Applications of Foundational Proof Certificates in theorem proving

Formal trust in an abstract property, be it a mathematical result or a quality of the behavior of a computer program or a piece of hardware, is founded on the existence of a proof of its correctness. Many different kinds of proofs are written by mathematicians or generated by theorem provers, with the common problem of ascertaining whether those claimed proofs are themselves correct. The recently proposed Foundational Proof Certificate (FPC) framework harnesses advances in proof theory to define the semantics of proof formats, which can be verified by an independent and trusted proof checking kernel written in a logic programming language. This thesis extends initial results in certification of first-order proofs in several directions. It covers various essential logical axes grouped in meaningful combinations as they occur in practice: first,classical logic without fixed points and proofs generated by automated theorem provers; later, intuitionistic logic with fixed points and equality as logical connectives and proofs generated by proof assistants. The role of proof certificates is no longer limited to representing complete proofs to enable independent checking, but is extended to model proof transformations where details can be added to or subtracted from a certificate. These transformations yield operationally simpler certificates, around which increasingly trustworthy and performant proof checkers are constructed. Another new role of proof certificates is writing high-level proof outlines, which can be used to represent standard proof patterns as written by mathematicians, as well as automated techniques like property-based testing. We apply these developments to fully certify results produced by two families of standard automated theorem provers: resolution- and satisfiability-based. Another application is the design of programmable proof description languages for a proof assistant.