mvN_runs (Generic counted loops over item arrays)

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

theorem NLogN.mvN_runs {X Y C : Fin NS} (hXY : X ≠ Y) (hXC : X ≠ C) (hYC : Y ≠ C) (hX3 : X ≠ NLogN.T3)
    (hY3 : Y ≠ NLogN.T3) (hC3 : C ≠ NLogN.T3) (s : IState NS) (A : List ℤ) (R LC : List Item) (m : ℕ)
    (hX : s.G X = ints A ++ R) (hC : s.G C = .int A.length :: LC) (hA : NLogN.Bnd m A) :
    IRuns I iprog (NLogN.mvN X Y C) s
      (((s.upd X R).upd Y ((ints A).reverse ++ s.G Y)).upd C (.int 0 :: LC))
      (A.length * (1000 * (m + zsz (A.length : ℤ) + 3)) + 30)

Context. The lemma belongs to the part Generic counted loops over item arrays.

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.