So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.
Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of addition.
I don't think so. Look more closely. (x * p) is repeated addition with p integer - we can do this no multiplication involved. Then dividing by q involves finding r such that r * q = x * p. Here, again, we can do this by finding r such that r, added to itself q times, equals x * p. Since q is integer, again, this again involves repeated addition. (It does need an algorithm to find the right value of r. I suppose that trying an arbitrary value, then increasing or decreasing it, would work.)
Yes, that's the "first bit" that I was not raising an issue with.
> dividing by q involves finding r such that r * q = x * p
Which, as you note, requires a trial and error algorithm to find r. But in any case, r is the answer being sought, not the thing we're supposed to be adding to itself some number of times to get the answer.
So now your definition of "repeated addition" is "repeated addition, plus a version of multiplication". Division is the inverse of multiplication, so your definition is circular: you're "defining" multiplication in terms of repeated addition and multiplication.
Similar objections apply to another poster's contention upthread that the "repeated addition" definition is justified because of the distributive law. The distributive law defines A times (b plus c) in terms of A times b plus A times c. So it's useless as a definition of multiplication in terms of addition.