rotsBK (KFUU §9a: the passes on managed states)

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

theorem NLogN.KFUU.rotsBK (d : ℕ) (sF μ : ℕ → ℕ) (hs : ∀ j < d, 1 ≤ sF j) (hμ : ∀ j < d, μ j < sF j)
    (B Bb w : ℕ) (hVB : zsz ((vol (KFU.shR d (KFU.sP d sF)) : ℕ) : ℤ) ≤ B) (hdB : zsz (d : ℤ) ≤ B)
    (hVb : zsz ((vol (KFU.shR d (KFU.sP d sF)) : ℕ) : ℤ) ≤ Bb)
    (xs : List ℤ) (hlen : xs.length = vol (KFU.shR d (KFU.sP d sF))) (hb : Bnd w xs)
    (s : IState HS) (F : Fin HS → List ℤ) (l lI lP lB rSL rSP rML lD : List ℤ)
    (h0 : F 0 = xs ++ l) (hI : F oI = 0 :: lI) (hPi : F oPi = 1 :: lP)
    (hBv : F oBv = (vol (KFU.shR d (KFU.sP d sF)) : ℤ) :: lB) (hSL : F oSL = sZ sF d ++ rSL)
    (hSP : F oSP = rSP) (hML : F oML = μZ μ d ++ rML) (hD : F oD = (d : ℤ) :: lD) :
    IRuns I iprog rotsBU (NLogN.KIL.encK s F)
      (NLogN.KIL.encK s (Function.update (Function.update (Function.update (Function.update
        (Function.update (Function.update (Function.update F 0 (pull (KFU.shR d (KFU.sP d sF))
          (KFU.shR d (KFU.sP d sF)) (KFU.Gb sF (KFU.csB sF μ) d) 0 xs ++ l)) oI ((d : ℤ) :: lI))
          oPi ((Pv sF d : ℤ) :: lP)) oBv (2 :: lB)) oSL rSL) oSP ((sZ sF d).reverse ++ rSP))
          oML rML))
      ((d + 1) * (cStepU d (vol ((List.range (d - 1)).map sF)) (vol (KFU.shR d (KFU.sP d sF))) w B Bb
        + 3000))

Context. The lemma belongs to the part KFUU §9a: the passes on managed states: The pass lemmas (padAllU_runs, unpadAllU_runs, revU_runs, rotsFU_runs, rotsBU_runs) restated on managed states encK s F (data xs ++ l on stack 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.