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Journal articles on the topic 'Theorem proving'

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Published: 4 June 2021

Last updated: 8 June 2024

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1

Gan, Wenbin, Xinguo Yu, Ting Zhang, and Mingshu Wang. "Automatically Proving Plane Geometry Theorems Stated by Text and Diagram." International Journal of Pattern Recognition and Artificial Intelligence 33, no. 07 (June 7, 2019): 1940003. http://dx.doi.org/10.1142/s0218001419400032.

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This paper presents an algorithm for proving plane geometry theorems stated by text and diagram in a complementary way. The problem of proving plane geometry theorems involves two challenging subtasks, being theorem understanding and theorem proving. This paper proposes to consider theorem understanding as a problem of extracting relations from text and diagram. A syntax–semantics (S2) model method is proposed to extract the geometric relations from theorem text, and a diagram mining method is proposed to extract geometry relations from diagram. Then, a procedure is developed to obtain a set of relations that is consistent with the given theorem with high confidence. Finally, theorem proving is conducted by using the existing proving methods which take the extracted geometric relations as input. The experimental results show that the proposed theorem proving algorithm can prove 86% of plane geometry theorems in the test dataset of 200 theorems, which is all the theorems in the popular textbook. The proposed algorithm outperforms the existing algorithms mainly because it can extract relations not only from text but also from diagram.

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2

Crouse, Maxwell, Ibrahim Abdelaziz, Bassem Makni, Spencer Whitehead, Cristina Cornelio, Pavan Kapanipathi, Kavitha Srinivas, Veronika Thost, Michael Witbrock, and Achille Fokoue. "A Deep Reinforcement Learning Approach to First-Order Logic Theorem Proving." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 7 (May 18, 2021): 6279–87. http://dx.doi.org/10.1609/aaai.v35i7.16780.

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Automated theorem provers have traditionally relied on manually tuned heuristics to guide how they perform proof search. Deep reinforcement learning has been proposed as a way to obviate the need for such heuristics, however, its deployment in automated theorem proving remains a challenge. In this paper we introduce TRAIL, a system that applies deep reinforcement learning to saturation-based theorem proving. TRAIL leverages (a) a novel neural representation of the state of a theorem prover and (b) a novel characterization of the inference selection process in terms of an attention-based action policy. We show through systematic analysis that these mechanisms allow TRAIL to significantly outperform previous reinforcement-learning-based theorem provers on two benchmark datasets for first-order logic automated theorem proving (proving around 15% more theorems).

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3

Xiao, Da, Yue Fei Zhu, Sheng Li Liu, Dong Xia Wang, and You Qiang Luo. "Digital Hardware Design Formal Verification Based on HOL System." Applied Mechanics and Materials 716-717 (December 2014): 1382–86. http://dx.doi.org/10.4028/www.scientific.net/amm.716-717.1382.

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This article selects HOL theorem proving systems for hardware Trojan detection and gives the symbol and meaning of theorem proving systems, and then introduces the symbol table, item and the meaning of HOL theorem proving systems. In order to solve the theorem proving the application of the system in hardware Trojan detection requirements, this article analyses basic hardware Trojan detection methods which applies for theorem proving systems and introduces the implementation methods and process of theorem proving about hardware Trojan detection.

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4

Bahodirovich, Hojiyev Dilmurodjon, Muhammadjonov Akbarshoh Akramjon Og`Li Og`Li, Muzaffarova Dilshoda Botirjon Qizi, Ibrohimjonov Islombek Ilhomjon O`G`Li, and Ahmadjonova Musharrafxon Dilmurod Qizi. "About One Theorem Of 2x2 Jordan Blocks Matrix." American Journal of Applied sciences 03, no. 06 (June 12, 2021): 28–33. http://dx.doi.org/10.37547/tajas/volume03issue06-05.

