We study a cyclic proof system C over regular expression types, inspired by linear logic and non-wellfounded proof theory. Proofs in C denote total computable functions; we analyse the relative strength of C and Gödel's system T, showing that contraction plays a crucial role. In the general case, we show that the two systems capture the same functions on natural numbers. In the affine case, we manage to give a direct and uniform encoding of C into T, translating cycles into explicit recursions. We also show that for functions on natural numbers, removing contraction reduces the expres-sivity precisely to primitive recursive functions-providing an alternative and more general proof of a result by Dal Lago. The two upper bounds on the expressivity of C w.r.t. functions on natural numbers are obtained by formalising weak normalisation of a small step reduction semantics in subsystems of second-order arithmetic: ACA0 and RCA0. Whether a direct and uniform translation from C to T can be given in the presence of contraction remains open.