Recently, the weakness of the canonical support-confidence framework for associations mining has been widely studied in the literature. One of the difficulties in applying association rules mining to real world applications is the setting of support constraint. A high support constraint avoids the combinatorial explosion in discovering frequent itemsets, but at the expense of missing interesting patterns of low support. Instead of seeking the way for setting the appropriate support constraint, all current approaches leave the users in charge of the support setting, which, however, puts the users in a dilemma. This paper is an effort to answer this long-standing open question. Based on the notion of confidence and lift measures, we propose an automatic support specification for mining high confidence and positive lift associations without consulting the users. Experimental results show that this specification is good at discovering the low support, but high confidence and positive lift associations, and is effective in reducing the spurious frequent itemsets.