### 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.

### License

BSD License

### Topics

- Mathematics/Algebra

### Session Farkas

- Farkas
- Matrix_Farkas
- Simplex_for_Reals

### Depends on

- Matrices, Jordan Normal Forms, and Spectral Radius Theory
- An Incremental Simplex Algorithm with Unsatisfiable Core Generation

### Used by

- Linear Programming

## Cite

×

Farkas-AFP

    @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   = {January},
      year    = {2019},
      note    = {\url{https://isa-afp.org/entries/Farkas.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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Older releases:

- Sep 13, 2023 : Isabelle2023
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- Jan 21, 2019 : Isabelle2018