Irrational numbers from THE BOOK

Lawrence C. Paulson ­čîÉ

January 8, 2022

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


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 Proofs from THE BOOK, 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). As for the theorem itself, note that a much stronger and more general result (the Hermite--Lindemann--Weierstra├č transcendence theorem) is already available in the AFP.


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Session Irrationals_From_THEBOOK