How can I solve the following problem:
Let $f, \pi, g$ accept one, two and three arguments respectively. If you know that $f, \pi, g$ are primitive recursive functions prove that $h$ defined as: $$ \begin{align*} h(0, y) &\simeq f(y) \\ h(x + 1, y) &\simeq g(x, y, h(x, \pi(x, y))) \end{align*} $$ is also primitive recursive function.
The definition of primitive recursion I know is: $$ \begin{align*} h(\bar{x}, 0) &\simeq f(\bar{x}) \\ h(\bar{x}, y + 1) &\simeq g(\bar{x}, y, h(\bar{x}, y)) \end{align*} $$