The decrement of the loop index

The decrement of the loop index.

The following holds (Lean statement, in the scope of module Solutions.KFV39LayOut2):

theorem NLogN.KFV.stepD_sp (s : IState HS) (f : Fin HS → List ℤ) {k B : ℕ} (hI : f NLogN.KFV.gI = [((k + 1 : ℕ) : ℤ)])
    (hB : k + 2 ≤ B) :
    KIL.Sp stepD (encK s f) (encK s (Function.update f NLogN.KFV.gI [((k : ℕ) : ℤ)]))
      ((((Ex.reg NLogN.KFV.gI).add (Ex.lit (-1))).size + 1) * 10 ^ 7 * (B + 2) ^ 2)

Context. The lemma belongs to the part KFV §42: phase 2 of the layout, backwards, as a fixed IL loop: Iteration j < b D (k = bD-1-j): gI ← gI - 1 (from k+1 to k), then the inverse move mvF puts the bit at nL + k back to nL + k + δ₂(k). ph2R_ok: the loop computes runArrR n 0 (ops2 …).

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.