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A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 112 (11 self)
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This section describes the structure of the proof of
Some considerations on the usability of Interactive Provers
"... Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to ana ..."
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Cited by 3 (1 self)
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Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1
Stateless HOL Dedicated to Roel de Vrijer, in the tradition of Automath
"... Abstract. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that ..."
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Abstract. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml – the implementation language of the system – can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system is about 1 6 version of HOL Light. th slower than the stateful
Chapter 1 of Kelley’s famous book General Topology introduces the most
"... fundamental concepts of a topological space. One such notion is defined as follows [1]: A topological space (X, τ) is connected if and ony if X is not the union of two nonvoid separated subsets, where A and B are separated in X if and only if A ∩ B = � and A ∩ B = �. As usual, Y denotes the closure ..."
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fundamental concepts of a topological space. One such notion is defined as follows [1]: A topological space (X, τ) is connected if and ony if X is not the union of two nonvoid separated subsets, where A and B are separated in X if and only if A ∩ B = � and A ∩ B = �. As usual, Y denotes the closure of a subset Y
Stateless HOL
"... Dedicated to Roel de Vrijer, in the tradition of Automath. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far h ..."
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Dedicated to Roel de Vrijer, in the tradition of Automath. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml – the implementation language of the system – can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system runs at about 85 % of the speed of the stateful version of HOL Light.
Under revision for publication in the American Mathematical Monthly. Preprint available at www.igt.unistuttgart.de/eiserm. THE FUNDAMENTAL THEOREM OF ALGEBRA MADE EFFECTIVE: AN ELEMENTARY REALALGEBRAIC PROOF VIA STURM CHAINS
"... ABSTRACT. Sturm’s theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm’s method to count and locate the complex roots of any complex polynomial. For holomorphic functions Cauchy’s index is b ..."
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ABSTRACT. Sturm’s theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm’s method to count and locate the complex roots of any complex polynomial. For holomorphic functions Cauchy’s index is based on contour integration, but in the special case of polynomials it can effectively be calculated via Sturm chains using euclidean division as in the real case. In this way we provide an algebraic proof of Cauchy’s theorem for polynomials over any real closed field. As our main tool, we formalize Gauss ’ geometric notion of winding number (1799) in the realalgebraic setting, from which we derive a realalgebraic proof of the Fundamental Theorem of Algebra. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the firstorder language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic rootfinding algorithm. L’algèbre est généreuse, elle donne souvent plus qu’on lui demande. (Jean le Rond d’Alembert) 1
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"... Abstract: According to the most popular non‐skeptical views about intuition, intuitions justify beliefs because they are based on understanding. More precisely: if intuiting that p justifies you in believing that p it does so because your intuition is based on your understanding of the proposition ..."
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Abstract: According to the most popular non‐skeptical views about intuition, intuitions justify beliefs because they are based on understanding. More precisely: if intuiting that p justifies you in believing that p it does so because your intuition is based on your understanding of the proposition that p. The aim of this paper is to raise some challenges for accounts of intuitive justification along these lines. I pursue this project from a non‐skeptical perspective. I argue that there are cases in which intuiting that p justifies you in believing that p, but such that there is no compelling reason to think this is because your intuition is based on your understanding of the proposition that p. A common idea about intuition is that it is based on understanding. 2 According to one way of developing the idea, in intuiting that p you simply draw on