This Coq library contains a subset of the work that was developed in the context of the ForMath EU FP7 project (2009-2013). It has two parts:

• theory, which contains developments in algebra including normal forms of matrices, and optimized algorithms on MathComp data structures.
• refinements, which is a framework to ease change of data representations during a proof.

• Author(s):
  • Guillaume Cano (initial)
  • Cyril Cohen (initial)
  • Maxime Dénès (initial)
  • Érik Martin-Dorel
  • Anders Mörtberg (initial)
  • Damien Rouhling
  • Pierre Roux
  • Vincent Siles (initial)
• Coq-community maintainer(s):
  • Cyril Cohen (@CohenCyril)
  • Pierre Roux (@proux01)
• License: MIT License
• Compatible Coq versions: 8.16 or later (use releases for other Coq versions)
• Additional dependencies:
  • Bignums same version as Coq
  • Paramcoq 1.1.3 or later
  • Hierarchy Builder 1.4.0 or later
  • MathComp ssreflect 2.1 or later
  • MathComp algebra 2.1 or later
  • MathComp Multinomials 2.0 or later
  • MathComp real-closed 2.0 or later
• Coq namespace: CoqEAL
• Related publication(s):
  • A refinement-based approach to computational algebra in Coq doi:10.1007/978-3-642-32347-8_7
  • Refinements for free! doi:10.1007/978-3-319-03545-1_10
  • A Coq Formalization of Finitely Presented Modules doi:10.1007/978-3-319-08970-6_13
  • Formalized Linear Algebra over Elementary Divisor Rings in Coq doi:10.2168/LMCS-12(2:7)2016
  • A refinement-based approach to large scale reflection for algebra
  • Interaction entre algèbre linéaire et analyse en formalisation des mathématiques
  • A formal proof of Sasaki-Murao algorithm doi:10.6092/issn.1972-5787/2615
  • Formalizing Refinements and Constructive Algebra in Type Theory
  • Coherent and Strongly Discrete Rings in Type Theory doi:10.1007/978-3-642-35308-6_21

The easiest way to install the latest released version of CoqEAL is via OPAM:

opam repo add coq-released https://coq.inria.fr/opam/released
opam install coq-coqeal

To instead build and install manually, do:

git clone https://github.com/coq-community/coqeal.git
cd coqeal
make   # or make -j <number-of-cores-on-your-machine> 
make install

The theory directory has the following content:

• ssrcomplements, minor mxstructure, polydvd, similar, binetcauchy, ssralg_ring_tac: Various extensions of the Mathematical Components library.

• dvdring, coherent, stronglydiscrete, edr: Hierarchy of structures with divisibility (from rings with divisibility, PIDs, elementary divisor rings, etc.).

• fpmod: Formalization of finitely presented modules.

• kaplansky: For providing elementary divisor rings from the Kaplansky condition.

• closed_poly: Polynomials with coefficients in a closed field.

• companion, frobenius_form, jordan, perm_eq_image, smith_complements: Results on normal forms of matrices.

• bareiss_dvdring, bareiss, gauss, karatsuba, rank, strassen, toomcook, smithpid, smith: Various efficient algorithms for computing operations on polynomials or matrices.

The refinements directory has the following content:

• refinements: Classes for refinements and refines together with operational typeclasses for common operations.

• binnat: Proof that the binary naturals of Coq (N) are a refinement of the MathComp unary naturals (nat) together with basic operations.

• binord: Proof that the binary natural numbers of Coq (N) are a refinement of the MathComp ordinals.

• binint: MathComp integers (ssrint) are refined to a new type parameterized by positive numbers (represented by a sigma type) and natural numbers. This means that proofs can be done using only lemmas from the MathComp library which leads to simpler proofs than previous versions of binint (e.g., N).

• binrat: Arbitrary precision rational numbers (bigQ) from the Bignums library are refined to MathComp's rationals (rat).

• rational: The rational numbers of MathComp (rat) are refined to pairs of elements refining integers using parametricity of refinements.

• seqmatrix and seqmx_complements: Refinement of MathComp matrices (M[R]_(m,n)) to lists of lists (seq (seq R)).

• seqpoly: Refinement of MathComp polynomials ({poly R}) to lists (seq R).

• multipoly: Refinement of MathComp multinomials and multivariate polynomials to Coq finite maps.

Files should use the following conventions (w.r.t. Local and Global instances):

(** Part 1: Generic operations *)
Section generic_operations.

Global Instance generic_operation := ...

(** Part 2: Correctness proof for proof-oriented types and programs *)
Section theory.

Local Instance param_correctness : param ...

(** Part 3: Parametricity *)
Section parametricity.

Global Instance param_parametricity : param ...
Proof. exact: param_trans. Qed.

End parametricity.
End theory.