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Optimising Problem Formulation for Cylindrical Algebraic Decomposition
, 2013
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Type classes for efficient exact real arithmetic
 IN COQ. CORR ABS/1106.3448
, 2011
"... Floating point operations are fast, but require continuous effort by the user to ensure correctness. This burden can be shifted to the machine by providing a library of exact analysis in which the computer handles the error estimates. Previously, we provided a fast implementation of the exact real ..."
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Floating point operations are fast, but require continuous effort by the user to ensure correctness. This burden can be shifted to the machine by providing a library of exact analysis in which the computer handles the error estimates. Previously, we provided a fast implementation of the exact real numbers in the Coq proof assistant. This implementation incorporates various optimizations to speed up the basic operations of O’Connor’s implementation by a 100 times. We implemented these optimizations in a modular way using type classes to define an abstract specification of the underlying dense set from which the real numbers are built. This abstraction does not hurt the efficiency. This article is a substantially expanded version of (Krebbers/Spitters, Calculemus 2011) in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq’s fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speedup by avoiding evaluation of termination proofs at runtime.
Author manuscript, published in "ITP 3rd International Conference on Interactive Theorem Proving 2012 (2012)" Construction of real algebraic numbers in Coq
, 2012
"... Abstract. This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete Archimedean real closed field. This construction hence implements an interface of real closed field. Instances of such an interface immediatel ..."
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Abstract. This paper shows a construction in Coq of the set of real algebraic numbers, together with a formal proof that this set has a structure of discrete Archimedean real closed field. This construction hence implements an interface of real closed field. Instances of such an interface immediately enjoy quantifier elimination thanks to a previous work. This work also intends to be a basis for the construction of complex algebraic numbers and to be a reference implementation for the certification of numerous algorithms relying on algebraic numbers in computer algebra.
manuscript No. (will be inserted by the editor) Formalization of Bernstein Polynomials and Applications to Global Optimization
"... Abstract This paper presents a formalization in higherorder logic of a practical representation of multivariate Bernstein polynomials. Using this representation, an algorithm for finding lower and upper bounds of the minimum and maximum values of a polynomial has been formalized and verified correc ..."
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Abstract This paper presents a formalization in higherorder logic of a practical representation of multivariate Bernstein polynomials. Using this representation, an algorithm for finding lower and upper bounds of the minimum and maximum values of a polynomial has been formalized and verified correct in the Prototype Verification System (PVS). The algorithm is used in the definition of proof strategies for formally and automatically solving polynomial global optimization problems. 1
Verifying an algorithm computing Discrete Vector Fields for digital imaging
"... In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological analysis of digital images since it provides a ..."
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In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological analysis of digital images since it provides a procedure to reduce the amount of information but preserving the homological properties. In particular, thanks to discrete vector fields, we are able to compute, inside Coq, homological properties of biomedical images which otherwise are out of the reach of this system.
Theorem of three circles in Coq
"... The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong to a certain area of the complex plane delimited by straight ..."
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The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong to a certain area of the complex plane delimited by straight lines. After applying a transformation involving inversion this area is mapped to an area delimited by circles. We provide a formalisation of this rather geometric proof in Ssreflect, an extension of the proof assistant Coq, providing versatile algebraic tools. They allow us to formalise the proof from an algebraic point of view. 1
Automated Theorem Proving For Special Functions: The Next Phase
"... [Symbolic and algebraic manipulation]: Symbolic and algebraic algorithms—Theorem proving algorithms; [Theory of computation]: Logic—Logic and verification General Terms theorem proving, verification, decision procedures 1. RESOLUTION THEOREM PROVING Automated theorem proving, in a nutshell, is the c ..."
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[Symbolic and algebraic manipulation]: Symbolic and algebraic algorithms—Theorem proving algorithms; [Theory of computation]: Logic—Logic and verification General Terms theorem proving, verification, decision procedures 1. RESOLUTION THEOREM PROVING Automated theorem proving, in a nutshell, is the combination of symbolic logic with syntactic algorithms. A formal proof calculus is chosen with two criteria in mind: expressiveness and ease of automation. These desiderata pull in opposite directions: Boolean logic and linear arithmetic are decidable, so the answers to all questions can simply be calculated, but these theories are not very expressive. At the other extreme, a dependent type theory such as the calculus of constructions used in Coq [6] is highly expressive and