Abstract. Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions

This thesis discusses the use of the real numbers in theorem proving. Typically, theorem provers only support a few `discrete' datatypes such as the natural numbers. However the availability of the real numbers opens up many interesting and important application areas, such as the verification

and Sturm sequences are given. Probabilistlc calculatton in real anthmetlc, prewously considered by Davis, is justified ngorously, but only in a special case. Theorems of elementary geometry can be proved much more efficiently by the techmques presented than by any known arttficml-mtelhgence approach

in a more gen eral setting? We compare the marginals com puted using loopy propagation to the exact ones in four Bayesian network architectures, including two real-world networks: ALARM and QMR. We find that the loopy beliefs of ten converge and when they do, they give a good approximation

We found out a new method for proving Fermat last theorem (FL T) on the afternoon of October 25, 1991. We proved FLT at one stroke for all prime exponents p>3, It led to the dis-covery to calculate n = 15,21, 35, 105, ••.•••. To this date, no one disprove this proof Anyone can not deny it

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 128 Konrad Slind The K Combinator as a Semantically TransparentTagging Mechanism:::::::::::::::::::::::::::: 139 Konrad Slind, Michael Norrish FCM 2002 Invited Talk Real Numbers in Real Applications ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 146 John Harrison v vi FCM 2002

In this thesis we present an automatic proof method on real valued formulas. It translates the formulas into interval arithmetic calculations on floating point numbers. The resulting formulas are then evaluated by utilizing the code generator. These computations are entirely verified in Isabelle

: This paper proves that one can not build a computer which can, for any physical system, take the specification of that system's state as input and then correctly predict its future state before that future state actually occurs. Loosely speaking, this means that one can not build a physical

near x. Based on the density of the Gauss proposed regional distribution of prime numbers theorem. And regional distribution of prime numbers theorem proved easy to understand way. The fundamental theorem to obtain the distribution of prime numbers. Thus proving that a new prime number theorem

Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally