A push of an evaluated expression onto an empty managed stack

A push of an evaluated expression onto an empty managed stack.

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

theorem NLogN.KFV.pushV (s : IState HS) (f : Fin HS → List ℤ) {i : Fin HS} {e : Ex HS} {v : ℤ} {B : ℕ}
    (hi : i.val < 56) (hc : e.Clean) (h : EV (encK s f) B e v) (hf : f i = []) :
    Sp (.push i e) (encK s f) (encK s (Function.update f i [v]))
      ((e.size + 1) * 10 ^ 7 * (B + 2) ^ 2)

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.