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A Little Challenge Problem for Real Number Theorem ... - Lemma 1!

A Little Challenge Problem for Real Number Theorem ... - Lemma 1!

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A Little Challenge Problem for Real Number Theorem Proving 6 ✬ The challenge problem as a trigonometric polynomial ✩ • γ is a trigonometric rational function. • If trigonometric polynomials are more to your taste, put s = x 1+x , and then γ ′′ (x) has the same sign as h(s) where h(s) = (1 + Ks + s 2 + sin s M − 2(1 + s)(K + 2s + cos s M )2 . sin s 2cos s )[(1 + s)(2 − ) + 2K + 4s + ] M M • If you forget the formulas, you can find these slides at http://www.lemma-one.com/papers/papers.html ✫ ✪

A Little Challenge Problem for Real Number Theorem Proving 7 ✬ Remarks ✩ • The classical method for this kind of problem is to find polynomial bounds, subdividing the interval if necessary. • Happily, we were free to use more convenient values for K and M. • Some types of problem should be well suited to automation . . . • . . . but what are the useful decidable fragments? • Paulson’s MetiTarski system is very promising, but didn’t solve the little challenge “out of the box”. ✫ ✪

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