Cut-Simulation and Impredicativity

We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for classical type theory -- is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.Comment: 21 page

Higher-Order Tarski Grothendieck as a Foundation for Formal Proof

We formally introduce a foundation for computer verified proofs based on higher-order Tarski-Grothendieck set theory. We show that this theory has a model if a 2-inaccessible cardinal exists. This assumption is the same as the one needed for a model of plain Tarski-Grothendieck set theory. The foundation allows the co-existence of proofs based on two major competing foundations for formal proofs: higher-order logic and TG set theory. We align two co-existing Isabelle libraries, Isabelle/HOL and Isabelle/Mizar, in a single foundation in the Isabelle logical framework. We do this by defining isomorphisms between the basic concepts, including integers, functions, lists, and algebraic structures that preserve the important operations. With this we can transfer theorems proved in higher-order logic to TG set theory and vice versa. We practically show this by formally transferring Lagrange\u27s four-square theorem, Fermat 3-4, and other theorems between the foundations in the Isabelle framework

Proofgold: Blockchain for Formal Methods

Proofgold is a peer to peer cryptocurrency making use of formal logic. Users can publish theories and then develop a theory by publishing documents with definitions, conjectures and proofs. The blockchain records the theories and their state of development (e.g., which theorems have been proven and when). Two of the main theories are a form of classical set theory (for formalizing mathematics) and an intuitionistic theory of higher-order abstract syntax (for reasoning about syntax with binders). We have also significantly modified the open source Proofgold Core client software to create a faster, more stable and more efficient client, Proofgold Lava. Two important changes are the cryptography code and the database code, and we discuss these improvements. We also discuss how the Proofgold network can be used to support large formalization efforts

Analytic Tableaux for Simple Type Theory and its First-Order Fragment

We study simple type theory with primitive equality (STT) and its first-order fragment EFO, which restricts equality and quantification to base types but retains lambda abstraction and higher-order variables. As deductive system we employ a cut-free tableau calculus. We consider completeness, compactness, and existence of countable models. We prove these properties for STT with respect to Henkin models and for EFO with respect to standard models. We also show that the tableau system yields a decision procedure for three EFO fragments

AIM Loops and the AIM Conjecture

In this article, we prove, using the Mizar [2] formalism, a number of properties that correspond to the AIM Conjecture. In the first section, we define division operations on loops, inner mappings T, L and R, commutators and associators and basic attributes of interest. We also consider subloops and homomorphisms. Particular subloops are the nucleus and center of a loop and kernels of homomorphisms. Then in Section 2, we define a set Mlt Q of multiplicative mappings of Q and cosets (mostly following Albert 1943 for cosets [1]). Next, in Section 3 we define the notion of a normal subloop and construct quotients by normal subloops. In the last section we define the set InnAut of inner mappings of Q, define the notion of an AIM loop and relate this to the conditions on T, L, and R defined by satisfies TT, etc. We prove in Theorem (67) that the nucleus of an AIM loop is normal and finally in Theorem (68) that the AIM Conjecture follows from knowing every AIM loop satisfies aa1, aa2, aa3, Ka, aK1, aK2 and aK3. The formalization follows M.K. Kinyon, R. Veroff, P. Vojtechovsky [4] (in [3]) as well as Veroff’s Prover9 files.This work has been supported by the European Research Council (ERC) Consolidator grant nr. 649043 AI4REASON and the Polish National Science Centre granted by decision no. DEC-2015/19/D/ST6/01473.Chad E. Brown - Český Institut Informatiky Robotiky a Kybernetiky, Zikova 4, 166 36 Praha 6, Czech RepublicKarol Pąk - Institute of Informatics, University of Białystok, PolandA. A. Albert. Quasigroups. I. Transactions of the American Mathematical Society, 54(3): 507–519, 1943.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Maria Paola Bonacina and Mark E. Stickel, editors. Automated Reasoning and Mathematics – Essays in Memory of William W. McCune, volume 7788 of Lecture Notes in Computer Science, 2013. Springer.Michael K. Kinyon, Robert Veroff, and Petr Vojtěchovský. Loops with abelian inner mapping groups: An application of automated deduction. In Bonacina and Stickel [3], pages 151–164.Christoph Schwarzweller and Artur Korniłowicz. Characteristic of rings. Prime fields. Formalized Mathematics, 23(4):333–349, 2015. doi:10.1515/forma-2015-0027.27432133

Acid-Suppressing Agents and Risk for Clostridium difficile Infection in Pediatric Patients

Background. Acid-suppressing agents have been associated with increased Clostridium difficile infection (CDI) in adults. The objective of this study was to evaluate the association of acid-suppressing therapy with the development of CDI in the pediatric population. Methods. This was a retrospective case-control study. Children aged 1 through 17 years with a positive C difficile polymerase chain reaction (PCR) result obtained between June 1, 2008, and June 1, 2012, were randomly matched to a control population selected from patients with negative PCR. Results. A total of 458 children were included. No difference was observed in acid-suppressive therapy prior to PCR in CDI-positive versus -negative patients (n = 131 [57.2%] vs n = 121 [52.8%], P = .348). Among patients receiving acid-suppressing therapy prior to obtaining a PCR, no difference was observed in proton pump inhibitor use (45% vs 46.3%, P = .843), but histamine-2 receptor antagonist (H2RA) use was greater in the CDI-positive patients (32.8% vs 14.9%, P = .001). Logistic regression analysis demonstrated that H2RA therapy at home (odds ratio = 4.6; 95% confidence interval = 1.5-14.5) was an independent CDI predictor. Conclusion. In this pediatric population, CDI risk in children receiving home acid-suppressive therapy with H2RAs is nearly 4.5 times greater than that of children not receiving H2RA therapy. These results suggest the need for continued monitoring and study of H2RA therapy in children

Limits on Quaoar's Atmosphere

Here we present high cadence photometry taken by the Acquisition Camera on Gemini South, of a close passage by the ~540 km radius Kuiper belt object, (50000) Quaoar, of a r' = 20.2 background star. Observations before and after the event show that the apparent impact parameter of the event was 0."019 ± 0."004, corresponding to a close approach of 580 ± 120 km to the center of Quaoar. No signatures of occultation by either Quaoar's limb or its potential atmosphere are detectable in the relative photometry of Quaoar and the target star, which were unresolved during closest approach. From this photometry we are able to put constraints on any potential atmosphere Quaoar might have. Using a Markov chain Monte Carlo and likelihood approach, we place pressure upper limits on sublimation supported, isothermal atmospheres of pure N_2, CO, and CH_4. For N_2 and CO, the upper limit surface pressures are 1 and 0.7 μbar, respectively. The surface temperature required for such low sublimation pressures is ~33 K, much lower than Quaoar's mean temperature of ~44 K measured by others. We conclude that Quaoar cannot have an isothermal N_2 or CO atmosphere. We cannot eliminate the possibility of a CH_4 atmosphere, but place upper surface pressure and mean temperature limits of ~138 nbar and ~44 K, respectively