Abstract

We compare the styles of several proof assistants for mathematics. We present Pythagoras' proof of the irrationality of 2 both informal and formalized in (1) HOL, (2) Mizar, (3) PVS, (4) Coq, (5) Otter /Ivy, (6) Isabelle/Isar, (7) Alfa/Agda, (8) ACL2, (9) PhoX, (10) IMPS, (11) Metamath, (12) Theorema, (13) Lego, (14) Nuprl, (15) mega. proof assistant author of proof page informal Henk Barendregt 3 1 HOL John Harrison, Konrad Slind 4 2 Mizar Andrzej Trybulec 8 3 PVS Bart Jacobs, John Rushby 12 4 Coq Laurent Thery 16 5 Otter/Ivy Michael Beeson, William McCune 22 6 Isabelle/Isar Markus Wenzel, Larry Paulson 27 7 Alfa/Agda Thierry Coquand 35 8 ACL2 Ruben Gamboa 40 9 PhoX Christophe Raalli, Paul Roziere 53 10 IMPS William Farmer 59 11 Metamath Norman Megill 75 12 Theorema Markus Rosenkranz, Tudor Jebelean, Bruno Buchberger 82 13 Lego Conor McBride 90 14 Nuprl Paul Jackson 99 15 15 Christoph Benzmuller, Armin Fiedler, Andreas Meier, Martin Pollet 111 2 Freek Wiedijk proof assistant HOL Mizar PVS Coq Otter/Ivy Isabelle/Isar Alfa/Agda ACL2 PhoX IMPS Metamath Theorema Lego Nuprl prl small proof kernel (`proof objects') calculations can be proved automatically + + + + + + + + + + + + extensible/programmable by the user + + + + + + powerful automation + + + + + + + + readable proof input les + + + + constructive logic supported + + + + + + logical framework + + typed + + + + + + + + + + + decidable types + + + + + + + + + dependent types + + + + + + based on higher order logic + + + + + + + + + + + based on ZFC set theory + + + large mathematical standard library + + + + + + + statement about R + + + + + + + + + + statement about () m = n = 0 G.H. Hardy and E.M. Wright, An

Citations