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In this paper, we have studied one theorem on 2x2 Jordan blocks matrix. There are 4 important statements which is used for proving other theorems such as in the differensial equations. In proving these statements, we have used mathematic induction, norm of matrix, Taylor series of

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5

Perron, Steven. "Examining Fragments of the Quantified Propositional Calculus." Journal of Symbolic Logic 73, no. 3 (September 2008): 1051–80. http://dx.doi.org/10.2178/jsl/1230396765.

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AbstractWhen restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory . Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas. In this paper, we prove a witnessing theorem for similar in style to the KPT witnessing theorem for . This witnessing theorem is then used to show that proves is sound with respect to formulas. Note that unless the polynomial-time hierarchy collapses is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi, p-simulates with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by plus axioms stating that the cut-free version of is sound. All together this shows that the connection between and does not extend to more complex formulas.

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6

Jupri, Al, Siti Fatimah, and Dian Usdiyana. "Dampak Perkuliahan Geometri Pada Penalaran Deduktif Mahasiswa: Kasus Pembelajaran Teorema Ceva." AKSIOMA : Jurnal Matematika dan Pendidikan Matematika 11, no. 1 (July 15, 2020): 93–104. http://dx.doi.org/10.26877/aks.v11i1.6011.

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Geometry is one of branches of mathematics that can develop deductive thinking ability for anyone, including students of prospective mathematics teachers, who learning it. This deductive thinking ability is needed by prospective mathematics teachers for their future careers as mathematics educators. This research therefore aims to investigate the influence of the learning process of a geometry course toward deductive reasoning ability of students of prospective mathematics teachers. To do so, this qualitative research was carried out through an observation of the learning process and assessment of the geometry course, involving 56 students of prospective mathematics teachers, in one of mathematics education program, in one of state universities in Bandung. A geometry topic observed in the learning process was the Ceva’s theorem, and the assessment was in the form of an individual written test on the application of the Ceva’s theorem in a proving process. The results showed that the learning process emphasizes on proving of theorems and mathematical statements. In addition, the test revealed that ten students are able to use the Ceva’s theorem in a proving process and different strategies of proving are found, including the use of properties of similarity between triangles and of the concept of trigonometry. This indicates a creativity of student deductive thinking in proving process. In conclusion, the geometry course that emphasizes on proving of theorems and mathematical statements has influenced on filexibility of student deductive thinking in proving processes.

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7

Stickel, M. E. "Resolution Theorem Proving." Annual Review of Computer Science 3, no. 1 (June 1988): 285–316. http://dx.doi.org/10.1146/annurev.cs.03.060188.001441.

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8

Gogate, Vibhav, and Pedro Domingos. "Probabilistic theorem proving." Communications of the ACM 59, no. 7 (June 24, 2016): 107–15. http://dx.doi.org/10.1145/2936726.

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9

Klein, Gerwin, and Ruben Gamboa. "Interactive Theorem Proving." Journal of Automated Reasoning 56, no. 3 (February 20, 2016): 201–3. http://dx.doi.org/10.1007/s10817-016-9363-7.

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10

Plaisted, David A. "Automated theorem proving." Wiley Interdisciplinary Reviews: Cognitive Science 5, no. 2 (January 17, 2014): 115–28. http://dx.doi.org/10.1002/wcs.1269.

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11

KANSO, KARIM, and ANTON SETZER. "A light-weight integration of automated and interactive theorem proving." Mathematical Structures in Computer Science 26, no. 1 (November 12, 2014): 129–53. http://dx.doi.org/10.1017/s0960129514000140.

