The inverse move: the single field q (at position |A| + |Mf|) in front of Mf

The inverse move: the single field q (at position |A| + |Mf|) in front of Mf.

The following holds (Lean statement, in the scope of module Solutions.KILLemma3):

theorem NLogN.KIL.moveFieldBack_spec (t : IState HS) (xs : List ℤ) (A Mf C : Shape) (q Bb : ℕ)
    (R L1 L2 L3 L4 : List Item)
    (hlen : xs.length = AModel.vol (A ++ Mf ++ [q] ++ C)) (hb : Bnd Bb xs)
    (hC1 : 1 ≤ AModel.vol C) (hM1 : 1 ≤ AModel.vol Mf) (hq1 : 1 ≤ q) (hA1 : 1 ≤ AModel.vol A)
    (hV : zsz ((AModel.vol A * (q * (AModel.vol Mf * AModel.vol C)) : ℕ) : ℤ) ≤ Bb)
    (h0 : t.G 0 = ints xs ++ R) (h1 : t.G 1 = .int (AModel.vol A) :: L1) (h2 : t.G 2 = .int q :: L2)
    (h3 : t.G 3 = .int (AModel.vol Mf) :: L3) (h4 : t.G 4 = .int (AModel.vol C) :: L4) :
    bwdP.ok t ∧ bwdP.run t = t.upd 0 (ints (AModel.pull (A ++ Mf ++ [q] ++ C) (A ++ [q] ++ Mf ++ C)
        (swapMid A.length Mf.length 1) 0 xs) ++ R) ∧
      bwdP.cost t ≤ cMove (AModel.vol A) q (AModel.vol Mf) (AModel.vol C) Bb

Context. The lemma belongs to the part Paper Lemma 3 (single-field move) in the framework's stack-disciplined form.

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.