upd (KFM SE: the axis-rotation descriptor (`TransNeed`))

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

theorem NLogN.KFM.TrRegs.upd {t : IState HS} {n w m : ℕ} {rw rm dst : Fin HS} (h : TrRegs t n w m rw rm)
    (hd : TrOk dst rw rm) (L : List Item) : TrRegs (t.upd dst L) n w m rw rm

Context. The lemma belongs to the part KFM SE: the axis-rotation descriptor (TransNeed): descTr w h bs = [w, h, k, b_0, …, b_{k−1}] (KFMNeed). In kmain the array has the padded shape [t_0, …, t_{d−1}] (t_j = 2^{h + b_j}, h = ℓ − 1, b_j = [d' ≤ j]). After m rotations its shape is that list rotated left by m, so the descriptor of the next rotation uses the bits bR n m j = [d' ≤ (j + m) mod d].

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.