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In this paper, aimed at dependently typed programmers, we present a novel connection between automated and interactive theorem proving paradigms. The novelty is that the connection offers a better trade-off between usability, efficiency and soundness when compared to existing techniques. This technique allows for a powerful interactive proof framework that facilitates efficient verification of finite domain theorems and guided construction of the proof of infinite domain theorems. Such situations typically occur with industrial verification. As a case study, an embedding of SAT and CTL model checking is presented, both of which have been implemented for the dependently typed proof assistant Agda.Finally, an example of a real world railway control system is presented, and shown using our proof framework to be safe with respect to an abstract model of trains not colliding or derailing. We demonstrate how to formulate safety directly and show using interactive theorem proving that signalling principles imply safety. Therefore, a proof by an automated theorem prover that the signalling principles hold for a concrete system implies the overall safety. Therefore, instead of the need for domain experts to validate that the signalling principles imply safety they only need to make sure that the safety is formulated correctly. Therefore, some of the validation is replaced by verification using interactive theorem proving.

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12

Hsiang, Jieh, and Michaël Rusinowitch. "Proving refutational completeness of theorem-proving strategies." Journal of the ACM 38, no. 3 (July 1991): 558–86. http://dx.doi.org/10.1145/116825.116833.

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13

Voronkov, A. A., and A. I. Degtyarev. "Automatic theorem proving. I." Cybernetics 22, no. 3 (May 1986): 290–97. http://dx.doi.org/10.1007/bf01069967.

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14

Bertrand, P. "Simply proving Pythagoras's theorem." Teaching Mathematics and its Applications 15, no. 1 (March 1, 1996): 10–11. http://dx.doi.org/10.1093/teamat/15.1.10.

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15

Nossum, Rolf. "Automated theorem proving methods." BIT 25, no. 1 (March 1985): 51–64. http://dx.doi.org/10.1007/bf01934987.

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16

Cooper, D. "Overview of Theorem Proving." ACM SIGSOFT Software Engineering Notes 10, no. 4 (August 1985): 53–54. http://dx.doi.org/10.1145/1012497.1012517.

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17

RUSSELL, STEPHEN, and TRACI WHEELER UNISYS. "On Automated Theorem Proving." Annals of the New York Academy of Sciences 661, no. 1 Frontiers of (December 1992): 160–73. http://dx.doi.org/10.1111/j.1749-6632.1992.tb26040.x.

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18

Voronkov, A. A., and A. I. Degtyarev. "Automatic theorem proving. II." Cybernetics 23, no. 4 (1988): 547–56. http://dx.doi.org/10.1007/bf01078915.

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19

Policriti, Alberto, and Jacob T. Schwartz. "T-Theorem Proving I." Journal of Symbolic Computation 20, no. 3 (September 1995): 315–42. http://dx.doi.org/10.1006/jsco.1995.1053.

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20

López-Renteria, Jorge-Antonio, Baltazar Aguirre-Hernández, and Fernando Verduzco. "The Boundary Crossing Theorem and the Maximal Stability Interval." Mathematical Problems in Engineering 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/123403.

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The boundary crossing theorem and the zero exclusion principle are very useful tools in the study of the stability of family of polynomials. Although both of these theorem seem intuitively obvious, they can be used for proving important results. In this paper, we give generalizations of these two theorems and we apply such generalizations for finding the maximal stability interval.

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21

Chen, Baoying. "The Related Extension and Application of the Ši'lnikov Theorem." Journal of Applied Mathematics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/287123.

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The traditional Ši'lnikov theorems provide analytic criteria for proving the existence of chaos in high-dimensional autonomous systems. We have established one extended version of the Ši'lnikov homoclinic theorem and have given a set of sufficient conditions under which the system generates chaos in the sense of Smale horseshoes. In this paper, the extension questions of the Ši'lnikov homoclinic theorem and its applications are still discussed. We establish another extended version of the Ši'lnikov homoclinic theorem. In addition, we construct a new three-dimensional chaotic system which meets all the conditions in this extended Ši'lnikov homoclinic theorem. Finally, we list all well-known three-dimensional autonomous quadratic chaotic systems and classify them in the light of the Ši'lnikov theorems.

