This standalone (one file) proof was implemented by Dominique Larchey-Wendling, see hydra.v. It is distributed under the terms of the MPL-2.0.

The Hydra is mainly a Rose tree, that is a finite tree with arbitrary branching. Hercules tries to kill the Hydra by chopping of heads (ie leaves of the Hydra) but the Hydra responds in growing copies of specific sub-trees determined by which head was chopped last.

The Hercules and Hydra game consists in Hercules chopping off heads of the Hydra until the Hydra has no more heads left. In response to a head cut, the Hydra may grow an arbitrary number of copies of the sub-tree rooted two levels below the head that was last chopped by Hercules.

Importantly the previous rule excludes root leaves (ie a leaf just above the root of the Hydra). If Hercules chops a root leaf, then the Hydra cannot morph in response: it must wait for the next move of Hercules. Absent of that exclusion, the game would possibly not end because nothing would force the number of nodes to (sometimes) decrease after the response of Hydra.

There are variants of the game where the number of copies that the Hydra makes is determined fully by the advancement of the game (eg the number of head that were chopped by Hercules so far) by this does not affect the termination property of the game.

In the code of hydra.v, we only model a round which is the combination of a head cut by Hercules and a conforming response by the Hydra (see above). We do not model the moves of Hercules and Hydra independently because:

• while a move of Hercules is nondeterministic and only depends on the current shape of the Hydra;
• the response from the Hydra (also possibly nondeterministic), is still constrained by which head was chopped off last. The growing of copies of sub-hydras depends on the position of this last chopped head.

Modeling this dependency would lengthen the code base because we would need to represent position of a head inside the Hydra so that Hercules can transmit that information. By combining Hercules move and the Hydra's response into the notion of round, the information can be transmitted without requiring external information.