digIU_spec (KFM U3: uniform digits in and carries out)

The following holds (Lean statement, in the scope of module Solutions.KFMU3):

theorem NLogN.KFM.digIU_spec (b e M S x : ℕ) (hx : x < 2 ^ (b * M)) (s : IState HS) (LX L1 L2 L3 L4 : List Item)
    (hX : s.G tX = .int (x : ℤ) :: LX) (hB2 : s.G uB2 = .int ((2 : ℤ) ^ b) :: L1)
    (hM : s.G uM = .int (M : ℤ) :: L2) (hZ : s.G uZ = .int ((2 * (S - M) : ℕ) : ℤ) :: L3)
    (hE : s.G uE = .int ((2 : ℤ) ^ e) :: L4) :
    IRuns I iprog digIU s
      ((s.upd tX LX).upd 0 (ints (gflat (digG b e M S x)) ++ s.G 0)) (cDigU b e M S)

Context. The lemma belongs to the part KFM U3: uniform digits in and carries out.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs22.