HvdH machine lemma BD.bapL_length (Crude magnitude bounds, part 1: scalars and the one-dimensional maps)

The following holds (Lean statement, in the scope of module NLogN.HBd1):

theorem NLogN.BD.bapL_length (p s t α : ℕ) (V : List (ℤ × ℤ)) : (bapL p s t α V).length = s

Context. The lemma belongs to the part Crude magnitude bounds, part 1: scalars and the one-dimensional maps: All bounds are on Int.natAbs of each component. ZB z B says both components of the Gaussian integer z are at most B in absolute value; LB L B says this for every entry of a list, FB f B for every value of a function ℕ → ℤ × ℤ.

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 $|z|$.