Axioms:

        G, a  |-  a

Rules:

        G |- a   G |- b
       -----------------
           G |- a & b


          G |- a & b
       ----------------
            G |- a


          G |- a & b
       ----------------
            G |- b


           G |- a
       ----------------
         G |- a + b


           G |- a
       ----------------
         G |- b + a


        G |- a->b  G |- a
       -------------------
             G |- b


            G, a |- b
        ------------------
            G |- a->b


            G |- 0
        ------------------
            G |- a