Trakhtenbrot's Theorem in Coq: Finite Model Theory through the Constructive Lens

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