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22

Coghetto, Roland. "Pascal’s Theorem in Real Projective Plane." Formalized Mathematics 25, no. 2 (July 1, 2017): 107–19. http://dx.doi.org/10.1515/forma-2017-0011.

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Summary In this article we check, with the Mizar system [2], Pascal’s theorem in the real projective plane (in projective geometry Pascal’s theorem is also known as the Hexagrammum Mysticum Theorem)1. Pappus’ theorem is a special case of a degenerate conic of two lines. For proving Pascal’s theorem, we use the techniques developed in the section “Projective Proofs of Pappus’ Theorem” in the chapter “Pappus’ Theorem: Nine proofs and three variations” [11]. We also follow some ideas from Harrison’s work. With HOL Light, he has the proof of Pascal’s theorem2. For a lemma, we use PROVER93 and OTT2MIZ by Josef Urban4 [12, 6, 7]. We note, that we don’t use Skolem/Herbrand functions (see “Skolemization” in [1]).

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23

Alsaadi, Ateq, Bijender Singh, Vizender Singh, and Izhar Uddin. "Meir–Keeler Type Contraction in Orthogonal M-Metric Spaces." Symmetry 14, no. 9 (September 6, 2022): 1856. http://dx.doi.org/10.3390/sym14091856.

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In this article, we prove fixed point results for a Meir–Keeler type contraction due to orthogonal M-metric spaces. The results of the paper improve and extend some recent developments in fixed point theory. The extension is assured by the concluding remarks and followed by the main theorem. Finally, an application of the main theorem is established in proving theorems for some integral equations and integral-type contractive conditions.

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24

Wan, Hai, Anping He, Zhiyang You, and Xibin Zhao. "Formal Proof of a Machine Closed Theorem in Coq." Journal of Applied Mathematics 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/892832.

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The paper presents a formal proof of a machine closed theorem of TLA+in the theorem proving system Coq. A shallow embedding scheme is employed for the proof which is independent of concrete syntax. Fundamental concepts need to state that the machine closed theorems are addressed in the proof platform. A useful proof pattern of constructing a trace with desired properties is devised. A number of Coq reusable libraries are established.

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25

Al-Hawasy, Jamil A. Ali, and Lamyaa H. Ali. "Constraints Optimal Control Governing by Triple Nonlinear Hyperbolic Boundary Value Problem." Journal of Applied Mathematics 2020 (April 10, 2020): 1–14. http://dx.doi.org/10.1155/2020/8021635.

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The focus of this work lies on proving the existence theorem of a unique state vector solution (Stvs) of the triple nonlinear hyperbolic boundary value problem (TNHBVP) when the classical continuous control vector (CCCVE) is fixed by using the Galerkin method (Galm), proving the existence theorem of a unique constraints classical continuous optimal control vector (CCCOCVE) with vector state constraints (equality EQVC and inequality INEQVC). Also, it consists of studying for the existence and uniqueness adjoint vector solution (Advs) of the triple adjoint vector equations (TAEqs) associated with the considered triple state equations (Tsteqs). The Fréchet Derivative (Frde.) of the Hamiltonian (HAM) is found. At the end, the theorems for the necessary conditions and the sufficient conditions of optimality (Necoop and Sucoop) are achieved.

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26

Dalinger, Victor Alekseevich. "Revisiting a Proof of the Sine Theorem." Development of education, no. 1 (7) (March 13, 2020): 16–18. http://dx.doi.org/10.31483/r-74764.

