A role permutation on role streams of records

A role permutation on role streams of records.

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

theorem NLogN.KFN.rImpl_perm (T : Butterfly.Op κ) (σ : Equiv.Perm (Fin W)) {r n : ℕ} (hr : 0 < r)
    (st : Fin m × κ → ℕ) (Z : ℕ) (hZ : zsz ((2 * (W * (n * r)) : ℕ) : ℤ) ≤ Z) :
    RImpl r n st ((RStep.perm σ : RStep (Fin W) m).den T) (swapsP (n * r) (permSw σ)) 0 0
      (fun w => W * cAtom w 0 (n * r) W Z + 1)

Context. The lemma belongs to the part KFN: gates and role permutations on role streams of records.

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.