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

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

theorem NLogN.Rsp.cStage_okc (M : MulSpec) (mm Φ B : ℕ) (hM : MulBnd M mm Φ) (s t kb p α : ℕ) (hs : 0 < s)
    (hst : s ≤ t) (L1 : List (ℤ × ℤ)) (hL1 : L1.length = t) (τ : IState HS) (RU : List Item)
    (hU : τ.G CU = gints L1 ++ RU) (hP : BParams τ p s t α kb)
    (hu : ∀ z ∈ L1, zsz z.1 ≤ mm ∧ zsz z.2 ≤ mm) (hmm : 1 ≤ mm)
    (hAcc : 2 * 2 ^ (2 * mm) ≤ 2 ^ B) (hmB : 2 * mm ≤ B) (hT : zsz ((t + 1 : ℕ) : ℤ) ≤ B) :
    ((stageP Ac).ren σc).ok M τ ∧
      ((stageP Ac).ren σc).cost M τ ≤ stCost t 1 s mm Φ B (t / s + 1) ∧
      ∀ z ∈ cselL s t L1, zsz z.1 ≤ B ∧ zsz z.2 ≤ B

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 $|z|$.