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Extending the HOL theorem prover with a Computer Algebra System to Reason about the Reals
 Higher Order Logic Theorem Proving and its Applications (HUG `93
, 1993
"... In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from ..."
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Cited by 33 (4 self)
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In this paper we describe an environment for reasoning about the reals which combines the rigour of a theorem prover with the power of a computer algebra system. 1 Introduction Computer theorem provers are a topic of research interest in their own right. However much of their popularity stems from their application in computeraided verification, i.e. proving that designs of electronic or computer systems, programs, protocols and cryptosystems satisfy certain properties. Such proofs, as compared with the proofs one finds in mathematics books, usually involve less sophisticated central ideas, but contain far more technical Supported by the Science and Engineering Research Council, UK. y Supported by SERC grant GR/G 33837 and a grant from DSTO Australia. details and therefore tend to be much more difficult for humans to write or check without making mistakes. Hence it is appealing to let computers help. Some fundamental mathematical theories, such as arithmetic, are usually requi...
The LaplaceJaynes approach to induction Being part II of “From ‘plausibilities of plausibilities ’ to stateassignment methods”
, 2007
"... An approach to induction is presented, based on the idea of analysing the context of a given problem into ‘circumstances’. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti’s representation theorem and on the notion o ..."
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Cited by 1 (1 self)
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An approach to induction is presented, based on the idea of analysing the context of a given problem into ‘circumstances’. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that based on de Finetti’s representation theorem and on the notion of infinite exchangeability. In particular, it gives an alternative interpretation of those formulae that apparently involve ‘unknown probabilities ’ or ‘propensities’. Various advantages and applications of the presented approach are discussed, especially in comparison to that based on exchangeability. Generalisations are also discussed. PACS numbers: 02.50.Cw,02.50.Tt,01.70.+w MSC numbers: 03B48,60G09,60A05 Note, to head off a common misconception, that this is in no way to introduce a “probability of a probability”. It is simply convenient to index our hypotheses by parameters [...] chosen to be numerically equal to the probabilities assigned by those hypotheses; this avoids a doubling of our notation. We could easily restate everything so that the misconception could not arise; it would only be rather clumsy notationally and tedious verbally.
ON THE CINTEGRAL
"... Let F: [a, b] → R be a differentiable function and let f be its derivative. The problem of recovering F from f is called problem of primitives. In 1912, the problem of primitives was solved by A. Denjoy with an integration process (called totalization) that includes the Lebesgue integral and the Ri ..."
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Let F: [a, b] → R be a differentiable function and let f be its derivative. The problem of recovering F from f is called problem of primitives. In 1912, the problem of primitives was solved by A. Denjoy with an integration process (called totalization) that includes the Lebesgue integral and the Riemann improper integral. Two years later, a second solution was obtained by O. Perron with a method based on the notions of major function and minor function. A third solution, based on a generalization of Riemann integral, is due to J. Kurzweil (1957) and R. Henstock (1963). It is surprising that, nevertheless the three integration processes are completely different, they produce the same integral (i.e. they have the same space of integrable functions and satisfy the same properties). In 1986, A.M. Bruckner, R.J. Fleissner and J. Foran [9] remarked that the solution provided by Denjoy, Perron, Kurzweil and Henstock possesses a generality which is not needed for this purpose. In fact the function 1 x sin (1) F (x) = x2, 0 < x ≤ 1 0, x = 0 is a primitive for the DenjoyPerronKurzweilHenstock integral (more precisely, F is a primitive for the Riemann improper integral), but it is neither a Lebesgue primitive, neither a differentiable function, nor a sum of a Lebesgue primitive and a differentiable function (see [9] for details). The question of providing a minimal constructive integration process which includes the Lebesgue integral and also integrates the derivatives of differentiable functions was solved by the following Riemanntype integral: