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
General Leibniz rule General Leibniz rule In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if formula_1 and formula_2 are formula_3-times differentiable functions, then the product formula_4 is also formula_3-times differentiable and its formula_3th derivative is given by where formula_8 is the binomial coefficient and formula_9 This can be proved by using the product rule and mathematical induction. If, for example, , the rule gives an expression for the second derivative of a product of two functions: The formula can be generalized to the product of "m"