Cost bound for attaching target indices to the rows of an array

Let $D$ be a list of $n\cdot m$ rows of $d$ integers ($m \ge 1$) stored flattened on stack 0, with stacks 1, 2 holding $n$, $m$, and let $X$ bound the bit lengths of all entries and of $nm+n$, $nm(d+1)$, $d+1$. Then the program attachP d (which prefixes each row with its index in the transposed order) costs at most $10^5(nm+1)(d+1)(X+2)$.

Formalization Note. The machine layer: IState HS is a state with $64$ stacks of items (Item = integer or mark), an input position and an output; RM programs (straight-line register code RS, loops rep, conditionals, calls of the multiplication procedure through a MulSpec $M$ giving its domain and cost) have a functional semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s; Spec M p s s' c abbreviates p.ok M s ∧ p.run s = s' ∧ p.cost M s ≤ c. ints L / gints U push an integer list / a list of pairs as items (head = top); zsz z is the bit length of $|z|$. MulBnd M m Φ says the multiplication procedure accepts all pairs of integers of bit length $\le m$ at cost $\le Φ$. Register names (GU, CR, KB, XN, …) are fixed stack indices.