Abstract. This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.
Abstract. Lean is a new open source theorem prover being developed at Microsoft Research and Carnegie Mellon University, with a small trusted kernel based on dependent type theory. It aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. Lean is an ongoing and long-term effort, but it already provides many useful components, integrated development environments, and a rich API which can be used to embed it into other systems. It is currently being used to formalize category theory, homotopy type theory, and abstract algebra. We describe the project goals, system architecture, and main features, and we discuss applications and continuing work.
The mean ergodic theorem is equivalent to the assertion that for every
function K and every epsilon, there is an n with the property that the ergodic
averages A_m f are stable to within epsilon on the interval [n,K(n)]. We show
that even though it is not generally possible to compute a bound on the rate of
convergence of a sequence of ergodic averages, one can give explicit bounds on
n in terms of K and || f || / epsilon. This tells us how far one has to search
to find an n so that the ergodic averages are "locally stable" on a large
interval. We use these bounds to obtain a similarly explicit version of the
pointwise ergodic theorem, and show that our bounds are qualitatively different
from ones that can be obtained using upcrossing inequalities due to Bishop and
Ivanov. Finally, we explain how our positive results can be viewed as an
application of a body of general proof-theoretic methods falling under the
heading of "proof mining."Comment: Minor errors corrected. To appear in Transactions of the AM
We show that certain model-theoretic forcing arguments involving subsystems of second-order arithmetic can be formalized in the base theory, thereby converting them to effective proof-theoretic arguments. We use this method to sharpen conservation theorems of Harrington and Brown-Simpson, giving an effective proof that W KL+0 is conservative over RCA0 with no significant increase in the lengths of proofs.
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