  We study finite first-order satisfiability (FSAT) in the constructive setting
of dependent type theory. Employing synthetic accounts of enumerability and
decidability, we give a full classification of FSAT depending on the
first-order signature of non-logical symbols. On the one hand, our development
focuses on Trakhtenbrot's theorem, stating that FSAT is undecidable as soon as
the signature contains an at least binary relation symbol. Our proof proceeds
by a many-one reduction chain starting from the Post correspondence problem. On
the other hand, we establish the decidability of FSAT for monadic first-order
logic, i.e. where the signature only contains at most unary function and
relation symbols, as well as the enumerability of FSAT for arbitrary enumerable
signatures. To showcase an application of Trakthenbrot's theorem, we continue
our reduction chain with a many-one reduction from FSAT to separation logic.
All our results are mechanised in the framework of a growing Coq library of
synthetic undecidability proofs.
