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Sudoku

A formalisation of Sudoku in Coq. It implements a naive Davis-Putnam procedure to solve Sudokus.

Meta

• Author(s):
  • Laurent Théry (initial)
• Coq-community maintainer(s):
  • Ben Siraphob (@siraben)
  • Laurent Théry (@thery)
• License: GNU Lesser General Public License v2.1 or later
• Compatible Coq versions: 8.12 or later
• Additional dependencies: none
• Coq namespace: Sudoku
• Related publication(s):
  • Sudoku in Coq

Building and installation instructions

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

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

To instead build and install manually, do:

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

Documentation

A Sudoku is represented as a mono-dimensional list of natural numbers. Zeros are used to represent empty cells. For example, the 3x3 Sudoku:

  -------------------------------------
  |   |   | 8 | 1 | 6 |   | 9 |   |   |
  -------------------------------------
  |   |   | 4 |   | 5 |   | 2 |   |   |
  -------------------------------------
  | 9 | 7 |   |   |   | 8 |   | 4 | 5 |
  -------------------------------------
  |   |   | 5 |   |   |   |   |   | 6 |
  -------------------------------------
  | 8 | 9 |   |   |   |   |   | 3 | 7 |
  -------------------------------------
  | 1 |   |   |   |   |   | 4 |   |   |
  -------------------------------------
  | 3 | 6 |   | 5 |   |   |   | 8 | 4 |
  -------------------------------------
  |   |   | 2 |   | 7 |   | 5 |   |   |
  -------------------------------------
  |   |   | 7 |   | 4 | 9 | 3 |   |   |
  -------------------------------------

is represented as

  0 :: 0 :: 8 :: 1 :: 6 :: 0 :: 9 :: 0 :: 0 ::
  0 :: 0 :: 4 :: 0 :: 5 :: 0 :: 2 :: 0 :: 0 ::
  9 :: 7 :: 0 :: 0 :: 0 :: 8 :: 0 :: 4 :: 5 ::
  0 :: 0 :: 5 :: 0 :: 0 :: 0 :: 0 :: 0 :: 6 ::
  8 :: 9 :: 0 :: 0 :: 0 :: 0 :: 0 :: 3 :: 7 ::
  1 :: 0 :: 0 :: 0 :: 0 :: 0 :: 4 :: 0 :: 0 ::
  3 :: 6 :: 0 :: 5 :: 0 :: 0 :: 0 :: 8 :: 4 ::
  0 :: 0 :: 2 :: 0 :: 7 :: 0 :: 5 :: 0 :: 0 ::
  0 :: 0 :: 7 :: 0 :: 4 :: 9 :: 3 :: 0 :: 0 :: nil

All functions are parametrized by the height and width of a Sudoku's subrectangles. For example, for a 3x3 Sudoku:

sudoku 3 3: list nat -> Prop

check 3 3: forall l, {sudoku 3 3 l} + {~ sudoku 3 3 l}

find_one 3 3: list nat -> option (list nat)

find_all 3 3: list nat -> list (list nat)

See Test.v.

Corresponding correctness theorems are:

find_one_correct 3 3
     : forall s,
       length s = 81 ->
       match find_one 3 3 s with
       | Some s1 => refine 3 3 s s1 /\ sudoku 3 3 s1
       | None =>
           forall s, refine 3 3 s s1 -> ~ sudoku 3 3 s1
       end

find_all_correct 3 3
     : forall s s1, refine 3 3 s s1 -> (sudoku 3 3 s1 <-> In s1 (find_all 3 3 s))

See Sudoku.v.

More about the formalisation can be found in a note.

The following files are included:

• ListOp.v some basic functions on list
• Sudoku.v main file
• Test.v test file
• Tactic.v contradict tactic
• Div.v division and modulo for nat
• Permutation.v permutation
• UList.v unique list
• ListAux.v auxillary facts on lists
• OrderedList.v ordered list

The Sudoku code can be extracted to JavaScript using js_of_ocaml:

make Sudoku.js

Then, point your browser at Sudoku.html.