provers

Provers are software systems designed to establish the validity of logical statements, either automatically or with human guidance. They play a central role in formal verification, automated reasoning, and the formalization of mathematics. A core goal is to determine whether a given statement is true within a formal logical system and, in many cases, to produce a formal proof or a proof object.

Automated theorem provers (ATPs) attempt to prove theorems without human input. They typically operate on first-order logic and use strategies based on resolution, rewriting, or natural deduction to search for proofs. ATPs may prove statements, refute them, or fail to decide within given resources. Notable examples include Vampire, E, and Prover9.

Interactive theorem provers, also called proof assistants, require human guidance to construct proofs, but validate each inference through a trusted kernel. They are designed to handle more expressive logics and rigorous formalization. Examples include Coq, Isabelle/HOL, Lean, and HOL Light. These systems often support rich type theories and allow the extraction of verified programs from formal proofs.

Satisfiability modulo theories (SMT) solvers specialize in deciding the satisfiability of logical formulas with respect to background theories such as integers, arrays, or bit-vectors. They are widely used in software and hardware verification. Prominent SMT solvers include Z3, CVC4, and MathSAT.

Applications and challenges: provers are used to verify software correctness, prove mathematical theorems, certify hardware design, and aid in formalizing complex arguments. Limitations include undecidability in many logics, potentially non-terminating searches, and the effort required to develop and maintain formal proofs.