top_encK (KIL base: register programs on "managed" stacks)

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

theorem NLogN.KIL.top_encK {s : IState HS} {f : Fin HS → List ℤ} {i : Fin HS} {a : ℤ} {l : List ℤ}
    (h : f i = a :: l) : (NLogN.KIL.encK s f).top i = a

Context. The lemma belongs to the part KIL base: register programs on "managed" stacks: encK s f is the state s with the list of integers f j pushed on top of every stack j (f j = [] for unmanaged stacks). All programs of the KIL files are specified on such states; updates are Function.updates of f, which keeps the bookkeeping small.

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.