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Polyhedral theory in mathlib

The goal of this project is to provide a flexible and useful implementation of polyhedral geometry/combinatorics in Lean, for mathlib, on which more advanced theory can be built. Currently we focus on polyhedral cones since convexity on affine spaces is not yet well developed in mathlib. There is a clear plan for how to move to polyhedra and polytopes eventually. To get to a point where we can implement and work with polyhedral cones comfortably, we also needed to implement duality theory, co-finitely generated subspaces, faces of cones and many more details.

Currently the project implements:

  • co-finitely generated submodules (CoFG)
  • duality for submodules w.r.t. a general bilinear pairing.
  • dual closed subspaces (DualClosed) which expresses that a subspace is its own double dual.
  • FGDual submodules, which are the dual of FG (finitely generated) submodules.
  • duality theory for FG submodules
  • dual closed pointed cones
  • FGDual pointed cones
  • duality theory for FG pointed cones, in particular, a version of the Minkowski-Weyl theorem that works to infinite dimensional modules.
  • polyhedral cones as cones that can be written as the sum of an FG cone and a submodule.
  • duality theory of polyhedral cones
  • faces and exposed faces of cones
  • the face lattice of a cone
  • face theory of polyhedral cones (all faces are exposed, the face lattice is graded, etc.)

On it's way

  • Face-lattices of cones: #33664
  • Duality operator for submodules: #34007
  • Instances for SeparatingLeft, SeparatingRight and Nondegenerate: #34487

Merged PRs

  • Co-finitely generated submodules: #34006