We present a new framework for formalizing mathematics in untyped set theory using auto2. Using this framework, we formalize in Isabelle FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.
We adopt the following convention: p, q, x, y are real numbers and n is a natural number. Next we state a number of propositions: (1) Let G, H be groups and h be a homomorphism from G to H. If h ·h−1 = idH and h−1 · h = idG, then h is an isomorphism. (2) For every subset X of I and for every point a of I such that X = ]a, 1] holds Xc = [0, a]. The paper was written during author’s post-doctoral fellowship granted by Shinshu University, Japan. This work has been partially supported by the CALCULEMUS grant HPRN-CT-200000102 and KBN grant 4 T11C 039 24.