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

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

theorem NLogN.twoAtanP_run (x₁ x₂ : ℕ) (σ c₁ c₂ : ℤ) (hx₁ : 0 < x₁) (hx₂ : 0 < x₂) (n : ℕ)
    (s : IState HS) (L3 : List Item) (h3 : s.G 3 = .int n :: L3) :
    (twoAtanP x₁ x₂ σ c₁ c₂).run s = s.upd 4 (.int ((c₁ * atanSum x₁ (2 * n + 9) σ (2 * n + 9) +
      c₂ * atanSum x₂ (2 * n + 9) σ (2 * n + 9) + 2 ^ (n + 8)) / 2 ^ (n + 9)) :: s.G 4)

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 $|z|$.