HvdH machine lemma AX.axisP_all2 (One resampling axis, part 1: `axisP` with separate input / output boun)

The following holds (Lean statement, in the scope of module NLogN.HAx1):

theorem NLogN.AX.axisP_all2 (ax : GL.Axis M Ctx B) (Bo : ℕ) (s : IState HS) (Q : ℕ) (hr : AxReady2 ax Bo s Q)
    (hC : Ctx s) (L : List (ℤ × ℤ)) (Rest : List Item) (hD : s.G gD = gints L ++ Rest)
    (hL : L.length = ax.n * ax.m) (hLB : GB B L) :
    (axisP ax).run s = s.upd gD (gints (axOut ax L) ++ Rest) ∧
    (axisP ax).ok M s ∧ (axisP ax).cost M s ≤ axCost ax Q

Context. The lemma belongs to the part One resampling axis, part 1: axisP with separate input / output bounds: GL.axisP_all needs the slice map to preserve one bound GB B. The resampling maps grow their entries, so here AxReady2 ax Bo s Q asks instead that the slice map sends GB B inputs (of length n) to GB Bo outputs (of length n'). The program is unchanged (GL.axisP ax).

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 $|z|$.