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French cyclist Aurelien Fontenoy has set a new record for the fastest climb of the Eiffel Tower stairs on a bike.

Advice from special relativity: avert your eyes as that bicycle whizzes by.

Advice from special relativity: avert your eyes as that bicycle whizzes by.

Advice from special relativity: avert your eyes as that bicycle whizzes by.

Oxygen uptake (VO.sub.2) is one of the most important measures of fitness and critical vital sign. Cardiopulmonary exercise testing (CPET) is a valuable method of assessing fitness in sport and clinical settings. There is a lack of large studies on athletic populations to predict VO.sub.2max using somatic or submaximal CPET variables. Thus, this study aimed to: (1) derive prediction models for maximal VO.sub.2 (VO.sub.2max) based on submaximal exercise variables at anaerobic threshold (AT) or respiratory compensation point (RCP) or only somatic and (2) internally validate provided equations. Four thousand four hundred twenty-four male endurance athletes (EA) underwent maximal symptom-limited CPET on a treadmill (n=3330) or cycle ergometer (n=1094). The cohort was randomly divided between: variables selection (n.sub.runners = 1998; n.sub.cyclist = 656), model building (n.sub.runners = 666; n.sub.cyclist = 219), and validation (n.sub.runners = 666; n.sub.cyclist = 219). Random forest was used to select the most significant variables. Models were derived and internally validated with multiple linear regression. Runners were 36.24[plus or minus]8.45 years; BMI = 23.94 [plus or minus] 2.43 kg.m.sup.-2; VO.sub.2max=53.81[plus or minus]6.67 mL.min.sup.-1.kg.sup.-1. Cyclists were 37.33[plus or minus]9.13 years; BMI = 24.34 [plus or minus] 2.63 kg.m.sup.-2; VO.sub.2max=51.74[plus or minus]7.99 mL.min.sup.-1.kg.sup.-1. VO.sub.2 at AT and RCP were the most contributing variables to exercise equations. Body mass and body fat had the highest impact on the somatic equation. Model performance for VO.sub.2max based on variables at AT was R.sup.2=0.81, at RCP was R.sup.2=0.91, at AT and RCP was R.sup.2=0.91 and for somatic-only was R.sup.2=0.43. Derived prediction models were highly accurate and fairly replicable. Formulae allow for precise estimation of VO.sub.2max based on submaximal exercise performance or somatic variables. Presented models are applicable for sport and clinical settling. They are a valuable supplementary method for fitness practitioners to adjust individualised training recommendations.

Cyclic induction is a powerful reasoning technique that consists in blocking the proof development of certain subgoals already encountered during the proof process. In the setting of first-order logic with inductive definitions and equality ([FOL.sub.ID]), cyclic proofs can be built automatically by the Cyclist prover, but their implementations are error-prone and the human validation may be tedious. On the other hand, cyclic induction is not yet integrated into certifying proof environments that support first-order logic and inductive definitions, such as Isabelle and Coq. We propose a solution to check, using Coq, the cyclic proofs produced by E-Cyclist, an extension of Cyclist that implements a more efficient soundness validation method, by using the general Noetherian induction principle integrated into Coq. Our work is based on a methodology for certifying first-order formula-based Noetherian induction proofs, such as those based on implicit induction. The advantages of our approach are threefold: -1) The certification of cyclic [FOL.sub.ID] proofs is mechanical. Coq can validate every single step from the E-Cyclist proofs, as well as the induction arguments; also, it helps to identify errors in a very precise way.--II) There is a great potential for automation. The methodology has already been used to automatically convert to Coq scripts implicit induction proofs.--III) Cyclic induction can be directly performed in Coq. Coq functions are provided to manage the induction part. Keywords Automated reasoning * Cyclic induction * First-order logic with inductive definitions * Proof certification * Coq * E-Cyclist