HvdH machine lemma ST.popB_of (The recursive step: numeric facts about `mkParams K N` derived from th)
ProvedNLogN.ST.popB_ofThe following holds (Lean statement, in the scope of module NLogN.HStNum):
theorem NLogN.ST.popB_of (K N : ℕ) (hK : 20 ≤ K) (hN : n0K K ≤ N) :
∀ r ∈ parRegs, zsz (parV K N r) ≤ BcK K N + N
Context. The lemma belongs to the part The recursive step: numeric facts about mkParams K N derived from the published lemmas.
Formalization Note. This is a lemma of the Lean development NLogN, a multitape-Turing-machine implementation of the Harvey–van der Hoeven 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 .
-- Harvey–van der Hoeven multitape-TM development (NLogN), generated by skeleton subtraction. import Definitions.Def_IntMul_HvdH_M_Defs8 set_option autoImplicit false set_option relaxedAutoImplicit false set_option maxHeartbeats 1000000 set_option linter.unusedSimpArgs false set_option linter.unusedVariables false open NLogN open NLogN.ST open IntMul NLogN IntMul.HvdH IntMul.HvdH.Ref
theorem NLogN.ST.popB_of (K N : ℕ) (hK : 20 ≤ K) (hN : n0K K ≤ N) :
∀ r ∈ parRegs, zsz (parV K N r) ≤ BcK K N + N := by sorry