Recent work has highlighted the importance of the constraint based mining paradigm in the context of frequent itemsets, associations, correlations, sequential patterns, and many other interesting patterns in large databases. The authors study constraints which cannot be handled with existing theory and techniques. For example, avg(S) /spl theta/ /spl nu/, median(S) /spl theta/ /spl nu/, sum(S) /spl theta/ /spl nu/ (S can contain items of arbitrary values) (/spl theta//spl isin/{/spl ges/, /spl les/}), are customarily regarded as "tough" constraints in that they cannot be pushed inside an algorithm such as a priori. We develop a notion of convertible constraints and systematically analyze, classify, and characterize this class. We also develop techniques which enable them to be readily pushed deep inside the recently developed FP-growth algorithm for frequent itemset mining. Results from our detailed experiments show the effectiveness of the techniques developed.