prodP_run (KFM S2: counting loops with an expression test, and products)

The following holds (Lean statement, in the scope of module Solutions.KFMS2):

theorem NLogN.KFM.prodP_run (n : Ex HS) (rd : Fin HS) (s : IState HS) (N : ℕ) (hd56 : rd.val < 56)
    (hdI : rd ≠ NLogN.KFM.sI) (hn : n.Clean) (hnd : rd ∉ n.regs) (hnok : n.ok s) (hN : n.eval s = N) :
    (prodP n rd).ok s ∧ (prodP n rd).run s = s.upd rd (.int (N.factorial : ℤ) :: s.G rd)

Context. The lemma belongs to the part KFM S2: counting loops with an expression test, and products.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs22.