A stream program on one role

A stream program on one role.

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

theorem NLogN.KFN.rImplI_onRole (hW : 0 < W) (w : Fin W) {ι : Type*} [Fintype ι] [DecidableEq ι]
    {r n : ℕ} (hr : 0 < r) (st' : ι → ℕ) (Tw : Matrix (ι → Fin 2) (ι → Fin 2) ℂ) {code : ℕ → IStmt HS Pp}
    {e dw : ℕ} {cost : ℕ → ℕ} (h : SImplI I iprog r st' n Tw code e dw cost) (st : Fin m × κ → ℕ)
    (A : ROp (Fin W) m κ)
    (hA : ∀ X : Fin W → List (Fin r → ℂ), (∀ v, (X v).length = n) →
      roleOpF st n A X = Function.update X w (selOpF st' n Tw (X w)))
    (Z : ℕ) (hZ : zsz ((2 * (W * (n * r)) : ℕ) : ℤ) ≤ Z) :
    RImplI I iprog r n st A (wrapCI W (n * r) w.val e code) e (dw + (e + 2) * W)
      (cWrapR W (n * r) Z e dw cost)

Context. The lemma belongs to the part KFN: implementations by item-language programs.

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_Defs20.