desc_line (KFM Cost: the cost of `bigE` in the unit `ZU`)
ProvedNLogN.KFM.desc_lineThe following holds (Lean statement, in the scope of module Solutions.KFMCost):
theorem NLogN.KFM.desc_line : ∀ x ∈ descLineN n, 0 ≤ x ∧ x ≤ 2 ^ bP n
Context. The lemma belongs to the part KFM Cost: the cost of bigE in the unit ZU.
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 : ℕ} (hn : n0 ≤ n)
include hn
theorem NLogN.KFM.desc_line : ∀ x ∈ descLineN n, 0 ≤ x ∧ x ≤ 2 ^ bP n := by sorry