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Partial Functions in ACL2
 Journal of Automated Reasoning
"... 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 ..."
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Cited by 35 (7 self)
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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.
NonStandard Analysis in ACL2
, 2001
"... 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 complexrational numbers and excluding the real and complex irrationals. In conjunction ..."
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Cited by 26 (11 self)
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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 complexrational 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 nonexistence 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 nonstandard analysis, which interacts better with the discrete flavor of ACL2 than does traditional analysis.
Real number calculations and theorem proving
 Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics, TPHOLs 2005, volume 3603 of Lecture Notes in Computer Science
, 2005
"... 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. ..."
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Cited by 16 (4 self)
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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
Continuity and differentiability in ACL2
 ComputerAided Reasoning: ACL2 Case Studies, chapter 18
, 2000
"... This case study shows how ACL2 can be used to reason about the real and complex numbers, using nonstandard 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 ..."
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Cited by 13 (8 self)
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This case study shows how ACL2 can be used to reason about the real and complex numbers, using nonstandard 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 intermediatevalue and meanvalue theorems.
Verified Real Number Calculations: A Library for Interval Arithmetic
, 2007
"... 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 est ..."
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Cited by 9 (1 self)
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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 establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theoremprover interaction.
Formal Verification of Divide and Square Root Algorithms using Series Calculation
, 2002
"... 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. ..."
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Cited by 9 (1 self)
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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.
Mechanical Verification of a Square Root Algorithm Using Taylor’s Theorem
 In Formal Methods in Computer Aided Design (FMCAD'02
, 2002
"... 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 theore ..."
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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 nonstandard 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
Proofplanning Nonstandard Analysis
 IN THE 7TH INTERNATIONAL SYMPOSIUM ON AI AND MATHEMATICS
, 2002
"... This paper presents work carried out in the Clam proofplanner (Richardson et al. 00) on automating mathematical proofs using induction and nonstandard analysis. The central idea is to show that the proofs we present are wellsuited to proofplanning, due to their shared common structure. The theor ..."
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Cited by 6 (3 self)
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This paper presents work carried out in the Clam proofplanner (Richardson et al. 00) on automating mathematical proofs using induction and nonstandard analysis. The central idea is to show that the proofs we present are wellsuited to proofplanning, 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 nonstandard analysis. We rst give an overview of the proofplanning paradigm, giving a brief exposition of rippling as a heuristic for guiding rewriting. We then present the basic notions of nonstandard 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
Taylor's Formula with Remainder
 In Proceedings of the Third International Workshop of the ACL2 Theorem Prover and its Applications
, 2002
"... In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder. ..."
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In this paper, we present a proof in ACL2(r) of Taylor's formula with remainder.