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This last decade, propositional reasoning and search has been one of the hottest topics of research in the A.I. community, as the Boolean framework has been recognized as a powerful setting for many reasoning paradigms thanks to dramatic improvements of the efficiency of satisfiability checking procedures. SAT, namely checking whether a set of propositional clauses is satisfiable or not, is the technical core of this framework. In the paper, a new linear-time pre-treatment of SAT instances is introduced. Interestingly, it allows us to discover a new polynomial-time fragment of SAT that can be recognized in linear-time, and show that some benchmarks from international SAT competitions that were believed to be difficult ones, are actually polynomial-time and thus easy-to-solve ones.