tabU_sp (KFUU §7d: the uniform table set-up of one shift)

The following holds (Lean statement, in the scope of module Solutions.KFUURot4):

theorem NLogN.KFUU.tabU_sp (neg : Bool) (sF : ℕ → ℕ) (μ Q i Bv : ℕ)
    (hs : ∀ j < i, 1 ≤ sF j) (hQ1 : 1 ≤ Q) (hμ : μ < Q) (B Bb : ℕ)
    (hB : zsz ((Q * Pv sF i : ℕ) : ℤ) ≤ B) (hiB : zsz (i : ℤ) ≤ B) (hBvB : zsz (Bv : ℤ) ≤ B)
    (hBb : zsz ((2 * (Q * Pv sF i)) : ℕ) ≤ Bb)
    (s : IState HS) {LQ LM LP LI LB : List Item}
    (hQ : s.G oQ = .int Q :: LQ) (hMu : s.G oMu = .int μ :: LM)
    (hPi : s.G oPi = .int (Pv sF i) :: LP) (hI : s.G oI = .int i :: LI)
    (hBv : s.G oBv = .int Bv :: LB) :
    Sp (NLogN.KFUU.tabU neg)
      (NLogN.KIL.encK s (Function.update (fun _ => []) oSP ((List.range i).map fun j => (sF j : ℤ)).reverse))
      (NLogN.KIL.encK s (tabEnd neg sF μ Q i (Pv sF i) Bv)) (NLogN.KFUU.cTab i (Pv sF i) B Bb)

Context. The lemma belongs to the part KFUU §7d: the uniform table set-up of one shift: tabU neg (register code): from the registers oI = i, oQ = Q = s_i, oMu = μ, oPi = P_i, oBv = B and the sizes s_{i-1}, …, s_0 on oSP, it pushes the offset table tbl Q (acL neg sF μ Q i) on stack 14, P_i, Q, B on stacks 1, 2, 4 (the parameters of KFC rotF) and -Q on sNQ; oSP is restored. tabU_sp, cost cTab.

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.