Guaranteed Proofs Using Interval Arithmetic - LRI

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Introduction Intervals Improvements Conclusion Rational intervals Proofs Elementary functions PVSNumerical proofs in PVSInstead of numerical tools, NIA and NASA use PVS, a proofassistant. http://pvs.csl.sri.com/Proofs are constructed by applying strategies to transform thehypotheses and goals of theorems until they match each other. Forproofs by approximation, the assistant will formally guarantee thecorrectness of the computations.Our strategy numerical does a proof by approximation: it appliesinterval arithmetic theorems to certify an inequality. The proofassistant will formally guarantee the correctness of thecomputations.Marc Daumas, Guillaume Melquiond, César MuñozGuaranteed Proofs Using Interval Arithmetic
16The NATO General, Sir Adrian Bradshaw, has warned that NATO mustmaintain a cohesive system of deterrence on its eastern borders, somethingthat would require help from the European Union, and has stressed thatNATO is pushing ahead with plans for a very high readiness joint taskforce,“in order to convince Russia, or any other state adversary, that any attack onone NATO member will inevitably lead them into a conflict with the wholealliance.” 21Thus, speaking geopolitically, the current buffer zone of the Euro -pean Union — the Eastern partnership countries — is rapidly eroding. Theborders of these countries, earlier showing a clear division of sovereignty,are now becoming frontiers which indicate anything but “a pre-modernworld of vaguer, more informal, overlapping divisions.” 22 Therefore, theexternal border of the European Union is de facto without the buffer zone,and Central Europe, together with the Baltic States, is face to face withRussia. Is Central Europe the next in line to become the crush zone, thusfulfilling the predictions of Mackinder? It depends upon whether the Westwill try to overreach by penetrating this region geostrategically. The geo -politics of Europe could shift. Less dependence on Russia would allow thevision of a truly independent, culturally vibrant Central and Eastern Europeto fully prosper. 23 But those who do not know history are doomed to repeat it.21Russian expansionism may pose existential threat, says Nato general. 20.02.2015. http://www.theguardian.com/world/2015/feb/20/russia-existential-threat-british-nato-general (Homepage of The Guardian; visited 21.02.2015.).22Kaplan R. D. The Revenge of Geography. What the Map Tells Us About Coming Conflicts andthe Battle Against Fate. NY: Random House Trade Paperbacks, 2012, pp. 352–353,23Ibid, p. 350.
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