Groupoid of Equational Proofs

An equational presentation P = h ; V; E i induces a graph G P of terms and equations, while proof gures in P induces an -algebra A P on groupoids. Although the construction of A P is quite syntactic, it is not free over G P . In order to make the meaning of freeness precise, we introduce a notion of -graph, and show that A P is freely generated by the -graph G P induced by P . Incidentally, this construction introduces a congruence on proof gures, by which `logically equivalent' proof gures are equalized. The authors gratefully acknowledge the support from Monbu-sho Kaken-hi, Suri-ronri oyobi sono shuhen-bun'ya no kenkyu (04302009). (axiom) s = t ax (\s = t" 2 E) (re exivity) t = t re (symmetricity) t = s s = t sym (transitivity) s = t t = u s = u trans (functionality) s 1 = t 1 : : : s n = t n !(s 1 ; : : : ; s n ) = !(t 1 ; : : : ; t n ) fun[!] (! 2 n ) (substitutivity) s = t [s] = [t] subst[ ] Table 1: Rules of inference of equational logic for an equational presentation h ; V; E i