HvdH machine lemma AX.sapSl_all (One resampling axis, part 2: the `sapL` slice program as a `GL.LProg`)

The following holds (Lean statement, in the scope of module NLogN.HAx2):

theorem NLogN.AX.sapSl_all (M : MulSpec) (p s t α Bi Bo mm Φ Bacc Q : ℕ) (hM : MulBnd M mm Φ)
    (H : SapHyp p s t α Bi Bo mm Bacc Q) (st : IState HS) (x : List (ℤ × ℤ)) (R : List Item)
    (hC : CtxS p s t α st) (hx : x.length = s ∧ GB Bi x) (hI : st.G aI = gints x ++ R) :
    sapSlP.run st = (st.upd aI R).upd aO (gints (sapL p s t α x) ++ st.G aO) ∧
      sapSlP.ok M st ∧ sapSlP.cost M st ≤ sapSlC p s t α mm Φ Bacc Q

Context. The lemma belongs to the part One resampling axis, part 2: the sapL slice program as a GL.LProg.

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 $|z|$.