Using A Partial Model
The unit clauses that are present in a set of clauses or can be derived from it can be stored in form of a partial model (this partial model may also contain other literals, depending on the application). In this case, unit propagation is performed based on the literals of the partial model, and unit clauses are removed if their literal is in the model. In the example above, the unit clause would be added to the partial model; the simplification of the set of clause would then proceed as above with the difference that the unit clause is now removed from the set. The resulting set of clauses is equivalent to the original one under the assumption of validity of the literals in the partial model.
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