The row operator from Lemma 13

The row operator from Lemma 13.

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

theorem NLogN.KFR.rowImpl_of_lem13 {iprog : Fin 1 → IStmt HS 1} {code} {Kc : ℝ}
    (H : Lem13Body iprog code Kc) (Isym : List Sym) {K m k r : ℕ} (hK : 6 ≤ K) (hm : 3 ≤ m)
    (hr : 1 ≤ r) (pos : Fin m × Fin (m ^ k) → ℕ) (t S0 B : ℕ) (hinj : Function.Injective pos)
    (hpos : ∀ p, pos p < t) (haff : AffK2 K t m k pos) (hS0 : 1 ≤ S0) (hB : 1 ≤ B)
    (i j : Fin m) (hij : i ≠ j) :
    ∃ codeR : ℕ → IStmt HS 1,
      SImplI Isym iprog r (fun p => 2 ^ pos p * S0) (B * 2 ^ t * S0)
        (Pm i j : Butterfly.Op (Fin m × Fin (m ^ k))) codeR 0 0
        (fun w => ⌈NLogN.KFR.bnd Kc K (m ^ k) (B * 2 ^ t * S0) r w⌉₊)

Context. The lemma belongs to the part KFR: permRecs realises the row operator Pm i j, and the statement Lem13ILP.

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_Defs22.