Sentence examples for modern proof from inspiring English sources

Observing that the problem of resolving composite numbers into prime factors is "one of the most important and useful in arithmetic," Gauss provided the first modern proof of the unique factorization theorem.

A short modern proof can be found in (Friz and Hairer 2014, Chapter 4).

We will therefore begin our account of the historical development of intuitionistic logic with Brouwer's ideas, and then show how, via Heyting and others, the modern Proof Interpretation was arrived at.

(A modern proof is presented in Göller, Lohrey and Lutz [2007].) The complexity gap between this double exponential-time upper bound for deciding (IPDL-SAT) and the simple exponential-time lower bound for deciding (PDL-SAT) obtained by Fischer and Ladner [1979] remained open for more than twenty years.

The publication was obscure, and it is unclear if his proof directly involved 0.999..., but at least one modern proof by W. G. Leavitt does.

To end, it presents three modern proofs of Siegel's theorem on integral points.

This proof is strikingly similar both in structure and in subject to modern proofs of the redundancy of axioms in a system.

Modern proofs of the Church Rosser theorem for CL proceed indirectly because one-step reduction fails to have the diamond property.

In a more modern proof-theoretical approach, however, definition rules often provide a more convenient alternative.

The new testing brings an element of modern scientific proof to one murder in a case that seemed likely never to be solved, especially after Mr. DeSalvo was killed in 1973 in prison, where he had been sentenced to life for unrelated sexual assault and robbery charges.

For the interested reader the essential steps of the modern probabilistic proof are sketched.