The free-variable tableau method has been widely used in order to automate proofs in multiple kinds of logics. Many automated theorem provers rely on this approach, either because it is the only available method - e.g. in certain modal logics - or because it facilitates the generation of proof certificates. However, as far as the authors know, its results have never been formalized in a proof assistant. In this paper, we present TableauxRocq, a deep embedding of free-variable first-order tableaux in the Rocq prover. The formalized calculus is proved sound and provides a modular Skolemization system that enables the use of Skolemization-based optimizations. Moreover, we show how TableauxRocq can be used as a certifier for automated theorem provers by adapting the Goeland prover - that can already output Rocq terms - to output proofs in the TableauxRocq format. By using the power of reflection, thereby providing a fully certified proof checker for free, we show that Goeland’s exported Rocq terms and TableauxRocq’s proof certificates can be checked in a similar time frame without proof optimizations, and that the latter has strictly better performances in presence of Skolemization-related optimizations.

Proof assistants based on dependent type theory—such as Agda , Lean , and Rocq —employ different universes to classify types, typically combining a predicative tower for computationally relevant types with a possibly impredicative universe for proof-irrelevant propositions. Several other universes with specific logical and computational principles have been explored in the literature. In general, a universe is characterized by its sort (e.g., Type, Prop, or SProp) and, in the predicative case, by its level. To improve modularity and better avoid code duplication, sort polymorphism has recently been introduced and integrated in the Rocq prover. However, we observe that, due to its unbounded formulation, sort polymorphism is currently insufficiently expressive to abstract over valid definitions with a single polymorphic schema. Indeed, to ensure soundness of a multi-sorted type theory, the interaction between different sorts must be carefully controlled, as exemplified by the forbidden elimination of irrelevant terms to produce relevant ones. As a result, generic functions that eliminate values of inductive types from one sort to another cannot be made polymorphic; dually, polymorphic records that encapsulate attributes of different sorts cannot be defined. This lack of expressiveness also breaks the possibility to infer principal types, which is highly desirable for both metatheoretical and practical reasons. To address these issues, we extend sort polymorphism with bounds that reflect the required elimination constraints on sort variables. We present the metatheory of bounded sort polymorphism, paying particular attention to the consistency of the resulting constraint graph. We implement bounded sort polymorphism in Rocq and illustrate its benefits through concrete examples. Bounded sort polymorphism with elimination constraints is a natural and general solution that effectively addresses current limitations and fosters the development of, and practical experimentation with, multi-sorted type theories.

In this paper, we present a general strategy that enables the translation of tableau proofs using different Skolemization rules into machine-checkable proofs. It is part of a framework that enables (i) instantiation of the strategy into algorithms for different sets of tableau rules (e.g., different logics) and (ii) easy soundness proof which relies on the local extensibility of user-defined rules. Furthermore, we propose an instantiation of this strategy for first-order tableaux that handles notably pre-inner Skolemization rules, which is, as far as the authors know, the first one in the literature. This deskolemization strategy has been implemented in the Goéland [17] automated theorem prover, enabling an export of its proofs to Coq [8] and Lambdapi [2]. Finally, we have evaluated the algorithm performances for inner and pre-inner Skolemization rules through the certification of proofs from some categories of the TPTP [39] library.