### Abstract

We formalize results about linear inqualities, mainly from Schrijver's book. The main results are the proof of the fundamental theorem on linear inequalities, Farkas' lemma, Carathéodory's theorem, the Farkas-Minkowsky-Weyl theorem, the decomposition theorem of polyhedra, and Meyer's result that the integer hull of a polyhedron is a polyhedron itself. Several theorems include bounds on the appearing numbers, and in particular we provide an a-priori bound on mixed-integer solutions of linear inequalities.

### License

BSD License

### History

July 3, 2022
  improved bounds on determinants to get smaller mixed integer solutions (revision 6cf62d8e618e)

### Topics

- Mathematics/Algebra

### Session Linear_Inequalities

- Missing_Matrix
- Missing_VS_Connect
- Basis_Extension
- Sum_Vec_Set
- Integral_Bounded_Vectors
- Cone
- Convex_Hull
- Normal_Vector
- Dim_Span
- Fundamental_Theorem_Linear_Inequalities
- Farkas_Lemma
- Farkas_Minkowsky_Weyl
- Decomposition_Theorem
- Mixed_Integer_Solutions
- Integer_Hull

### Depends on

- A verified LLL algorithm

### Used by

- Duality of Linear Programming
- Linear Programming

## Cite

×

Linear_Inequalities-AFP

    @article{Linear_Inequalities-AFP,
      author  = {Ralph Bottesch and Alban Reynaud and René Thiemann},
      title   = {Linear Inequalities},
      journal = {Archive of Formal Proofs},
      month   = {June},
      year    = {2019},
      note    = {\url{https://isa-afp.org/entries/Linear_Inequalities.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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Older releases:

- Sep 13, 2023 : Isabelle2023
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- Dec 14, 2021 : Isabelle2021-1
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- Apr 18, 2020 : Isabelle2020
- Jun 24, 2019 : Isabelle2019