In the context of static analysis based on Abstract Interpretation, we propose a lightweight pre-analysis step which is meant to suggest, at each program point, which program variables are likely to be unconstrained for a specific class of abstract properties. Using the outcome of this pre-analysis as an oracle, we simplify the statements of the program being analyzed by propagating this lack of information, aiming at fine-tuning the precision/efficiency trade-off of the analysis. A preliminary experimental evaluation shows that the idea underlying the approach is promising, as it improves the efficiency of the more costly analysis, while having a limited effect on its precision.