Calculus in Coinductive Form

Coinduction is often seen as a way of implementing innnite objects 8, 44. Since real numbers are typical innnite objects, it may not come as a surprise that calculus, when presented in a suitable way, is permeated by coinductive reasoning. What is surprising is that mathematical techniques, recently developed in the context of computer science, seem to be shedding a new light on some basic methods of calculus. We i n troduce a coinductive formalization of elementary calculus that can be used as a tool for symbolic computation, and geared towards computer algebra and theorem proving. So far, we have c o vered parts of ordinary diierential and diierence equations, Taylor series, Laplace transform and the basics of the operator calculus. Our point of departure is the observation that the algebraic structure of streams, given by the equations heada :: = a 1 taila :: = 2 head :: tail = 3 captures much of calculus. Given an analytic function f, deene headf = f0 tailf = f 0 a :: f = x 7 ! a + Z x 0 f Equation 3 now expresses the so-called Fundamental Theorem of Calculus, whereas equations 2 and 1 normalize the integral with respect to the subintegral function and the interval of integration. 1 Reapplying equation 3 yields the Taylor Maclaurin expansion Unfolding the above deenition of ::, which amounts to iterated integration, one nally gets fx = f0 + f 0 0x + + f n 0x n =n! + The idea of innnitely applying 3 is formally captured by the notion of a stream coalgebra. The set of innnite sequences forms a nal stream coal-gebra. Taylor expansions are then obtained using the unique homomorphism from the stream coal-gebra of analytic functions. From another point of view, a :: f is the unique solution of the diierential equation g 0 = f with the initial value g0 = a. The above derivation of Taylor series now leads to the usual power series method for solving diierential equations 2, Solving it amounts to running a corecursive program , which outputs the stream of Taylor coee-cients corresponding, in this case, to f = sin. Following these ideas, we i n troduce in sections 1 and 2 a formal setting for studying and implementing analytic structures by coalgebraic methods. Section 3 proceeds from our stream algebras 1 This example may be suggested by Hoare's …