In this paper, we introduce a general method for obtaining more flexible new distributions by compounding the extended Weibull and power series distributions on a latent complementary risk problem base. The properties of the proposed class are discussed, including a formal proof of an expension for its density function and explicit formulae for its hazard rate function, quantiles, ordinary and ...

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by constructing a set of clauses from a proof with cuts. Any resolution refutation of this set can then serve as a skelet...

We propose proof script refactorings as a robust tool for constructing, restructuring, and maintaining formal proof developments. We argue that a formal approach is vital, and illustrate by defining and proving correct a number of valuable refactorings in a simplified proof script and declarative proof language of our own design.

Probabilistic coupling is a powerful tool for analyzing pairs of probabilistic processes. Roughly, coupling two processes requires finding an appropriate witness process that models both processes in the same probability space. Couplings are powerful tools proving properties about the relation between two processes, include reasoning about convergence of distributions and stochastic dominance—a...

Recent years have seen increasing success in building large formal proof developments using interactive theorem provers (ITPs). Some proofs have involved many authors, years of effort, and resulted in large, complex interdependent sets of proof “source code” files. Developing these in the first place, and maintaining and extending them afterwards, is a considerable challenge. It has prompted th...

A List of 100 Theorems Today highly nontrivial mathematics is routinely being encoded in the computer, ensuring a reliability that is orders of a magnitude larger than if one had just used human minds. Such an encoding is called a formalization, and a program that checks such a formalization for correctness is called a proof assistant. Suppose you have proved a theorem and you want to make cert...

Usually we teach formal methods relying for a large part on one kind of reasoning technique about a formal model. For instance, we either use formal proof or we use model-checking. It would appear that it is already hard enough to learn one technique and having to cope with two puts just another burden on the students. This is not our experience. Especially model-checking is easily used to comp...

This article is the first in a series of two Mizar articles constituting a formal proof of the Gödel Completeness theorem [17] for uncountably large languages. We follow the proof given in [18]. The present article contains the techniques required to expand formal languages. We prove that consistent or satisfiable theories retain these properties under changes to the language they are formulate...

A proof by mathematical induction demonstrates that a general theorem is necessarily true for all natural numbers. It has been suggested that some theorems may also be proven by a ‘visual proof by induction’ (Brown, 2010), despite the fact that the image only displays particular cases of the general theorem. In this study we examine the nature of the conclusions drawn from a visual proof by ind...