= Solution
A <Dehn presentation> is a finite <group presentation> with the following strict shortening property: every nonempty <freely reduced word> representing the identity contains a consecutive segment $u$ of a <relator> $r$ satisfying
$$
\boxed{|u|>\tfrac12|r|.}
$$
<Relators> are taken as <cyclically reduced words> with a <symmetrized relator set>: their inverse words and cyclic permutations are included. This convention allows the matched segment to start anywhere on either orientation of a <relator>. The symmetrized set remains finite and has the same maximum <relator> length.
If $r=uv$, the relation replaces $u$ by $v^{-1}$, of length $|r|-|u|<|u|$. The <Dehn algorithm> alternates this replacement with <free reduction>. Every step shortens the word; it terminates at the empty word exactly for words representing the identity. The strict inequality matters: a half-perimeter match alone need not shorten anything.
Finiteness is part of the convention used here and is needed for the maximum and counting argument in the last part. With an unrestricted infinite <relator> set, merely imposing the shortening property would not by itself justify that conclusion.
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