Phase 1 of the layout, backwards, uniform

Phase 1 of the layout, backwards, uniform.

The following holds (Lean statement, in the scope of module Solutions.KFV38LayOut1):

theorem NLogN.KFV.ph1R_ok {L : LP} {mk p W B : ℕ} (hLn : L.n = nLong D long + D * (ℓ - 1)) (hLD : L.D = D)
    (hLnl : L.nL = nLong D long) (hmk : mk = mkOf (flOf D long)) (hℓ : 2 ≤ ℓ)
    (hB : L.n + ℓ + 5 ≤ B) :
    LayerU (r := 2 ^ ℓ) I iprog (fun s => LRg s L mk W ∧ L.ℓ = ℓ) ph1R
      (runArrR L.n 0 (ops1 D ℓ long)) (2 ^ L.n) p W (cPh1R D ℓ long L.n W B)

Context. The lemma belongs to the part KFV §41: phase 1 of the layout, backwards, as a fixed IL loop: Iteration j < D (k = D-1-j): the registers gI = k+1, gRk = rank(k+1) are first decremented (gI ← gI - 1, gRk ← gRk - flag), then the inverse move mvF puts the bit at position rank back to rank + flag·k(ℓ-1). ph1R_ok: the loop computes runArrR n 0 (ops1 …).

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.