The B₀ axis splits as Ctr + mainB + overN
ProvedNLogN.KFY.NG.axB_leThe B₀ axis splits as Ctr + mainB + overN.
The following holds (Lean statement, in the scope of module Solutions.KFY19N):
theorem NLogN.KFY.NG.axB_le (Kc : ℝ) (Ctr : ℕ) :
axBC Kc n (tN n k) (NLogN.KFY.BN n) 30 Ctr
(bap0AxCostQ (pP n) (KFM.srcN n k) (tN n k) (αP n) (NLogN.KFY.mN n k) (mmN n) (ΦN n) (BaccN n) (QN n)
(BqN n)) ≤ Ctr + mainB n + overN n
Context. The lemma belongs to the part KFY 19: the cost of the two passes for n (natural-number part).
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
set_option linter.unusedSectionVars false
set_option maxRecDepth 100000
set_option exponentiation.threshold 2048
open NLogN
open NLogN.KFY
open IntMul NLogN IntMul.OpenAI.Glue NLogN.GL NLogN.AX NLogN.Rsp
variable {n : ℕ} (h : NG n) {k : ℕ} (hk : k < dP n)
include h hk
set_option maxHeartbeats 4000000
theorem NLogN.KFY.NG.axB_le (Kc : ℝ) (Ctr : ℕ) :
axBC Kc n (tN n k) (NLogN.KFY.BN n) 30 Ctr
(bap0AxCostQ (pP n) (KFM.srcN n k) (tN n k) (αP n) (NLogN.KFY.mN n k) (mmN n) (ΦN n) (BaccN n) (QN n)
(BqN n)) ≤ Ctr + mainB n + overN n := by sorry