A node: implementations of the steps of an interface give an implementation of tens Cm on the concatenated…

A node: implementations of the steps of an interface give an implementation of tens Cm on the concatenated role streams.

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

theorem NLogN.KFN.sImpl_node {s : ℕ} (J : Prop8IfaceR W m s) (hW : 0 < W) {r n : ℕ} (hr : 0 < r)
    {st : Fin m × κ → ℕ} (hs : SLaws st n) (ha : ∀ i, 2 * st i ∣ n)
    (code : RStep (Fin W) m → ℕ → IStmt HS Pp) (e dw : RStep (Fin W) m → ℕ)
    (cost : RStep (Fin W) m → ℕ → ℕ)
    (h : ∀ a ∈ J.net, RImplI Isym iprog r n st (a.den (Butterfly.tens Cm)) (code a) (e a) (dw a) (cost a)) :
    SImplI Isym iprog r st (W * n) (Butterfly.tens Cm) (seqLI code dw J.net) (J.net.map e).sum
      (J.net.map dw).sum (costL cost dw J.net)

Context. The lemma belongs to the part KFN: one node of the recursive C-layer: A node runs the steps of an interface J : Prop8IfaceR W m s on W role streams of n records; its bits are Fin m × κ (rows × the bits of a child), at record strides st.

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.