## A Constructive Formalization of the Fundamental Theorem of Calculus (0)

Citations: | 8 - 0 self |

### BibTeX

@MISC{Cruz-Filipe_aconstructive,

author = {Luís Cruz-Filipe},

title = {A Constructive Formalization of the Fundamental Theorem of Calculus},

year = {}

}

### 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. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalization in some detail, focusing on how some of Bishop's original proofs had to be refined, adapted or redone from scratch.

### Citations

431 | Constructive Analysis
- Bridges, Bishop
- 1985
(Show Context)
Citation Context ... the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop’s work (=-=[4]-=-). In this paper, we describe the formalization in some detail, focusing on how some of Bishop’s original proofs had to be refined, adapted or redone from scratch. 1 Introduction In this paper we desc... |

283 |
Foundations of Modern Analysis
- Dieudonne
- 1960
(Show Context)
Citation Context ...o the righthandside then we have an upper bound for the expression on the lefthandside). The fact that some authors state and prove it directly in this form (Dieudonné is one of them, see for example =-=[8]-=- and [9]) is evidence that at least for some people it is the best formulation of the Mean Law. 9s10 6 Integration Integration turned out to be by far the most difficult process to formalize following... |

275 |
Foundation of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...aranteed to be actually in the image of f, and may therefore lie outside of J. For a model of Bishop style mathematics where a function which doesn’t satisfy this property see Theorem 8.1 on p. 71 of =-=[2]-=- 7s8 to y. This is achieved through dividing the interval [a, b] into n subintervals of length ε 2 (appealing to the archimedian axiom); then, using a tricky induction argument and the properties of t... |

92 | Theorem Proving with the Real Numbers
- Harrison
- 1998
(Show Context)
Citation Context ... Coq, starting with an axiomatic characterization of the reals, and showed how this formalization can be used to prove correctness of programs in numerical analysis (see [20] and [21]). John Harrison =-=[17]-=- has also formalized real numbers and differential calculus on his HOL-light system. This has been used together with his formalization of floating point arithmetic, described in [18], to prove correc... |

82 |
An introduction
- Intuitionism
- 1956
(Show Context)
Citation Context ...w results include a straightforward definition of subsequence and its main properties, which we will not discuss. 3 This axiomatization is based on Heyting’s work on algebraic structures presented in =-=[19]-=-. � � 4 A regular sequence is a Cauchy sequence such that ∀m,n∈IN |xm − xn| ≤ � 1 1 − � m n 3s4 To study series, we begin by associating to each sequence the sequence of its partial sums in the obviou... |

45 | A new representation for exact real numbers
- Edalat, Potts
- 2000
(Show Context)
Citation Context ...ults obtained by these. On the other hand, work has been done on exact real number arithmetic. Some representations of real numbers are presented and briefly discussed by A. Edalat and P. J. Potts in =-=[11]-=-; Edalat and Krznaric further show in [12] 17s18 how one specific representation can be used to compute integrals. It would be interesting to examine how well these real number representations fit wit... |

37 |
Calcul infinitésimal
- Dieudonné
- 1968
(Show Context)
Citation Context ...ghthandside then we have an upper bound for the expression on the lefthandside). The fact that some authors state and prove it directly in this form (Dieudonné is one of them, see for example [8] and =-=[9]-=-) is evidence that at least for some people it is the best formulation of the Mean Law. 9s10 6 Integration Integration turned out to be by far the most difficult process to formalize following Bishop’... |

35 | A Machine-Checked Theory of Floating Point Arithmetic
- Harrison
- 1999
(Show Context)
Citation Context ...). John Harrison [17] has also formalized real numbers and differential calculus on his HOL-light system. This has been used together with his formalization of floating point arithmetic, described in =-=[18]-=-, to prove correctness of floating point algorithms Similarly, Bruno Dutertre has developed a library of real analysis (see [10]) which was later extended by Hanne Gottliebsen to include the elementar... |

15 | Constructive reals in Coq: axioms and categoricity
- Geuvers, Niqui
- 2001
(Show Context)
Citation Context ...uch that x < n (where n denotes the image of n in the ring). 3 Then, a concrete structure, the set of Cauchy sequences of rational numbers with equality defined as equality of limits, is defined (see =-=[13]-=-) and proved both to satisfy these axioms and to be isomorphic to every other structure that satisfies these axioms. However, in our work we suppose an arbitrary real number structure. This allows us ... |

