Completeness and Decidability of de Bruijn Substitution Algebra in Coq

We consider a two-sorted algebra over de Bruijn terms and de Bruijn substitutions equipped with the constants and operations from Abadi et al.'s sigma-calculus. We consider expressions with term variables and substitution variables and show that the semantic equivalence obtained with the algebra coincides with the axiomatic equivalence obtained with finitely many axioms based on the sigma-calculus. We prove this result with an informative decision algorithm for axiomatic equivalence, which in the negative case returns a variable assignment separating the given expressions in the algebra. The entire development is formalized in Coq.