## Proof Checking Euclid## Joint work with Julien Narboux and Freek WiedijkWe used computer proof-checking methods to verify the correctness of our proofs of the propositions in Euclid Book I. We used axioms as close as possible to those of Euclid, in a language closely related to that used in Tarski's formal geometry. We used proofs as close as possible to those given by Euclid, but filling Euclid's gaps and correcting errors. Euclid Book I has 48 propositions; we proved 213 theorems. The extras were partly ``Book Zero'', preliminaries of a very fundamental nature, partly propositions that Euclid omitted but were used implicitly, partly advanced theorems that we found necessary to fill Euclid's gaps, and partly just variants of Euclid's propositions. We wrote these proofs in a simple fragment of first-order logic corresponding to Euclid's logic, debugged them using a custom software tool, and then checked them in the well-known and trusted proof checkers HOL Light and Coq. This web page provides (below) links to all the files needed to reproduce this work. The paper that describes the work can be found here: Persons wishing to download the files necessary to reproduce our work may download them all at once in the following archive: The steps to reproduce our work are as follows:(1) Unpack the archive mentioned above into a directory CheckEuclid(or another name of your choice). The subdirectory `proofs` contains .prf files. These are the formal proofs we wrote by hand. In the directory `CheckEuclid` are auxiliary files needed to check the proofs in HOL Light.
(2) Ensure that PHP is installed on your system, then run ./Checkproofs.phpfrom directory `CheckEuclid/proofs` .
This runs our custom proof-checker, or proof debugger, on all the proofs. That one PHP file
contains all the source code for our custom proof-checker.
(3) Ensure the HOL Light is running on your system. (4) Start HOL Light and load the files proofs.mland michael.ml. That will check all the proofs in HOL Light. It will take several minutes, but you will get a lot of reassuring output as it runs. (5) The file Axioms.phpthat contains the list of our axioms for Euclid and the list of proofs to be checked. If you wish to regenerate `michael.ml` >, the procedure is as follows, starting from the `CheckEuclid` directory.
cd proofs ../FreekFiles.php cd .. ./FreekFiles.pl (6) The script CoqExport.php has already been run to produce the .v files. Those are included in GeoCoq 2.4.0 library. GeoCoq 2.4.0 compiles at least with Coq versions 8.6.1, 8.7.2 and 8.8. See instructions given here. - The Coq version of Euclidean axioms
- Some adhoc tactics
- Book 1
- Going from Tarski's axiom to Euclid's axioms
- Going from Euclid's axioms to Tarski's axioms
(7) We also used HOL Light to verify that our axioms hold in the usual Cartesian plane. That verification required thousands of lines of HOL Light code. That code can be found in the following files. To run those proofs, load the three files in order into HOL Light. They will take some minutes but eventually you will see a list of the axioms of our theory, as objects of type thm in HOL Light. (8) There is also a subdirectory |