divF_runs (KFUU §2i: phase 2 in IL)

The following holds (Lean statement, in the scope of module Solutions.KFUURev9):

theorem NLogN.KFUU.divF_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) :
    IRuns I iprog (divE i j k) t (t.upd i (.int ((a / 2 ^ m : ℕ) : ℤ) :: Li)) (202 * (m + 1))

Context. The lemma belongs to the part KFUU §2i: phase 2 in IL: p2U computes vol ctB · cr (from vol (shP u bs), U^n and cz = 2 cr) and ⌊n/2⌋, then runs the ⌊n/2⌋ swaps. The main fields are reversed: the layout lay1 bg n 0 becomes rv n (n/2) (lay1 bg n 0).

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.