J. Harrison. Constructing the real numbers in HOL. In L.J.M. Claesen and M.J.C. Gordon, editors, Higher Order Logic Theorem Proving and its Applications (A-20), pages 145--164. Elsevier Science Publications BV North Holland, IFIP, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1 ) image:equiv: l2s:tl 2 ) ISL:v 1 ) map:equiv:tl 1 = map:equiv:tl 2 (v 1 = INR: INR: INL:one) 9t : v 1 = INR: INR: INR:t) Proving that the relation equiv is an equivalence relation is tedious but straightforward. Using the very nice way to represent equivalence relations from [Har93] we have: 5 Thm 5.2 equiv.t 1 .t 2 = equiv.t 1 = equiv.t 2 ) equiv EQUIV REL The subset predicate P , specifying a non empty subset of equivalence classes of ltrees, can now be de ned as the quotient set of an appropriate subset Q of ltrees and the equivalence relation equiv. Looking at the ....

J. Harrison. Constructing the real numbers in HOL. In L.J.M. Claesen and M.J.C. Gordon, editors, Higher Order Logic Theorem Proving and its Applications (A-20), pages 145-164. Elsevier Science Publications BV North Holland, IFIP, 1993.

....object within SVC. The Theory of Rationals SVC assumes that all numbers appearing in arithmetical expressions are rational numbers. There is no full theory of rationals in HOL although John Harrison provided a theory of half rationals to support his construction of a theory of real numbers [15]. It is preferable to provide de nitional theories for HOL (i.e. constructing a new theory from an old one by de nitions and then proving the axioms of the new theory) However it is possible and in many cases simpler to provide an axiomatic theory which we chose to do in this case. Our theory of ....

John Harrison, Constructing the real numbers in HOL, University of Cambridge.

....2 ) # ISL.v 1 ) # ( map.equiv.tl 1 = map.equiv.tl 2 ) # (v 1 = INR. INR. INL.one) # #t : v 1 = INR. INR. INR.t) J Proving that the relation equiv is an equivalence relation is tedious but straightforward. Using the very nice way to represent equivalence relations from [Har93], we have: Theorem 3.2 equiv is an equivalence relation equiv EQUIV REL equiv.t 1 .t 2 = equiv.t 1 = equiv.t 2 ) J The subset predicate P that has to specify a non empty subset of equivalence classes of ltrees can now be defined as the quotient set of an appropriate subset Q of ltrees by the ....

J. Harrison. Constructing the real numbers in HOL. In L.J.M. Claesen and M.J.C. Gordon, editors, Higher Order Logic Theorem Proving and its Applications (A-20), pages 145--164. Elsevier Science Publications BV North Holland, IFIP, 1993.

....on a project to support the ELLA hardware description language in hol [1] As part of that work he carried out a simple verification of a floating point square root algorithm. His subsequent PhD research involved developing substantial parts of useful mathematics (real analysis etc. inside hol [3]. This has, we believe, laid a foundation for the verification of industrially significant floating point algorithms. 2 Part II Description of Proposed Research A Abstract This project aims to demonstrate that it is practical, using existing theorem proving technology, to formally verify ....

John Harrison. Constructing the real numbers in HOL. Formal Methods in System Design, 5:35--59, 1994. 8

....tuple of potential and current waveforms at the ports of the device. In order to express the specification, it is assumed that the waveforms are differentiable functions and that a combinator, diff, is available that yields the derivative of a function. The presentation of the Reals described in [Har92] would provide a suitable theory in which to reason about such specifications) The specification is parametrized by the primary, secondary and mutual inductances of the transformer. Using the symmetric convention (with the potentials of the primary and secondary ports being denoted by v p , v 0 ....

John Harrison. Constructing the real numbers in hol. In Luc J. M. Claesen and Michael J. C. Gordon, editors, Higher order logic theorem proving and its applications, volume A-20 of IFIP Transactions A: Computer Science and Technology, pages 145--164. North Holland, September 1992.

....of functions, continuity, and differentiation, and contains various lemmas and theorems for manipulating these notions. Applications to hybrid systems were our prime motivation for developing such a library but integrating mathematical analysis to theorem proving can have other interests. Harrison [8] cites applications in areas such as floating point verification [9] or the combination of theorem provers and computer algebra systems [10] The work presented in this paper is an example of use of PVS in a slightly uncommon domain, different from the traditional computer related applications. It ....

....which sometimes rely on elaborate arguments could be performed without difficulty. As far as we are aware, analysis is not a very common domain of application for mechanical theorem provers. The work the most closely related is due to Harrison who developed a large fragment of analysis in HOL [8]. There is also an extensive formalization of analysis and calculus in IMPS [4, 6] Our own construction is modest in comparison: the HOL library for reals covers notions such as power series and transcendental functions and IMPS provides rich theories for metric and normed spaces. There are ....

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J. Harrison. Constructing the real numbers in HOL. Formal Methods in System Design, 4(1/2):35--59, July 1994.

....infinite sequences of situations. A situation has a state component and a time component. States are mappings from variables to values; times are non negative numbers (actually natural numbers in this work, but they could just as well be non negative reals using an embedding of the reals in HOL [3]) Specifically, the terms of our theory have the following types: 1 Terms Type Definition Times time natural numbers or positive reals Variables var names (e.g. tagged strings) Data values val union of all types in data domain States state var val Situations situation state Theta time ....

J. R. Harrison. Constructing the real numbers in HOL. In L. J. M. Claesen and M. J. G. Gordon, editors, Higher Order Logic Theorem Proving and its Applications, pages 145--164. North-Holland, 1992.

.... true) This applies, for example, to HOL s implementation of rewriting and definition of inductively defined relations, as well as numerous special purpose rules that users write in the course of their work, e.g. the conversion to differentiate algebraic expressions by proof described by Harrison [14]. The second reason harks back to the introduction above: the processes of proof discovery and proof checking can be quite different, not just for the community of human mathematicians, but for a theorem proving system like HOL. It is not necessary for each step of a higher level algorithm or ....

John Harrison, `Constructing the real numbers in HOL', Formal Methods in System Design, 5, 35--59, (1994).

....things too. In particular there is a definitional construction of the real numbers and associated theories of elementary real analysis, including sequences, series, limits, continuity, differentiation and integration, as well as the properties of common transcendental functions such as sin and log [20]. The integration theory we formalized, the gauge integral [26, 22] obeys the Fundamental Theorem of Calculus without additional differentiability assumptions; it also has the same attractive limit behaviour as the Lebesgue integral, which it properly includes. This analysis development has ....

J. Harrison. Constructing the real numbers in HOL. Formal Methods in System Design, 5:35--59, 1994.

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J. Harrison. Constructing the real numbers in HOL. In L.J.M. Claesen and M.J.C. Gordon, editors, Higher Order Logic Theorem Proving and its Applications (A-20), pages 145--164. Elsevier Science Publications BV North Holland, IFIP, 1993.

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John Harrison. Constructing the real numbers in HOL. Formal Methods in System Design, 5:35--59, 1994.

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J. Harrison. Constructing the real numbers in HOL. Formal Methods in System Design, 5:35--59, 1994.

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[Har93 J. Harrison. Constructing the real numbers in HOL. In L.J.M. Claesen and M.J.C. Gordon, editors, Higher Order Logic Theorem Prov35 and its Applications (A-20), pages 145--164. Elsevier Science Publications BV North Holland, IFIP,1993

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