Explicit Substitution Internal Languages for Autonomous and *-Autonomous Categories

We introduce a family of explicit substitution type theories as internal languages for autonomous (or symmetric monoidal closed) and-autonomous categories, in the same sense that the simply-typed-calculus with surjective pairing is the internal language for cartesian closed categories. We show that the eight equality and three commutation congruence axioms of the-autonomous type theory char-acterise-autonomous categories exactly. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also connu-ent. As a corollary, we solve a Coherence Problem a la Lambek 12]: the equality of maps in any-autonomous category freely generated from a discrete graph is decidable.