Deriving Induction Principles Using Parametricity
Roadmap to deriving induction principles using parametricity, challenges involved, logic of correct reasoning, study of proof assistants, and generating induction principles automatically in Coq. Exploring parametricity translation from types to relations and methods for deriving induction principles utilizing parametricity.
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ALGEBRAIC COMBINATIONS We have seen that it is fairly easy to compute the derivative of a simple function using the definition of the derivative. More complicated functions can be difficult or impossible to differentiate using this method. So we ask . . . If we know the derivatives of two fairly simple functions, can we deduce the derivative of some algebraic combination (e.g. the sum or difference) of these functions without going back to the difference quotient?
Note: We are going to assume that all of the functions we talk about in this PowerPoint presentation are differentiable.
THE DERIVATIVE OF A CONSTANT TIMES A FUNCTION ( ) ( h ) + ( ) ( ) kf x h kf x d dx ( ) = ( ) lim h kf x 0 Can we do this? Why? ( ) + ( ) ( ) f x k f x h h = lim h 0 ( ) + ( ) ( ) f x f x h = lim h k h 0 ( ) = kf x
WHAT DOES THIS MEAN? 23 = ( ) f x 3 x ) ) ( ( d dx d dx 2 3x 2 2 13 13 = = = 3 3 x x 2x 3 3 3
THE DERIVATIVE OF THE SUM OF TWO FUNCTIONS ( ) ( ) + + + + ( ) ( ) ( ) f x ( ) g x f x h g x h h d dx ( ) + = ( ) f x ( ) g x lim h 0 + + + ( ) ( ) ( ) f x ( ) g x f x h g x h = lim h h 0 Can we do this? + + ( ) ( ) f x ( ) ( ) g x f x h g x h = + lim h h h 0 + + ( ) ( ) f x ( ) ( ) g x f x h g x h ( ) ( ) = + f x g x = + lim h lim h h h 0 0
THE DERIVATIVE OF THE DIFFERENCE OF TWO FUNCTIONS ( ) ( ) + + ( ) ( ) ( ) f x ( ) g x f x h g x h h d dx ( ) = ( ) f x ( ) g x lim h 0 + + + ( ) ( ) ( ) f x ( ) g x f x h g x h = lim h h 0 + + ( ) ( ) f x ( ) ( ) g x f x h g x h = lim h h h 0 + + ( ) ( ) f x ( ) ( ) g x f x h g x h ( ) ( ) = f x g x = lim h lim h h h 0 0
