}.
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\paragraph*{Acknowledgements}
The author was supported by the ERC Advanced Grant ALEXANDRIA (Project 742178) funded by the European Research Council. 

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