HvdH machine lemma Rsp.psP_run (The `bapL` slice program)

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

theorem NLogN.Rsp.psP_run (s t kb : ℕ) (hs : 0 < s) (hco : Nat.Coprime t s) (hkb : s ≤ 2 ^ kb)
    (W : List (ℤ × ℤ)) (hW : W.length = s) (τ : IState HS) (RU : List Item)
    (hU : τ.G CU = gints W ++ RU) (h2 : ∃ L, τ.G 2 = .int (kb : ℤ) :: L)
    (hk : ∃ R, τ.G KS = natInts (psKeys s t) ++ R) (hps : ∃ L, τ.G PS = .int (s : ℤ) :: L) (B : ℕ)
    (hB : ∀ z ∈ W, zsz z.1 ≤ B ∧ zsz z.2 ≤ B) (hsB : zsz (s : ℤ) ≤ B) :
    psP.ok τ ∧ psP.run τ = τ.upd CU (gints (psL s t W) ++ RU) ∧
      psP.cost τ ≤ 2 * 2000 * (B + 2) +
        10 ^ 6 * (kb + 1) * (s + 1) * (2 + 1) * (B + kb + 2 + 4) + (2 * B + 2)

Context. The lemma belongs to the part The bapL slice program: bapP = P̃_s⁻¹ ∘ D̃' ∘ J̃' ∘ C̃ ∘ P̃_t on the slice stack CU:

• P̃_t, P̃_s⁻¹: permP 2 relocated (stack 0 ↔ CU, key stack 3 ↔ KS);
• C̃: a gresP stage with one all-ones weight row (tables relocated BW ↔ W1, BD ↔ D1);
• J̃': JP (the E parameters / tables are the native ones);
• D̃': a gresP stage with the diagonal table (relocated BW ↔ W3).

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 $|z|$.