Traditional data mining algorithms identify patterns in data that are not explicit. These patterns are denoted in the form of IF-THEN rules (IF antecedent THEN consequent), where the antecedent and consequent are logical conjunctions of propositions or first-order predicates. Generally, the mined rules apply to all time periods and specify no temporal interval between antecedent detection and consequent firing. Cycle mining algorithms identify meta-patterns of these associations depicting inferences forming cyclic chains of rule dependencies. Because traditional rules comprise these cycles, the mined cycles also apply to all time periods and do not currently possess the temporal interval of applicability. An active database is one that responds to stimuli in real time, operating in the event-condition-action (ECA) paradigm where a specific event is monitored, a condition is evaluated, and one or more actions are taken. The actions often involve real-time modification of the database. In this paper, we introduce the concepts and present algorithms for mining rules with firing intervals, and intervals of applicability. Using an active database environment, we describe a real time framework that incorporates the active database concept in order to ascertain previously undefined cycles in data over a specific time interval and thereby introduce the concept of interval of discovery. Comprised of discovered rules with firing intervals and intervals of applicability, the encompassing discovered cycles also possess a variation of these attributes. We illustrate this framework with an example from an E-commerce endeavor where data is mined for rules with firing intervals and intervals of applicability, which amalgamate to form a cycle in its interval of discovery. We describe the computer system INDED, the author's implementation of cycle mining, which we are currently interfacing to an active Oracle database using triggers and PL/SQL stored procedures.