Countable strict reverse mathematics

We investigate subsystems $CO{M}_{fcn}$, $COM{I}_{fcn}$ and $PR{A}_{fcn}$ of the elementary theory of functions $ETF$, the base theory for countable strict reverse mathematics. We show that inductions on any variable for unary, binary and ternary functions are pairwise equivalent over $CO{M}_{fcn}$. We prove that weakened primitive recursion axiom $WPRA$ is equivalent to primitive recursion axiom $PRA$ over $COM{I}_{fcn}$. We show that permutation axiom and minimization axioms $MI{N}^1$, $MI{N}^2$, $MI{N}^3$ are pairwise equivalent over $PR{A}_{fcn}$. Thus, we present several equivalent axiomatizations of $ETF$.