A push of an arbitrary (e.g. multiplying) expression, with the quadratic cost bound

A push of an arbitrary (e.g. multiplying) expression, with the quadratic cost bound.

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

theorem NLogN.KIL.Sp.pushQ {i : Fin HS} {e : Ex HS} {B : ℕ} (hi : i.val < 56) (hc : e.Clean)
    (hok : e.ok (NLogN.KIL.encK s f)) (hB : e.Bnd (NLogN.KIL.encK s f) B) :
    Sp (.push i e) (NLogN.KIL.encK s f) (NLogN.KIL.encK s (Function.update f i (e.eval (NLogN.KIL.encK s f) :: f i)))
      ((e.size + 1) * 10 ^ 7 * (B + 2) ^ 2)

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.