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The author of the article outlines, that in school geometry course, the sine theorem and the cosine theorem are well known. In this course, they are proved by the authors of the textbook in a way different from the one that is presented in the article. The article considers the author's method of proving the sine theorem unknown in the literature sources and based on the vector-coordinate method; two more theorems are also proved, one of which concerns the calculation of the inscribed angle in the circle, and the other concerns the calculation of viewing angles of the chord of the circle; one task is proposed by the author for independent work, in which the object of research is the chord of a circle. Research results. A proof of the sine theorem, presented by the author on the basis of his technique, undoubtedly quicken public interest. The material can serve as a basis for organizing educational and research activities of students in mathematics. It is concluded that educational and research activities of students in mathematics can be effectively organized when: establishing the essential properties of concepts; identifying the relationship of this concept with other concepts; searching for other methods of proving theorems; formulating the inverse theorem and establishing its truth; classification of mathematical objects and relations between them; solving mathematical problems in various ways and methods; drawing up new tasks that result from already solved ones; providing examples and counterexamples that illustrate a particular fact, etc.

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27

Jureczko, Joanna. "Strong sequences and partition relations." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 16, no. 1 (December 1, 2017): 51–59. http://dx.doi.org/10.1515/aupcsm-2017-0004.

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AbstractThe first result in partition relations topic belongs to Ramsey (1930). Since that this topic has been still explored. Probably the most famous partition theorem is Erdös-Rado theorem (1956). On the other hand in 60’s of the last century Efimov introduced strong sequences method, which was used for proving some famous theorems in dyadic spaces. The aim of this paper is to generalize theorem on strong sequences and to show that it is equivalent to generalized version of well-known Erdös-Rado theorem. It will be also shown that this equivalence holds for singulars. Some applications and conclusions will be presented too.

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28

Dawson, C. Bryan. "L-correspondences: the inclusionLp(μ,X)⊂Lq(ν,Y)." International Journal of Mathematics and Mathematical Sciences 19, no. 4 (1996): 723–26. http://dx.doi.org/10.1155/s0161171296000993.

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In order to study inclusions of the typeLp(μ,X)⊂Lq(ν,Y), we introduce the notion of anL-correspondence. After proving some basic theorems, we give characterizations of some types ofL-correspondences and offer a conjecture that is similar to an equimeasurability theorem.

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29

Hamdani, Deni, Ketut Sarjana, Ratna Yulis Tyaningsih, Ulfa Lu’luilmaknun, and J. Junaidi. "Exploration of Student Thinking Process in Proving Mathematical Statements." Prisma Sains : Jurnal Pengkajian Ilmu dan Pembelajaran Matematika dan IPA IKIP Mataram 8, no. 2 (December 27, 2020): 150. http://dx.doi.org/10.33394/j-ps.v8i2.3081.

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A mathematical statement is not a theorem until it has been carefully derived from previously proven axioms, definitions and theorems. The proof of a theorem is a logical argument that is given deductively and is often interpreted as a justification for statements as well as a fundamental part of the mathematical thinking process. Studying the proof can help decide if and why our answers are logical, develop the habit of arguing, and make investigating an integral part of any problem solving. However, not a few students have difficulty learning it. So it is necessary to explore the student's thought process in proving a statement through questions, answer sheets, and interviews. The ability to prove is explored through 4 (four) proof schemes, namely Scheme of Complete Proof, Scheme of Incomplete Proof, Scheme of unrelated proof, and Scheme of Proof is immature. The results obtained indicate that the ability to prove is influenced by understanding and the ability to see that new theorems are built on previous definitions, properties and theorems; and how to present proof and how students engage with proof. Suggestions in this research are to change the way proof is presented, and to change the way students are involved in proof; improve understanding through routine proving new mathematical statements; and developing course designs that can turn proving activities into routine activities.

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30

Thakur, Ramkrishna, and S. K. Samanta. "A Study on Some Fundamental Properties of Continuity and Differentiability of Functions of Soft Real Numbers." Advances in Fuzzy Systems 2018 (2018): 1–8. http://dx.doi.org/10.1155/2018/6429572.

