In [1] (Hybrid Conjugate Gradient Algorithm for Unconstrained Optimization J. Optimization. Theory Appl. 141 (2009) 249 - 264), an efficient hybrid conjugate gradient algorithm, the CCOMB algorithm is proposed for solving unconstrained optimization problems. However, the proof of Theorem 2.1 in [1] is incorrect due to an erroneous inequality which used to indicate the descent property for the s...

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.

In this paper we give the rst example of a signiicant piece of formal mathematics conducted in a hybrid of two diierent interactive systems. We constructively prove a theorem in Nuprl, from which a program can be extracted, but we use classical mathematics imported from HOL, and a connection to some of HOL's deenitional packages, for parts of the proof that do not contribute to the program.

Mathematicians rarely present proofs in all their detail; usually they give just an outline or abstraction of the proof. This paper describes our attempt to reproduce such activity within a computer proof system.

The Frankl’s conjecture, formulated in 1979. and still open, states that in every family of sets closed for unions there is an element contained in at least half of the sets. FC-families are families for which it is proved that every union-closed family containing them satisfies the Frankl’s condition (e.g., in every union-closed family that contains a oneelement set a, the element a is contain...