D ← D − H as IL code

D ← D − H as IL code.

The following holds (Lean statement, in the scope of module Solutions.KIL2Level):

theorem NLogN.KIL.subSteps_il [NeZero Q] (jOf : Fin (m + m) → Fin (m + m) → Fin Pp)
    (hpos : ∀ a ∈ sh, 1 ≤ a) (t : IState HS) (R : List Item) (Bb : ℕ)
    (hV : zsz ((AModel.vol sh : ℕ) : ℤ) ≤ Bb) (hQ2 : zsz ((2 * Q : ℕ) : ℤ) ≤ Bb)
    (xs : List ℤ) (hx : DOK sh Bb xs) :
    IRuns I iprog (runCode (scodeU2 Λ) jOf (subSteps m)) (t.upd 0 (ints xs ++ R))
      (t.upd 0 (ints (AModel.runArr Λ 0 (subSteps m) xs) ++ R))
      (m * cShiftU sh.length (AModel.vol sh) Bb + 1)

Context. The lemma belongs to the part KIL2: the pieces of one Prop. 11 level as IL code.

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_Defs18.