In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching; we show how ensuring syntactical exhaustivity and non-overlapping of clauses is enough to ensure reversibility. The language allows to represent any Primitive Recursive Function. We then give a Curry-Howard correspondence with the logic µMALL: linear logic extended with least xed points allowing inductive statements. The critical part of our work is to show how primitive recursion yields circular proofs that satisfy µMALL validity criterion and how the language simulates the cut-elimination procedure of µMALL.