We describe a macro for introducing \partial functions" into ACL2, i.e., functions not dened everywhere. The function \denitions" are actually admitted via the encapsulation principle. We discuss the basic issues surrounding partial functions in ACL2 and illustrate theorems that can be proved about such functions.

ACL2 refers to a mathematical logic based on applicative Common Lisp, as well as to an automated theorem prover for this logic. The numeric system of ACL2 reflects that of Common Lisp, including the rational and complex-rational numbers and excluding the real and complex irrationals. In conjunction with the arithmetic completion axioms, this numeric type system makes it possible to prove the non-existence of specific irrational numbers, such as √2. This paper describes ACL2(r), a version of ACL2 with support for the real and complex numbers. The modifications are based on non-standard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.

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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. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs. 1

This case study shows how ACL2 can be used to reason about the real and complex numbers, using non-standard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems of analysis, such as the intermediate-value and mean-value theorems.

IBM Power4 processor uses series approximation to calculate divide and square root. We formally verified that the algorithms with a series of rigorous error bound analysis using the ACL2 theorem prover.

This paper presents work carried out in the Clam proof-planner (Richardson et al. 00) on automating mathematical proofs using induction and non-standard analysis. The central idea is to show that the proofs we present are well-suited to proof-planning, due to their shared common structure. The theorems presented in this paper belong to standard analysis, and have been proved using induction and techniques from non-standard analysis. We rst give an overview of the proof-planning paradigm, giving a brief exposition of rippling as a heuristic for guiding rewriting. We then present the basic notions of non-standard analysis and explain our axiomatisation. We then go on to explain the theorems we intend to prove and sketch their proofs. Finally we show the parts of the proofs which have been planned automatically in Clam and draw some conclusions from the work completed so far

Abstract. The IBM Power4 TM processor uses series approximation to calculate square root. We formally verified the correctness of this algorithm using the ACL2(r) theorem prover. The proof requires the analysis of the approximation error on a Chebyshev series. This is done by proving Taylor’s theorem, and then analyzing the Chebyshev series using Taylor series. Taylor’s theorem is proved by way of non-standard analysis, as implemented in ACL2(r). Since Taylor series of a given order have less accuracy than Chebyshev series in general, we used hundreds of Taylor series generated by ACL2(r) to evaluate the error of a Chebyshev series. 1

In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder.

In this paper we use ACL2 to formally verify the correctness of an algorithm used in the analysis of dynamical systems. The algorithm uses interval arithmetic to check that a given vector field is transverse (non-tangential) to an edge (line segment). Instead of operating on numbers, interval operations operate on intervals, and they are guaranteed to return an over-approximation of the actual answer, thereby allowing us to use floating point arithmetic in a safe way. In this paper we prove that if the algorithm identifies an edge as transverse, then it is in fact transverse, as long as the underlying interval arithmetic operations are correctly implemented.