HvdH machine lemma expBits_zsz (Tools for composing numeric register programs)

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

theorem NLogN.expBits_zsz (n : ℕ) (x y : ℤ) :
    zsz (expBits n x y).1 ≤ expB n (zsz x + zsz y) + 1 ∧
    zsz (expBits n x y).2 ≤ expB n (zsz x + zsz y) + 1

Context. The lemma belongs to the part Tools for composing numeric register programs: * general integer division Ex.ediv, remainder Ex.emod, and Ex.max0 (toNat),

• "top-based" wrappers of piP, log2P, expP usable as conditional simp lemmas,
• crude size bounds on piBits, log2Bits, expBits.

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 $|z|$.