The chase

The chase.

The following holds (Lean statement, in the scope of module Solutions.KFV34LayLoop3):

theorem NLogN.KFV.chase_sp (s : IState HS) (f0 : Fin HS → List ℤ) {t B : ℕ}
    (hV : f0 NLogN.KFV.gV = []) (hT2 : f0 gT2 = []) (hT : f0 NLogN.KFV.gT = [((D * qq ℓ K : ℕ) : ℤ)])
    (hI : f0 NLogN.KFV.gI = [(t : ℤ)]) (hD : RT s NLogN.KFV.gD D) (hq : RT s gQq (qq ℓ K))
    (hfD : f0 NLogN.KFV.gD = []) (hfq : f0 gQq = [])
    (ht : t < D * qq ℓ K) (hNB : D * qq ℓ K + D + 1 ≤ B) :
    KIL.Sp chase (encK s f0) (encK s (Function.update f0 NLogN.KFV.gV [(xT (NLogN.KFV.srcP D ℓ K) (D * qq ℓ K) t : ℤ)]))
      (cChase (D * qq ℓ K) B)

Context. The lemma belongs to the part KFV §37: phase 3 of the layout (the chunk exchanges) as a fixed IL loop: Iteration t < N = D q: the chase x ← src t; N times (x ← src x while x < t) runs in the register gV (an RS counter loop on gT2), then, unless x = t, the chunks t and x (after the prefix Lst) are exchanged by the Lemma 12 entry (xchC). ph3_ok: the loop computes runArr n 0 (ops3 …).

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.