The rowAdd step as a fine IL run

The rowAdd step as a fine IL run.

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

theorem NLogN.KFB.rowAdd_iruns {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    (s : IState HS) (wb : ℕ) {W m n : ℕ} (hW : 1 ≤ W) (T : Butterfly.Op (Fin n)) (w : Fin W)
    (i j : Fin m) (hij : i ≠ j) (f : Fin HS → List ℤ) (zs : List GI) (l : List ℤ)
    (x : Fin W × (Fin m × Fin n → Fin 2) → ℂ) (hx : Rep zs x) (h0 : f 0 = gflat zs ++ l)
    (hC : Cl f) (hb : NLogN.KFB.GB wb zs) :
    ∃ zs', Rep zs' ((P8Lift.RStep.den T
        (P8Lift.RStep.rowAdd w i j hij : P8Lift.RStep (Fin W) m)).mulVec x) ∧ NLogN.KFB.GB wb zs' ∧
      IRuns I iprog (rowP W m n w.val i j n) (encK s f)
        (encK s (Function.update f 0 (gflat zs' ++ l))) (n * cColU W (m * n) wb + 1)

Context. The lemma belongs to the part KFB: the rowAdd step as a fine IL program.

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.