ZU_le (KFM Final: `PhiNeed ∧ LineNeed ∧ ChirpNeedR → KappaBound (1/2^182)`)
ProvedNLogN.KFM.ZU_leThe following holds (Lean statement, in the scope of module Solutions.KFMFinal):
theorem NLogN.KFM.ZU_le {A : ℕ} (hA : 2 ≤ A) : (ZU n : ℝ) ≤ Kz * cLineU n A
Context. The lemma belongs to the part KFM Final: PhiNeed ∧ LineNeed ∧ ChirpNeedR → KappaBound (1/2^182).
Formalization Note. This lemma belongs to layer 5 of the Lean formalization (integer-multiplication-kappa campaign) of the OpenAI preprint Integer multiplication below : the uniform item-language machine (register programs over the NLogN multitape framework) assembling the Proposition 14 / Lemma 13 / Proposition 18 pieces, the resampling and chirp stages and the cost recurrences into one fixed machine, culminating in NLogN.KFTop.kappa_bound : IntMul.KappaBound (1 / 2 ^ 182) (the bound of Def_IntMul_MultitapeModel). All names refer to the definition bundles up to IntMul_Kappa_K_Defs33.
-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_K_Defs33
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.KFM
open Finset IntMul KIL KFB KFT KFU RS IntMul.OpenAI.Glue IntMul.OpenAI.RCost
variable {n D : ℕ} (hn : n0 ≤ n) (hD : dP n = D + 1)
include hn hD
theorem NLogN.KFM.ZU_le {A : ℕ} (hA : 2 ≤ A) : (ZU n : ℝ) ≤ Kz * cLineU n A := by sorry