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We introduce a new type of functions from a soft set to a soft set and study their properties under soft real number setting. Firstly, we investigate some properties of soft real sets. Considering the partial order relation of soft real numbers, we introduce concept of soft intervals. Boundedness of soft real sets is defined, and the celebrated theorems like nested intervals theorem and Bolzano-Weierstrass theorem are extended in this setting. Next, we introduce the concepts of limit, continuity, and differentiability of functions of soft sets. It has been possible for us to study some fundamental results of continuity of functions of soft sets such as Bolzano’s theorem, intermediate value property, and fixed point theorem. Because the soft real numbers are not linearly ordered, several twists in the arguments are required for proving those results. In the context of differentiability of functions, some basic theorems like Rolle’s theorem and Lagrange’s mean value theorem are also extended in soft setting.

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31

Cao, Daming, Lin Zhou, and Vincent Y. F. Tan. "A Strong Converse Theorem for Hypothesis Testing Against Independence over a Two-Hop Network." Entropy 21, no. 12 (November 29, 2019): 1171. http://dx.doi.org/10.3390/e21121171.

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By proving a strong converse theorem, we strengthen the weak converse result by Salehkalaibar, Wigger and Wang (2017) concerning hypothesis testing against independence over a two-hop network with communication constraints. Our proof follows by combining two recently-proposed techniques for proving strong converse theorems, namely the strong converse technique via reverse hypercontractivity by Liu, van Handel, and Verdú (2017) and the strong converse technique by Tyagi and Watanabe (2018), in which the authors used a change-of-measure technique and replaced hard Markov constraints with soft information costs. The techniques used in our paper can also be applied to prove strong converse theorems for other multiterminal hypothesis testing against independence problems.

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32

Wu, Shuchen. "Proof and Application of the Mean Value Theorem." Highlights in Science, Engineering and Technology 72 (December 15, 2023): 565–71. http://dx.doi.org/10.54097/nw2nd028.

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In calculus, mean value theorem (MVT) connects a function's derivative and its rate of change over a certain interval. This paper delves into the mathematical intricacies of the MVT and its multifaceted applications. Through rigorous proofs and illustrative examples, this study establishes the MVT's fundamental role in calculus and its relevance in understanding the behavior of functions. The paper extends its exploration to encompass related theorems, including extreme value theorem, which connects function’s continuity and extrema, Intermediate Value Theorem, which states that the function value within an interval of a continuous function must be between the maximum and minimum values, local extreme value theorem, Rolle’s theorem, a specific situation of the theorem, and the integral MVT, an application in integral aspect of MVT, further enriching the comprehension of these pivotal concepts. These theorems provide powerful tools for understanding the properties of continuous functions, identifying critical points, and establishing relationships between function values and their derivatives. This paper highlights the significance of proving these theorems and solving mathematical problems as applications. Through a systematic exploration of the mathematical foundations, this paper contributes to a deeper comprehension of the core principles underlying calculus and their applied theorems in different contexts.

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33

Leslie-Hurd, Joe, and G. McC Haworth. "Computer Theorem Proving and HoTT." ICGA Journal 36, no. 2 (June 1, 2013): 100–103. http://dx.doi.org/10.3233/icg-2013-36204.

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34

Jones, C. B. "Theorem proving and software engineering." Software Engineering Journal 3, no. 1 (1988): 2. http://dx.doi.org/10.1049/sej.1988.0001.

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35

Chen, Chiyan, and Hongwei Xi. "Combining programming with theorem proving." ACM SIGPLAN Notices 40, no. 9 (September 12, 2005): 66–77. http://dx.doi.org/10.1145/1090189.1086375.

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36

Murthy, S., and K. Sekharam. "Software Reliability through Theorem Proving." Defence Science Journal 59, no. 3 (May 25, 2009): 314–17. http://dx.doi.org/10.14429/dsj.59.1527.

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37

Finger, M. "Towards structurally-free theorem proving." Logic Journal of IGPL 6, no. 3 (May 1, 1998): 425–49. http://dx.doi.org/10.1093/jigpal/6.3.425.

