What does Bellantoni-Cook say about Cook-Reckov?
A polynomially bounded proof system is a function $M$ computable by an FP machine with the following properties:

For any $x$, the output $M(x)$ is a propositional tautology.
There exists a polynomial $p(n)$ such that for any propositional tautology $y$ there exists $x$ of size at most $p(|y|)$ such that $M(x) = y$.

Now suppose that you found a sophisticated way to describe all functions in FP, call it System R. This lets you come up with the following equivalent definition.

A polynomially bounded proof system is a function $M$ expressible in System R with the following properties:

For any $x$, the output $M(x)$ is a propositional tautology.
There exists a polynomial $p(n)$ such that for any propositional tautology $y$ there exists $x$ of size at most $p(|y|)$ such that $M(x) = y$.


As you can see, nothing has really been gained. We only changed &quot;computable by an FP machine&quot; with &quot;expressible in System R&quot;.