Structural rule A famous structural rule is known as cut. Considerable effort is spent by proof theorists in showing that cut rules are superfluous in various logics. More precisely, what is shown is that cut is only (in a sense) a tool for abbreviating proofs, and does not add to the theorems that can be proved. The successful 'removal' of cut rules, known as "cut elimination", is directly related to the philosophy of "computation as normalization" (see Curry–Howard correspondence); it often gives a good indication of the complexity of deciding a given logic. Structural rule In proof theory, a structural rule is
Triple product rule Triple product rule The triple product rule, known variously as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, is a formula which relates partial derivatives of three interdependent variables. The rule finds application in thermodynamics, where frequently three variables can be related by a function of the form "f"("x", "y", "z") = 0, so each variable is given as an implicit function of the other two variables. For example, an equation of state for a fluid relates temperature, pressure, and volume in this manner. The triple product rule for such interrelated variables "x", "y", and "z"