takeD_all (KFY 7: the prologue of the line passes)

The following holds (Lean statement, in the scope of module Solutions.KFY7Pro):

theorem NLogN.KFY.takeD_all (r : Fin HS) (e : Ex HS) (t : IState HS) (v : ℤ) (R : List Item) (B : ℕ)
    (hD : t.G NLogN.KFY.dS = .int v :: R) (hr : r ≠ NLogN.KFY.dS) (hr56 : r.val < 56) (hc : e.Clean) (hok : e.ok t)
    (hB : e.Bnd t B) (hv : zsz v ≤ B) :
    RSAll (takeD r e) t ((t.upd r (.int (e.eval t) :: t.G r)).upd NLogN.KFY.dS R)
      ((e.size + 1) * 10 ^ 7 * (B + 2) ^ 2 + (2 * B + 2))

Context. The lemma belongs to the part KFY 7: the prologue of the line passes: The line-map descriptor [p, α, h, d, b_0..b_{d−1}, s_0..s_{d−1}] (top first) lies on the descriptor stack dS = 26. The prologue proP takes p, α, ℓ = h + 1, d into registers (yP = 11, yA = 12, KFM's zEl = 29, zD = 25), counts the zero bits into d' (KFM's zDp = 31, the bits are [d' ≤ j]), and pushes L = dℓ − d' (yL = 13, the base-2 logarithm of the array length). The sources stay on dS (proP_all).

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.