The Picard Algorithm for Ordinary Differential Equations in Coq

Makarov, Evgeny; Spitters, Bas

doi:10.1007/978-3-642-39634-2_34

Cited by 31 publications

(27 citation statements)

References 11 publications

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“…We could also have tried a much more general method, that is, solving a differential equation built from the integrand, as we did when using VNODE-LP. Again, there has been some work done for Coq in the setting of exact real arithmetic [10], but the performances are not good enough in practice. Much closer to actual numerical methods is Immler's work in Isabelle/HOL [8], which uses an arithmetic on affine forms.…”

Section: Resultsmentioning

confidence: 99%

Formally Verified Approximations of Definite Integrals

2018

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Finding an elementary form for an antiderivative is often a difficult task, so numerical integration has become a common tool when it comes to making sense of a definite integral. Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy. Yet numerical integration is still missing from the toolbox when performing formal proofs in analysis.This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system. Our approach is not based on traditional quadrature methods such as Newton-Cotes formulas. Instead, it relies on computing and evaluating antiderivatives of rigorous polynomial approximations, combined with an adaptive domain splitting. Our approach also handles improper integrals, provided that a factor of the integrand belongs to a catalog of identified integrable functions. This work has been integrated to the CoqInterval library.

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Section: Resultsmentioning

confidence: 99%

Formally Verified Approximations of Definite Integrals

2018

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“…The primary task is to define a constructive semantics for differential equations and to give constructive interpretations to the differential equation rules of dGL. Previous work on formalizations of differential equations [34] suggests differential equations can be treated constructively. In principle, existing proofs in dGL might happen to be constructive, but this does not obviate the present work.…”

Section: Discussionmentioning

confidence: 99%

Constructive Game Logic

Bohrer¹,

Platzer²

2020

Programming Languages and Systems

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Game Logic is an excellent setting to study proofs-aboutprograms via the interpretation of those proofs as programs, because constructive proofs for games correspond to effective winning strategies to follow in response to the opponent's actions. We thus develop Constructive Game Logic, which extends Parikh's Game Logic (GL) with constructivity and with first-order programsà la Pratt's first-order dynamic logic (DL). Our major contributions include: 1. a novel realizability semantics capturing the adversarial dynamics of games, 2. a natural deduction calculus and operational semantics describing the computational meaning of strategies via proof-terms, and 3. theoretical results including soundness of the proof calculus w.r.t. realizability semantics, progress and preservation of the operational semantics of proofs, and Existence Properties on support of the extraction of computational artifacts from game proofs. Together, these results provide the most general account of a Curry-Howard interpretation for any program logic to date, and the first at all for Game Logic.This work is at the intersection of game logic and constructive modal logics. Individually, they have a rich literature, but little work has been done at their intersection. Of these, we are the first for GL and the first with a proofs-asprograms interpretation for a full first-order program logic. Games in Logic.Parikh's propositional GL [38] was followed by coalitional GL [39]. A first-order version of GL is the basis of differential game logic dGL [42] for hybrid games. GL's are unique in their clear delegation of strategy to the proof language rather than the model language, crucially allowing succinct game specifications with sophisticated winning strategies. Succinct specifications are important: specifications are trusted because proving the wrong theorem would not ensure correctness. Relatives without this separation include Strategy Logic [15], Alternating-Time Temporal Logic (ATL) [4], CATL [30], Ghosh's SDGL [24], Ramanujam's structured strategies [44], Dynamic-epistemic logics [7,11,48], evidence logics [10], and Angelic Hoare logic [35]. and terms are rational-valued (Q); we also write B for the set of Boolean values {0, 1} for false and true respectively.

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“…First, we admit the existence and uniqueness of solutions of (2) and the continuity of solutions relative to initial conditions on K. A way to prove these properties is to use the Cauchy-Lipschitz Theorem (also known as the Picard-Lindelöf Theorem). However, the only formalization of this theorem in the Coq proof-assistant [39] we are aware of is the one by Makarov and Spitters [27]. It is based on the CoRN library of constructive real numbers [9], while our formalization of the inverted pendulum is based on the Coqelicot library [5], which extends Coq's standard library on classically axiomatized real numbers [29].…”

Section: Verification Of the Hypotheses Of Lasalle'smentioning

confidence: 99%

A formal proof in Coq of a control function for the inverted pendulum

Rouhling¹

2018

Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018

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Control theory provides techniques to design controllers, or control functions, for dynamical systems with inputs, so as to grant a particular behaviour of such a system. The inverted pendulum is a classic system in control theory: it is used as a benchmark for nonlinear control techniques and is a model for several other systems with various applications. We formalized in the Coq proof assistant the proof of soundness of a control function for the inverted pendulum. This is a first step towards the formal verification of more complex systems for which safety may be critical. CCS Concepts • Mathematics of computing → Ordinary differential equations; Topology; • Computer systems organization → Robotic control; • Software and its engineering → Software verification;

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The Picard Algorithm for Ordinary Differential Equations in Coq

Cited by 31 publications

References 11 publications

Formally Verified Approximations of Definite Integrals

Formally Verified Approximations of Definite Integrals

Constructive Game Logic

A formal proof in Coq of a control function for the inverted pendulum

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