ndiv (KFV §24: the leaf rounds with register-computed parameters)

The following holds (Lean statement, in the scope of module Solutions.KFV20Leaf):

theorem NLogN.KFV.EV.ndiv {a b : Ex HS} {x y : ℕ} (ha : EV st B a (x : ℤ)) (hb : EV st B b (y : ℤ)) (hy : 0 < y)
    (hv : zsz ((x / y : ℕ) : ℤ) ≤ B) : EV st B (.ndiv a b) ((x / y : ℕ) : ℤ)

Context. The lemma belongs to the part KFV §24: the leaf rounds with register-computed parameters: leafR eP eD eS pushes the eight leaf registers (LRegs) of the layer [2^a] × [2]^d × [2^c] (records of 2^ℓ coefficients), computed at run time from expressions eP, eD, eS over the layout registers (a, d, c their values, ℓ on gLl), runs the uniform leaf leafU (KFV12Leaf), and pops the registers. leafR_runs.

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.