C $, such that for all $ i \geq 0 $ $ x_1 x_2^i x_3 x_4^i x_5 y $ and $ x_1 x_2^i x_3 x_4^i x_5 z $ are in $ L $;

4.  there exist factorizations $ x = x_1 x_2 x_3 $, $ y = y_1 y_2 y_3 $ and $ z = z_1 z_2 z_3 $, $ |x_2| \geq 1 $ and $ |x_2 x_3| \leq C $, such that for all $ i \geq 0 $ $ x_1 x_2^i x_3 y_1 y_2^i y_3 $ and $ x_1 x_2^i x_3 z_1 z_2^i z_3 $ are in $ L $.

\[theorem:impossibilityDPA\] The language $ {\mathtt{END}}$ cannot be recognized by any 1DPA.

We assume that $ {\mathtt{END}}$ is a DCFL and let $ C $ be the constant for $ {\mathtt{END}}$ in Lemma \[lemma:pumping\]. Choose $ w = 2^p 1 0^{p-1} \in \tt END $ and $ w' = 2^p 1 0^{p-1} 1 0^{p-1} $ for some integer $ p>C+1 $, and set $ x = 2^p 1 0^{p-2}, y = 0 $ and $ z = 0 1 0^{p-1} $. Then, $ w = xy $ and $ w' = xz $ satisfy (1) and (2) in Lemma 