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 for short, The derivative function is therefore: Another way of computing this derivative is to view the composite function as the composite of and "h". Applying the chain rule to this situation gives: This is the same as what was computed above. This should be expected because . Sometimes it is necessary to differentiate an arbitrarily long composition of the form formula_23. In this case, define where formula_25 and formula_26 when formula_27. Then the chain rule takes the form or, in the Lagrange notation, The chain rule can be used to derive some well-known differentiation rules. For example, the quotient