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
Chain rule a variable "z" depends on the variable "y", which itself depends on the variable "x", so that "y" and "z" are therefore dependent variables, then "z", via the intermediate variable of "y", depends on "x" as well. The chain rule then states, The two versions of the chain rule are related; if formula_4 and formula_5, then In integration, the counterpart to the chain rule is the substitution rule. The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative of formula_7 as the composite of the square root function and the