 Solution

For each table index $b$, let $C_b=L(table(b))$ be the set of non-null nodes reachable by following next. A sufficient representation invariant is:

The abstraction function maps these nodes to the total functions of part (a): if the unique node for $u$ exists, $I(u)$ and $O(u)$ are its fields; otherwise both are zero. A stored record with two zero counters is harmless and has the same observable meaning as an absent record. This intentionally permits multiple concrete representations of the same abstract program state.

Initially every bucket is null, so the representation invariant holds and the abstraction function yields all zeros. The successful-update branch preserves every structural condition; the insertion branch creates a fresh node with a previously absent identity in its correct bucket. Both branches realize exactly the incoming-increment postcondition, demonstrating correct operations through an abstraction function.