HvdH machine lemma AX.sapAxP_all (One resampling axis, part 4: the `S̃` axis program `sapAxP`)

The following holds (Lean statement, in the scope of module NLogN.HAx4):

theorem NLogN.AX.sapAxP_all (M : MulSpec) (p s t α m kb Bi Bo mm Φ Bacc Q Bq : ℕ) (rest : List ℕ)
    (hM : MulBnd M mm Φ) (H : SapHyp p s t α Bi Bo mm Bacc Q) (HT : TabHyp p s t α Bq)
    (HQ : AxQ s t m kb Bi Bo Q) (hmr : m = rest.prod)
    (st : IState HS) (L : List (ℤ × ℤ)) (Rest : List Item) (hR : SapRegs p s t α m kb st)
    (hD : st.G gD = gints L ++ Rest) (hL : L.length = s * m) (hLB : GB Bi L) :
    sapAxP.run st = st.upd gD (gints (tensorGo [(t, sapL p s t α)] (s :: rest) L) ++ Rest) ∧
      sapAxP.ok M st ∧ sapAxP.cost M st ≤ sapAxCost p s t α m kb Bi mm Φ Bacc Q Bq ∧
      (tensorGo [(t, sapL p s t α)] (s :: rest) L).length = m * t ∧
      GB Bo (tensorGo [(t, sapL p s t α)] (s :: rest) L)

Context. The lemma belongs to the part One resampling axis, part 4: the S̃ axis program sapAxP: sapAxP on gD = gints L ++ R (with L of dims s :: rest, m = rest.prod) computes gints (tensorGo [(t, sapL p s t α)] (s :: rest) L) ++ R: it builds the S weight table on aW = 39 and the step table on aD = 40, runs GL.axisP with the slice program sapSl, and pops the two tables again. Every stack except gD is restored.

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 $|z|$.