## Formalizing Real Calculus in Coq (2002)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Cruz-Filipe02formalizingreal,

author = {Lus Cruz-Filipe},

title = {Formalizing Real Calculus in Coq},

year = {2002}

}

### OpenURL

### Abstract

We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. This formalization is built upon the library of constructive algebra created in the FTA (Fundamental Theorem of Algebra) project, which is extended with results about the real numbers, namely about (power) series. Two important issues that arose in this formalization and which will be discussed in this paper are partial functions (different ways of dealing with this concept and the advantages of each different approach) and the high level tactics that were developed in parallel with the formalization (which automate several routine procedures involving results about real-valued functions).

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Citation Context ...solve the problem of integrating rational functions, providing both an answer and a proof that this answer is correct. The basis for this work was chapter 2 of Bishop’s book on constructive analysis (=-=[3]-=-). The formalization was built upon the algebraic hierarchy developed at the University of Nijmegen, described in [7] and available in the Coq library, which included most of the results about real nu... |

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Citation Context ...ot necessarily true that e.g. point-wise continuity in a compact interval implies uniform continuity in that interval (a counterexample can be constructed with some extra assumptions, see for example =-=[1]-=-), and so in practice it is more natural to begin with the uniform concept altogether. The other important kind of domain is the interval. In practice, it is difficult to find examples where we really... |

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Citation Context ...ly with Computer Algebra systems to guarantee the correctness of applications of the Fundamental Theorem of Calculus.sFormalizing Real Calculus in Coq 9 John Harrison’s HOL-light system (described in =-=[10]-=-) is another proof assistant that comes with a library of real analysis; once again, the reasoning in this system is classical. The results included in this library include the usual results on preser... |

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Citation Context .... The basis for this work was chapter 2 of Bishop’s book on constructive analysis ([3]). The formalization was built upon the algebraic hierarchy developed at the University of Nijmegen, described in =-=[7]-=- and available in the Coq library, which included most of the results about real numbers that were needed, namely most of sections 1 to 3 of [3] (where real numbers are defined and their main properti... |

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Citation Context ...e allowed to exist (see [4], pp. 81-83). On the other hand, Coq comes with a program extraction mechanism 1 An approach following the first alternative was independently chosen by Micaela Mayero, see =-=[11]-=-.sFormalizing Real Calculus in Coq 3 (briefly described in Chapter 17 of [4]) which allows programs to be derived from informative proofs; for efficiency reasons, this mechanism assumes that proof obj... |

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Citation Context ...loped a classical theory of real analysis, including the main definitions in differential calculus. Building upon this work, Hanne Gottliebsen built a library of transcendental functions described in =-=[9]-=-, where she defines exponential, logarithmic and trigonometric functions, proving similar results to ours. She then defines an automatic procedure to prove continuity of a large class of functions, wh... |

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Citation Context ... will use f2 as a derivative for f1); however, the proof is still left to be done by the user. Another, and more powerful, approach is to use reflection (a method which is described in full detail in =-=[8]-=-). We select among the class of all partial functions those whose derivative we know how to compute, and model this as an inductive type PF. This type will have not only constructors for constant and ... |

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Citation Context ...s, f(x) instead of f(x, H) or something similar) it does have the disadvantage that you can write down things like 1 0 , although it is not clear what they mean. In the PVS system, Bruno Dutertre (in =-=[5]-=-) has also developed a classical theory of real analysis, including the main definitions in differential calculus. Building upon this work, Hanne Gottliebsen built a library of transcendental function... |

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Citation Context ...simplification procedure very time consuming. Thus, a different approach is needed, and we turn to a common alternative which has already been used for example in the Automath system (see for example =-=[2]-=-). As before, we associate to every partial function f the predicate domf , but now we identify f with a function of two arguments: a setoid element and the proof that it satisfies domf . That is, our... |