Abstract

We formalize a proof of Motzkin’s transposition theorem and Farkas’ lemma in Isabelle/HOL. Our proof is based on the formalization of the simplex algorithm which, given a set of linear constraints, either returns a satisfying assignment to the problem or detects unsatisfiability. By reusing facts about the simplex algorithm we show that a set of linear constraints is unsatisfiable if and only if there is a linear combination of the constraints which evaluates to a trivially unsatisfiable inequality.

BibTeX

@article{Farkas-AFP,
  author  = {Ralph Bottesch and Max W. Haslbeck and René Thiemann},
  title   = {Farkas' Lemma and Motzkin's Transposition Theorem},
  journal = {Archive of Formal Proofs},
  month   = jan,
  year    = 2019,
  note    = {\url{https://isa-afp.org/entries/Farkas.html},
            Formal proof development},
  ISSN    = {2150-914x},
}