The low procedure

The low procedure.

The following holds (Lean statement, in the scope of module Solutions.KFWC6):

theorem NLogN.KFW.lowQ (jW jL : Fin Pp) (rowC : IStmt HS Pp) (hW : tbl jW = lowC jW jL)
    (hL : tbl jL = levC jL rowC) {K S r H P D ρ k q0 Bp Z Zr Mb Bz : ℕ} {Kc : ℝ}
    (LV : ∀ j, LvQ Isym tbl jL rowC K r Z Zr H j (cbW Kc K j))
    (ctx : LowCtx K S r H P D ρ k q0 Bp Z Zr Mb Bz) :
    ∀ j R o (σ : Fin HS → Option ℤ), j ≤ k → CoreR σ K S r H P D ρ k q0 Bp → LowR σ j R o →
      o + R ≤ D - q0 → R < mC ^ (j + 1) →
      LowS Isym tbl jW K S r H ρ (D - q0) Bp j R o σ
        (lowF (cLowP Kc K S r H (D - q0) Bp Zr Bz) j R + (j + 1) * cLowS Bz)

Context. The lemma belongs to the part KFW C-layer 6: the low bits by digit pieces (uniform): On the low D' bits (strides 2^{ρ + K (D' - 1 - x)} S, B' 2^{K D'} S records), the recursive procedure lowC with loop registers 11 j, 12 R, 13 o, 14 m^j, 15 ES j covers the bits o ≤ x < o + R by pieces: while R ≥ m^j a level-j call on the bits o … o + m^j - 1, then j ↦ j - 1. lowF g j R sums g over the pieces (lowF ES: the scale, lowF DWb: the width, lowF 1: the count).

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.