fsub_runs (Schönhage–Strassen in the item language: stack layout and Fermat-ring arithmetic)

The following holds (Lean statement, in the scope of module NLogN.SSBase):

theorem NLogN.fsub_runs {X : Fin NS} (h1 : X ≠ RF) (h2 : X ≠ T1) (h3 : X ≠ T2) (s : IState NS)
    (w F : ℤ) (m : ℕ) (L LF : List Item) (hX : s.G X = .int w :: L) (hF : s.G RF = .int F :: LF)
    (hwm : zsz w ≤ m) (hFm : zsz F ≤ m) :
    IRuns I iprog (fsub X) s
      (s.upd X (.int (if ¬ (w + -F < 0) then w + -F else w) :: L)) (3000 * (m + 2))

Context. The lemma belongs to the part Schönhage–Strassen in the item language: stack layout and Fermat-ring arithmetic: The program uses NS item stacks with fixed roles, so that stack (in)equalities are decided by simp. The current ring ℤ/(2^n+1) is given by the ruler 2^n on top of RN and the modulus 2^n+1 on top of RF.

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_Defs18.