divE_runs (KFUU §2g: phase 2 in IL (one iteration))

The following holds (Lean statement, in the scope of module Solutions.KFUURev7):

theorem NLogN.KFUU.divE_runs {i j k : Fin HS} (hij : i ≠ j) (hik : i ≠ k) (hjk : j ≠ k) (t : IState HS)
    {a m : ℕ} {Li Lj : List Item} (hi : t.G i = .int (a : ℤ) :: Li)
    (hj : t.G j = .int ((2 ^ m : ℕ) : ℤ) :: Lj) (hdiv : 2 ^ m ∣ a) :
    IRuns I iprog (divE i j k) t (t.upd i (.int ((a / 2 ^ m : ℕ) : ℤ) :: Li)) (200 * (m + 1) + 2)

Context. The lemma belongs to the part KFUU §2g: phase 2 in IL (one iteration): p2B divides M by U², calls the swap entry with the descriptor [w, u, 1, 3, 5, U^k, U, U^{n-2-2k}, U, U^k·Bh], and multiplies A = U^k and C = U^k·Bh by U.

Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below $n\log n$: the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.