## Section: New Results

### Proof Certificates for First-Order Equational Logic

Participants : Dale Miller, Zakaria Chihani.

The kinds of inference rules and decision procedures that one writes
for proofs involving equality and rewriting are rather different from
proofs that one might write in first-order logic using, say, sequent
calculus or natural deduction. For example, equational logic proofs
are often chains of replacements or applications of oriented rewriting
and normal forms. In contrast, proofs involving logical connectives
are trees of introduction and elimination rules. Chihani and Miller
have shown [13] how it is possible to check
various equality-based proof systems with a programmable proof checker
(the *kernel* checker) for first-order logic. That proof
checker's design is based on the implementation of *focused proof
search* and on making calls to (user-supplied) *clerks and
experts* predicates that are tied to the two phases found in focused
proofs. This particular design is based on the work of Chihani,
Miller, and Renaud [14].

The specification of these clerks and experts provide a formal definition of the structure of proof evidence and they work just as well in the equational setting as in the logic setting where this scheme for proof checking was originally developed. Additionally, executing such a formal definition on top of a kernel provides an actual proof checker that can also do a degree of proof reconstruction. A number of rewriting based proofs have been defined and checked in this manner.