smulBody_runs (Schoolbook multiplication in the item language (base case))

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

theorem NLogN.smulBody_runs {A X Y W Q R T : Fin S} (hd : [A, X, Y, W, Q, R, T].Nodup)
    (s : IState S) (L : List Item) (a b c : ℤ) (hb : b ≠ 0) (N : ℕ)
    (ha : zsz a ≤ N) (hc : zsz c ≤ N) :
    IRuns I iprog (smulBody X Y W Q R T) (schoolSt s A X Y W Q L a b c)
      (schoolSt s A X Y W Q L (a * 2) (sg b (b.natAbs / 2))
        (c + a * sg b (b.natAbs % 2))) (schoolIter N)

Context. The lemma belongs to the part Schoolbook multiplication in the item language (base case).

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.