Final answer:
As the grammar rules are not provided, it is not possible to determine which input string proves the given condition by the grammar. A grammar typically involves a set of production rules, which define how strings in the language are formed. Without the grammar, one cannot perform language recognition.
Step-by-step explanation:
To determine which input string proves a given condition by the grammar, we need to know the rules of the grammar itself, which are not provided in the question. In typical formal language theory, a grammar consists of a set of production rules that define which strings of symbols can be generated in the language. The strings provided are likely to be tested against these grammar rules to see which one is valid.
Since the specific grammar rules and conditions have not been given, it is impossible to answer definitively which of the input strings (A) 010101, (B) 0011, (C) 111000, or (D) 1001 would prove the given condition. For a proper examination, the grammar definition should be provided. Once the rules are known, each string would be checked against the grammar to determine compliance.
Without the grammar, this problem falls under the purview called language recognition, which is a significant concept in the study of formal languages and automata within the field of computer science.