HvdH machine lemma log2P_spec (`piBits` and `log2Bits` as register programs)

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

theorem NLogN.log2P_spec (n : ℕ) (s : IState HS) (L3 : List Item) (h3 : s.G 3 = .int n :: L3) :
    log2P.ok s ∧ log2P.run s = s.upd 4 (.int (IntMul.HvdH.log2Bits n) :: s.G 4) ∧
      log2P.cost s ≤ piCost 3 3 n

Context. The lemma belongs to the part piBits and log2Bits as register programs: Input n on register 3; the result is pushed on register 4. Uses registers 0, 1, 2, 40 … 47.

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 $|z|$.