Challenge to the proving of theorems of Euclid's elements of as ATP

Automated theorem proving (ATP) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. An Isabelle/HOL is a generic proof assistant. We perform the challenge for the proving theorems of Euclid's elements of geometry. We could prove some theorems of Euclid's elements of geometry. Technique of programing and mental conception interact. The mathematics education which prove theorems of the Euclidean geometry by using the Isabelle/HOL can correct the present weak point

An Implementation of The Scalable and Correct Time-Stamped Stack

Many concurrent data structures impose real time ordering over their elements. This is needed only if the insertions modifying the data structure ran sequentially. A new approach using time stamps was proposed to avoid unneeded ordering. Our implementation is based on that time-stamped (TS) stack. Concurrent insertions can be left unordered and then ordered as necessary at removal. Because of this weak ordering, using linearizability to establish correctness is not possible. The original paper presents a new approach to proving correctness for the TS stack. This proof technique is a new, generic proof for correctness on concurrent data structures. In this paper, we highlight our general approach to re-implementing the Time-Stamped stack, discuss our modifications made to to the stack, give an overview of our implementation of a stack using software transactional memory, and analyze comparative performance graphs based on our experimental data.

Critical densities for Korteweg–de Vries-like acoustic solitons in multi-ion plasmas

A generic proof has been given that, for the acoustic mode with the highest velocity in a plasma comprising a number of fluid species and one kind of inertialess electrons, even though there can be critical densities (making the coefficient of the quadratic nonlinearity in a Korteweg–de Vries equation vanish), no supercritical densities exist (requiring the simultaneous annulment of both the quadratic and cubic nonlinearities in a reductive perturbation treatment). Similar conclusions hold upon expansion of the corresponding Sagdeev pseudopotential treatment. When there is only one (hot) electron species, the highest-velocity mode is an ion-acoustic one, but if there is an additional cool electron species, with its inertia taken into account, the highest-velocity mode is an electron-acoustic mode in a two-temperature plasma. The cool fluid species can have various polytropic pressure–density relations, including adiabatic and/or isothermal variations, whereas the hot inertialess electrons are modelled by extensions of the usual Boltzmann description that include non-thermal effects through Cairns, kappa or Tsallis distributions. Together, in this way quite a number of plasma models are covered. Unfortunately, there seems to be no equivalent generic statement for the slow modes, so that these have to be studied on a case-by-case basis, which for models with more than three species is far from straightforward, given the parameter ranges to be discussed.

Ergo 6: A Generic Proof Engine that Uses Prolog Proof Technology

To support formal reasoning in mathematical and software engineering applications, it is desirable to have a generic prover that can be instantiated with a range of logics. This allows the prover to be applied to a wider variety of reasoning tasks than a fixed-logic prover. This paper describes the design principles and the architecture of the latest version of the Ergo proof engine, Ergo 6. Ergo 6 is a generic interactive theorem prover, similar to Isabelle, but with better support for proving schematic theorems with user-defined constraints, and with a different approach to handling variable scoping. A major theme of the paper is that Prolog implementation technology can be generalized to obtain efficient implementations of generic proof engines. This is demonstrated via a Qu-Prolog implementation of Ergo 6.