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38

Fleuriot, J. "Theorem proving in infinitesimal geometry." Logic Journal of IGPL 9, no. 3 (May 1, 2001): 447–74. http://dx.doi.org/10.1093/jigpal/9.3.447.

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39

Ginsberg, Matthew L. "The MVL theorem proving system." ACM SIGART Bulletin 2, no. 3 (June 1991): 57–60. http://dx.doi.org/10.1145/122296.122304.

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40

Peltier, Nicolas, and Viorica Sofronie-Stokkermans. "First-order theorem proving: Foreword." Journal of Symbolic Computation 47, no. 9 (September 2012): 1009–10. http://dx.doi.org/10.1016/j.jsc.2011.12.030.

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41

Dehornoy, Patrick, and Abderrahim Marzouk. "Theorem proving by chain resolution." Theoretical Computer Science 206, no. 1-2 (October 1998): 163–80. http://dx.doi.org/10.1016/s0304-3975(97)00128-x.

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42

Indo, Kenryo. "Proving Arrow’s theorem by PROLOG." Computational Economics 30, no. 1 (March 13, 2007): 57–63. http://dx.doi.org/10.1007/s10614-007-9086-2.

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43

Ramsay, Allan. "Theorem proving for intensional logic." Journal of Automated Reasoning 14, no. 2 (1995): 237–55. http://dx.doi.org/10.1007/bf00881857.

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44

Pastre, Dominique. "Automated theorem proving in mathematics." Annals of Mathematics and Artificial Intelligence 8, no. 3-4 (September 1993): 425–47. http://dx.doi.org/10.1007/bf01530801.

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45

Sari, C. K., M. Waluyo, C. M. Ainur, and E. N. Darmaningsih. "Logical errors on proving theorem." Journal of Physics: Conference Series 948 (January 2018): 012059. http://dx.doi.org/10.1088/1742-6596/948/1/012059.

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46

Simpson, Carlos. "Computer Theorem Proving in Mathematics." Letters in Mathematical Physics 69, no. 1-3 (July 2004): 287–315. http://dx.doi.org/10.1007/s11005-004-0607-9.

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47

Hübner, M., S. Autexier, C. Benzmüller, and A. Meier. "Interactive Theorem Proving with Tasks." Electronic Notes in Theoretical Computer Science 103 (November 2004): 161–81. http://dx.doi.org/10.1016/j.entcs.2004.02.021.

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48

Humphreys, A. James, and Stephen G. Simpson. "Separation and Weak König's Lemma." Journal of Symbolic Logic 64, no. 1 (March 1999): 268–78. http://dx.doi.org/10.2307/2586763.

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AbstractWe continue the work of [14, 3, 1, 19, 16, 4, 12, 11, 20] investigating the strength of set existence axioms needed for separable Banach space theory. We show that the separation theorem for open convex sets is equivalent to WKL0 over RCA0. We show that the separation theorem for separably closed convex sets is equivalent to ACA0 over RCA0. Our strategy for proving these geometrical Hahn–Banach theorems is to reduce to the finite-dimensional case by means of a compactness argument.

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Fuhs, Carsten, Jürgen Giesl, Michael Parting, Peter Schneider-Kamp, and Stephan Swiderski. "Proving Termination by Dependency Pairs and Inductive Theorem Proving." Journal of Automated Reasoning 47, no. 2 (January 15, 2011): 133–60. http://dx.doi.org/10.1007/s10817-010-9215-9.

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50

Hovhannisyan, Gro. "On Oscillations of Solutions of Third-Order Dynamic Equation." Abstract and Applied Analysis 2012 (2012): 1–15. http://dx.doi.org/10.1155/2012/715981.

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We are proving the new oscillation theorems for the solutions of third-order linear nonautonomous differential equation with complex coefficients. In the case of real coefficients we derive the oscillation criterion that is invariant with respect to the adjoint transformation. Our main tool is a new version of Levinson's asymptotic theorem.

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