nD1_facts (KFUU §8e: clearing registers and lists)

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

theorem NLogN.KFUU.nD1_facts {s : IState HS} {F : Fin HS → List ℤ} {d B : ℕ} (hD : F oD = [(d : ℤ)])
    (hB : zsz ((d + 1 : ℕ) : ℤ) ≤ B) (hB1 : 1 ≤ B) :
    nD1.Clean ∧ nD1.ok (NLogN.KIL.encK s F) ∧ nD1.Lin ∧ nD1.Bnd (NLogN.KIL.encK s F) B ∧
      nD1.eval (NLogN.KIL.encK s F) = ((d + 1 : ℕ) : ℤ)

Context. The lemma belongs to the part KFUU §8e: clearing registers and lists.

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.