  Translating expressions between different logics and theorem provers is
notoriously and often prohibitively difficult, due to the large differences
between the logical foundations, the implementations of the systems, and the
structure of the respective libraries. Practical solutions for exchanging
theorems across theorem provers have remained both weak and brittle.
Consequently, libraries are not easily reusable across systems, and substantial
effort must be spent on reformalizing and proving basic results in each system.
Notably, this problem exists already if we only try to exchange theorem
statements and forgo exchanging proofs.
  In previous work we introduced alignments as a lightweight standard for
relating concepts across libraries and conjectured that it would provide a good
base for translating expressions. In this paper, we demonstrate the feasibility
of this approach. We use a foundationally uncommitted framework to write
interface theories that abstract from logical foundation, implementation, and
library structure. Then we use alignments to record how the concepts in the
interface theories are realized in several major proof assistant libraries, and
we use that information to translate expressions across libraries. Concretely,
we present exemplary interface theories for several areas of mathematics and -
in total - several hundred alignments that were found manually.