14 | The algebraic hierarchy of the FTA Project
- Geuvers, Pollack, et al.
- 2002
(Show Context)
Citation Context ...eorems in calculus of one variable: Taylor’s Theorem and the Fundamental Theorem of Calculus. This formalization was developed using the algebraic hierarchy developed for the FTA project described in =-=[14]-=- and extending it whenever necessary. Working in this way, we intend to show that it is possible to formalize large pieces of mathematics in a modular way—that is, such that new blocks can be built on... |

14 | Transcendental functions and continuity checking in pvs
- Gottliebsen
- 2000
(Show Context)
Citation Context ... developed a library of real analysis (see [10]) which was later extended by Hanne Gottliebsen to include the elementary transcendental functions and their properties. Gottliebsen proceeds to show in =-=[16]-=- how this system can be used interactively with computer algebra systems to ensure (greater) correctness of the results obtained by these. On the other hand, work has been done on exact real number ar... |

14 |
Formalisation et automatisation de preuves en analyse réelle et numérique
- Mayero
- 2001
(Show Context)
Citation Context ...nd transcendental functions in Coq, starting with an axiomatic characterization of the reals, and showed how this formalization can be used to prove correctness of programs in numerical analysis (see =-=[20]-=- and [21]). John Harrison [17] has also formalized real numbers and differential calculus on his HOL-light system. This has been used together with his formalization of floating point arithmetic, desc... |

11 | Equational reasoning via partial reflection
- Geuvers, Wiedijk, et al.
- 2000
(Show Context)
Citation Context ...sed in the FTA project [14]; in the end, we obtained a much larger library without having to change any of its original content. We did not discuss automation in this paper, as it was already done in =-=[15]-=- and [7]. In those papers, it was shown how several frequently occurring goals – including proofs of algebraic identities and checking that a function is continuous – can be automatically solved or, a... |

9 |
Elements of Mathematical Analysis
- Dutertre
- 1996
(Show Context)
Citation Context ...ether with his formalization of floating point arithmetic, described in [18], to prove correctness of floating point algorithms Similarly, Bruno Dutertre has developed a library of real analysis (see =-=[10]-=-) which was later extended by Hanne Gottliebsen to include the elementary transcendental functions and their properties. Gottliebsen proceeds to show in [16] how this system can be used interactively ... |

9 | Equational reasoning via partial re - Geuvers, Wiedijk, et al. - 2000 |

5 |
Checking Landau’s ‘Grundlagen
- Jutting
- 1979
(Show Context)
Citation Context ... the point where we are trying to evaluate it. This process, which is described in detail in [7], is very similar to the approach which was originally followed in the Automath system (see for example =-=[3]-=-). Of course, total functions simply correspond to the case when the predicate is always true. For generality’s sake, we decided to work constructively (following Bishop, see [4]). This means essentia... |

5 | Calcul in - Dieudonne - 1980 |

4 |
Generation and Presentation of Formal Mathematical Documents
- Oostdijk
- 2001
(Show Context)
Citation Context ...the work on integration underlined the need for a much higher level of automation, which may probably be efficiently achieved only through communication with computer algebra systems, as described in =-=[22]-=-. Still, we feel that this work is a significant step toward the building of a useful library of formalized analysis that can be actually used in the building of interactive proofs. Finally, we feel t... |

3 | A machine-checked theory of point arithmetic - Harrison - 1999 |

2 |
Numerical integration with exact arithmetic
- Edalat, Krznaric
(Show Context)
Citation Context ... work has been done on exact real number arithmetic. Some representations of real numbers are presented and briefly discussed by A. Edalat and P. J. Potts in [11]; Edalat and Krznaric further show in =-=[12]-=- 17s18 how one specific representation can be used to compute integrals. It would be interesting to examine how well these real number representations fit with our axiomatization of the reals, but we ... |

1 |
Formalizing Real Calculus
- Cruz-Filipe
- 2002
(Show Context)
Citation Context ... by a binary function which takes a proof term as a second argument—a proof that the function is defined at the point where we are trying to evaluate it. This process, which is described in detail in =-=[7]-=-, is very similar to the approach which was originally followed in the Automath system (see for example [3]). Of course, total functions simply correspond to the case when the predicate is always true... |

1 |
Using Theorem Proving for Numerical Analysis
- Mayero
(Show Context)
Citation Context ...endental functions in Coq, starting with an axiomatic characterization of the reals, and showed how this formalization can be used to prove correctness of programs in numerical analysis (see [20] and =-=[21]-=-). John Harrison [17] has also formalized real numbers and differential calculus on his HOL-light system. This has been used together with his formalization of floating point arithmetic, described in ... |

1 | Formalizing Real Calculus in Coq, in Theorem Proving in Higher Order Logics - Cruz-Filipe - 2002 |