HvdH machine lemma GL.xferP_all (Gaussian list primitives, part 3: moves, transfers, the zip product pr)

The following holds (Lean statement, in the scope of module NLogN.HGl3):

theorem NLogN.GL.xferP_all (M : MulSpec) (i j : Fin HS) (n : Ex HS) (s : IState HS) (l : List ℤ)
    (Ri : List Item) (B : ℕ) (hij : i ≠ j) (hiT : i ≠ gZT) (hjT : j ≠ gZT) (hiV : i ≠ gCV)
    (hjV : j ≠ gCV) (hi55 : i.val < 55) (hj55 : j.val < 55)
    (hi : s.G i = ints l ++ Ri) (hn : n.eval s = l.length) (hnc : n.Clean) (hno : n.ok s)
    (hl : n.Lin) (hnB : n.Bnd s B) (hB : ∀ x ∈ l, zsz x ≤ B)
    (hnr : ∀ t : IState HS, (∀ x, x ≠ i → x ≠ gZT → t.G x = s.G x) →
      n.eval t = n.eval s ∧ n.ok t ∧ n.Bnd t B) :
    (xferP i j n).run s = (s.upd i Ri).upd j (ints l ++ s.G j) ∧
    (xferP i j n).ok M s ∧
    (xferP i j n).cost M s ≤ 2 * ((n.size + 1) * 2000 * (B + 2) +
      l.length * (14000 * (B + 2) + 30) + 32)

Context. The lemma belongs to the part Gaussian list primitives, part 3: moves, transfers, the zip product program: * mv_all: mvList as an RM step (run / ok / cost in one statement).

• xferP i j n: move n integers from i to j keeping their order (via gZT).
• zipP σ cN: gD := zipWith (gzF σ p) A U (A consumed from gZA), cN ∈ {gPN, gPM} holds the length.

Formalization Note. This is a lemma of the Lean development NLogN, a multitape-Turing-machine implementation of the Harvey–van der Hoeven $O(n\log n)$ integer multiplication algorithm, used in the proof of NLogN.FIN.theorem_1_1_nlogn : IntMul.MulTimeBound fun n => n * Real.log n. All names refer to the definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8 (namespace NLogN), built on the platform model Def_IntMul_MultitapeModel and the published reference Def_IntMul_HvdH_MachineRef. The machine layer: IState HS is a state of 64 item stacks; RM programs have a semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s relative to a multiplication spec M : MulSpec; zsz z is the bit length of $|z|$.