encK_o0_upd (KFV §28: the uniform rounds and Lemma 16 (forward and opposite) as fixed IL code)

The following holds (Lean statement, in the scope of module Solutions.KFV24Proc):

theorem NLogN.KFV.encK_o0_upd (S : IState HS) {G : Fin HS → List ℤ} (hG0 : G 0 = []) {j : Fin HS}
    (hj : j ≠ 0) (x v : List ℤ) :
    encK (encK S (Function.update G j x)) (o0 v) =
      encK S (Function.update (Function.update G 0 v) j x)

Context. The lemma belongs to the part KFV §28: the uniform rounds and Lemma 16 (forward and opposite) as fixed IL code.

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.