Padding

Padding.

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

theorem NLogN.KFN.sImpl_pad {Pp : ℕ} {Isym : List Sym} {iprog : Fin Pp → IStmt HS Pp}
    {ι : Type*} [Fintype ι] [DecidableEq ι] {r : ℕ} {st : ι → ℕ} {n np : ℕ}
    (hs : SLaws st n) (hnp : n ≤ np) {T : Matrix (ι → Fin 2) (ι → Fin 2) ℂ}
    {code : ℕ → IStmt HS Pp} {e dw : ℕ} {cost : ℕ → ℕ}
    (h : SImplI Isym iprog r st np T code e dw cost) (Z : ℕ)
    (hZ : zsz ((2 * ((np - n) * r) : ℕ) : ℤ) ≤ Z) :
    SImplI Isym iprog r st n T (padC (2 * (n * r)) (2 * ((np - n) * r)) code) e dw
      (fun w => cPadC (2 * (n * r)) (2 * ((np - n) * r)) Z w dw cost)

Context. The lemma belongs to the part KFN: running a program on a zero-padded stream: padC L0 Q code: park the L0 entries of the stream (stack qP), push Q zeros, unpark; run code; park the first L0 entries, drop the next Q, unpark.

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.