In the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.

/pdf/formalization-of-real-analysis-a-survey-of-proof-assistants-450r9fm3mr.pdf

Formalization of real analysis: a survey of proof assistants and libraries †

Verification of numerical analysis programs requires dealing with derivatives and integrals. High confidence in this process can be achieved using a formal proof checker, such as Coq. Its standard library provides an axiomatization of real numbers and various lemmas about real analysis, which may be used for this purpose. Unfortunately, its definitions of derivative and integral are unpractical as they are partial functions that demand a proof term. This proof term makes the handling of mathematical formulas cumbersome and does not conform to traditional analysis. Other proof assistants usually do not suffer from this issue; for instance, they may rely on Hilbert's epsilon to get total operators. In this paper, we propose a way to define total operators for derivative and integral without having to extend Coq's standard axiomatization of real numbers. We proved the compatibility of our definitions with the standard library's in order to leverage existing results. We also greatly improved automation for real analysis proofs that use Coq standard definitions. We exercised our approach on lemmas involving iterated partial derivatives and differentiation under the integral sign, that were missing from the formal proof of a numerical program solving the wave equation.

/pdf/improving-real-analysis-in-coq-a-user-friendly-approach-to-2a42a5gc1f.pdf

Improving real analysis in coq: a user-friendly approach to integrals and derivatives

Let X be a real normed space, let A be a closed-interval subset of R, let f be a function from A into the carrier of X, and let D be a Division of A. A finite sequence of elements of X is said to be a middle volume of f and D if it satisfies the conditions (Def. 1). (Def. 1)(i) len it = lenD, and (ii) for every natural number i such that i ∈ domD there exists a point c of X such that c ∈ rng(f divset(D, i)) and it(i) = vol(divset(D, i)) · c.

https://repozytorium.uwb.edu.pl/jspui/bitstream/11320/3590/1/v10037-011-0003-8.pdf

Riemann Integral of Functions from R into Real Normed Space

Let A be a closed-interval subset of R, let f be a function from A into R, let S be a non empty Division of A, and let D be an element of S. A finite sequence of elements of R is said to be a middle volume of f and D if it satisfies the conditions (Def. 1). (Def. 1)(i) len it = lenD, and (ii) for every natural number i such that i ∈ domD there exists an element r of R such that r ∈ rng(f divset(D, i)) and it(i) = r · vol(divset(D, i)). Let A be a closed-interval subset of R, let f be a function from A into R, let S be a non empty Division of A, let D be an element of S, and let F be a middle volume of f andD. The functor middle sum(f, F ) yielding a real number is defined as follows: (Def. 2) middle sum(f, F ) = ∑ F.

/pdf/riemann-integral-of-functions-from-r-into-rn-mp7i0l1v7y.pdf

Riemann Integral of Functions from R into Rn

One can prove the following propositions: (2) For all real numbers a, b, r such that 0 ¬ r and r ¬ 1 and a ¬ b holds a ¬ (1 − r) · a + r · b and (1 − r) · a + r · b ¬ b. (3) For all real numbers a, b such that a ­ 0 and b > 0 or a > 0 and b ­ 0 holds a + b > 0. (4) For all real numbers a, b such that −1 ¬ a and a ¬ 1 and −1 ¬ b and b ¬ 1 holds a · b ¬ 1. (5) For all real numbers a, b such that a ­ 0 and b ­ 0 holds a· √ b = √ a · b. (6) For all real numbers a, b such that −1 ¬ a and a ¬ 1 and −1 ¬ b and b ¬ 1 holds (−b) · √ 1 + a ¬ √ 1 + b and − √ 1 + b ¬ b · √ 1 + a.

/pdf/general-fashoda-meet-theorem-for-unit-circle-and-square-faa9bfkgw4.pdf

General Fashoda Meet Theorem for Unit Circle and Square

For simplicity, we follow the rules: a, b are real numbers, F , G, H are finite sequences of elements of R, i, j, k are natural numbers, X is a non empty set, and x1 is a set. Let I1 be a subset of R. We say that I1 is closed-interval if and only if: (Def. 1) There exist real numbers a, b such that a ≤ b and I1 = [a, b]. Let us mention that there exists a subset of R which is closed-interval. In the sequel A is a closed-interval subset of R. One can prove the following propositions:

The Definition of the Riemann Definite Integral and some Related Lemmas

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The Definition of the Riemann Definite Integral and some Related Lemmas