A gate step on role streams of records

A gate step on role streams of records.

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

theorem NLogN.KFN.rImpl_gate (T : Butterfly.Op κ) (as : List KFG.Atom) (hok : ∀ a ∈ as, a.Ok W ∧ a.h ≤ 1)
    (hts : NoTS as) {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.gate (gateMat W as) : RStep (Fin W) m).den T) (gateC W (n * r) as)
      (gδ as) (3 * as.length + 3 * W) (cGateC W (n * r) Z as.length)

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.