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

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

theorem NLogN.Rsp.cStage_run (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) :
    ((stageP Ac).ren σc).run τ = τ.upd CU (gints (cselL s t L1) ++ RU)

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 $|z|$.