HvdH machine lemma GL.ftlP_all (`convL` and `FtL` programs (parametric in the `fermatW` program))

The following holds (Lean statement, in the scope of module NLogN.HGl7):

theorem NLogN.GL.ftlP_all (fw : LProg2 M Ctx PreW (fermatW q) CW) (hfw : FwOK fw) {m Φm : ℕ}
    (hM : MulBnd M m Φm) (s : IState HS) (hC : Ctx s) (U : List (ℤ × ℤ))
    (R RC LN LT LP : List Item) (T Q : ℕ)
    (hU : s.G gD = gints U ++ R) (hCH : s.G gCH = gints (chirpL q) ++ RC) (hUl : U.length = T)
    (hAl : (chirpL q).length = T) (hN : s.G gPN = .int T :: LN)
    (hT : s.G gPT = .int (2 ^ (q.p + q.logT)) :: LT) (hP : s.G gPP = .int (2 ^ q.p) :: LP)
    (hUb : GB m U) (hAb : GB m (chirpL q)) (hPre : PreW (ftlV q U) (chirpL q))
    (hWl : (fermatW q (ftlV q U) (chirpL q)).length = T)
    (hWb : GB Q (fermatW q (ftlV q U) (chirpL q))) (hCb : GB m (convL q (ftlV q U) (chirpL q)))
    (hmQ : 2 * m + 1 ≤ Q) (hk : q.p + q.logT + 1 ≤ Q) (hNQ : zsz ((2 * T : ℕ) : ℤ) ≤ Q) :
    (ftlP fw).run s = s.upd gD (gints (FtL q U) ++ R) ∧
    (ftlP fw).ok M s ∧
    (ftlP fw).cost M s ≤ CW + 8 * T * Φm + 1000000 * (T + 1) * (Q + 2)

Context. The lemma belongs to the part convL and FtL programs (parametric in the fermatW program): * convP fw: gD : gints U ++ R, gZA : gints A ++ R₁ ↦ gD : gints (convL q U A) ++ R, gZA : R₁ (transfers to the abstract fermatW program fw, then tdiv 2^(p+logT)).

• ftlP fw: gD : gints U ++ R ↦ gD : gints (FtL q U) ++ R with the chirp table chirpL q kept on gCH.

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 $|z|$.