The meta-theorem

The meta-theorem.

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

theorem NLogN.KFW.FSp.of_sub {σ : Fin HS → Option ℤ} (hσ : SClean σ) {p : RS} (hw : NoW σ p)
    {s t : IState HS} (hs : Agree σ s) {c : ℕ} (h : FSp (RS.sub σ p) s t c) : FSp p s t (400 * c)

Context. The lemma belongs to the part KFW: literals read from registers (the uniformity meta-theorem for register programs): A uniform register program U reads its run-time parameters from registers (tops of stacks). RS.sub σ U replaces each register i with σ i = some v by the literal v; the result is the (non-uniform) program with baked-in literals, for which a spec is usually already proved.

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.