Functional Pearl: the Proof Search Monad

We present the proof search monad, a set of combinators that allows one to write a proof search engine in a style that resembles the formal rules closely. The user calls functions such as premise, prove or choice; the library then takes care of generating a derivation tree. Proof search engines written in this style enjoy: first, a one-to-one correspondence between the implementation and the derivation rules, which makes manual inspection easier; second, proof witnesses “for free”, which makes a verified, independent validation approach easier too. 1 Theory and practice 1.1 A minimal theory We are concerned with proving the validity of logical formulas; that is, with writing a search procedure that determines whether a given goal is satisfiable. To get started, we consider a system made up of conjunctions of equalities, along with existential quantifiers. Any free variables are assumed to be universally quantified. For instance, one may want to prove the following formula: