Program Extraction in Constructive Analysis [chapter]

Helmut Schwichtenberg
2009 Logicism, Intuitionism, and Formalism  
We sketch a development of constructive analysis in Bishop's style, with special emphasis on low type-level witnesses (using separability of the reals). The goal is to set up things in such a way that realistically executable programs can be extracted from proofs. This is carried out for (1) the Intermediate Value Theorem and (2) the existence of a continuous inverse to a monotonically increasing continuous function. Using the Minlog proof assistant, the proofs leading to the Intermediate Value
more » ... Theorem are formalized and realizing terms extracted. It turns out that evaluating these terms is a reasonably fast algorithm to compute, say, approximations of √ 2. 2 Real Numbers Reals, Equality of Reals We shall view a real as a Cauchy sequence of rationals with a separately given modulus. Definition. A real number x is a pair ((a n ) n∈N , M ) with a n ∈ Q and M : N → N such that (a n ) n is a Cauchy sequence with modulus M , that is |a n − a m | ≤ 2 −k for n, m ≥ M (k) and M is weakly increasing. M is called a Cauchy modulus of x.
doi:10.1007/978-1-4020-8926-8_13 fatcat:nyhngtujmrdadhifdonrhintzi