The Hydra Battle Revisited

Showing termination of the Battle of Hercules and Hydra is a challenge. We present the battle both as a rewrite system and as an arithmetic while program, provide proofs of their termination, and recall why their termination cannot be proved within Peano arithmetic. As a second labour he ordered him to kill the Lernaean hydra. That creature, bred in the swamp of Lerna, used to go forth into the plain and ravage both the cattle and the country. Now the hydra had a huge body, with nine heads, eight mortal, but the middle one immortal. . . . By pelting it with fiery shafts he forced it to come out, and in the act of doing so he seized and held it fast. But the hydra wound itself about one of his feet and clung to him. Nor could he effect anything by smashing its heads with his club, for as fast as one head was smashed there grew up two. – Pausanias, Description of Greece, 2.37.4