keyProg_fsp (KFL §10: the destination key of a record, in registers)

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

theorem NLogN.KFL.keyProg_fsp {K ρ f : ℕ} (hK : 1 ≤ K) (hρK : ρ < K) (hf : 1 ≤ f) {g : Geo} (hg : g.Pos)
    (hQe : g.Q = 2 ^ (f * K)) {Bb : ℕ} (hB2 : 2 ≤ Bb) (hbig : 4 * g.V < 2 ^ Bb)
    (s : IState HS) (G : Fin HS → List ℤ) (hGc : G rC = []) {m : ℕ} (hm : m < g.V)
    {lm : List ℤ} (hGm : G NLogN.KFL.rm = (m : ℤ) :: lm) :
    FSp (keyProg K ρ f g) (encK s G)
      (encK s (Function.update G NLogN.KFL.rK ((phi g (keyc K ρ f) m : ℤ) :: G NLogN.KFL.rK))) (cKey K f Bb)

Context. The lemma belongs to the part KFL §10: the destination key of a record, in registers.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$-per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs18.