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Use the following context-free grammar, with start symbol $S$:The symbol $B$ generates the balanced members of the suffix-nonnegative binary language, while $S$ permits extra zeros. To prove the claim and prepare the ambiguity argument, reverse every generated string. The reversed productions areHere $U$ represents the reversals of the $S$ strings, and $D$ represents the reversals of the $B$ strings. A string belongs to the desired formal language precisely when its reversal has no string prefix with more ones than zeros. Associate a path with each reversed string, counting a zero as a step $+1$ and a one as a step $-1$.

The nonterminal $D$ generates exactly the paths that stay nonnegative and finish at zero. The production $0D1D$ consists of an initial up-step, a balanced path one level higher, a down-step to zero, and another balanced path. Conversely, any nonempty balanced nonnegative path has a first return to zero, and splitting there gives exactly this form. Mathematical induction on length proves the characterization.

The productions for $U$ generate nonnegative paths: $0U$ stays strictly above zero after its first step, while $0D1U$ first executes a balanced excursion and then another nonnegative path. Conversely, a nonempty nonnegative path starts with zero. If it never returns to height zero, removing that initial zero leaves a nonnegative path, giving $0U$. If it does return, split at its first return to obtain $0D1U$. These shorter components can be generated by mathematical induction. This proves that the original context-free grammar generates exactly the required formal language.

For instance, in the original context-free grammar,while $S\Rightarrow S1B0\Rightarrow S1B01B0{\Rightarrow }^{\ast }1010$. The string $110$ is excluded since its entire length is a string suffix having two ones and only one zero.