 the primitive recursive defining equations for each function constant, in particular for addition:  \[ \forall x (x + 0 = x),\]   \[ \forall x \forall y (x + S(y) = S(x + y));\]  and multiplication:  \[ \forall x (x \cdot 0 = 0),\]   \[ \forall x \forall y (x \cdot S(y) = (x \cdot y) + x);\]  and the (universal closure of the) schema of mathematical induction, for arbitrary formulas \(A(x)\): 