HvdH machine lemma Ex.zsz_eval_le (A uniform cost bound for expressions)

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

theorem NLogN.Ex.zsz_eval_le (s : IState S) (B : ℕ) : ∀ e : Ex S, e.Bnd s B → zsz (e.eval s) ≤ B

Context. The lemma belongs to the part A uniform cost bound for expressions: If every value an expression computes (and every register it reads) has at most B bits, its code costs at most size · 10^7 · (B+2)^2.

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 $|z|$.