Round-six lemma `cY_le`
ProvedJ6Seg.ChB.cY_lecomplexity-theoryinteger-multiplicationkappa-improvement
An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.
The following holds (Lean statement, in the scope of module Solutions.J6SegAxBud7):
theorem J6Seg.ChB.cY_le :
(cY P n src k ap : ℝ) ≤ Kch P ap * ((pP6 P n : ℝ) * (Real.log (IntMul.lg n) + 1) ^ 2)
Context. B-line axis budget 7: chunkBudget6_all
cY_le:cY ≤ Kch · p · (log lg n + 1)²;chunk_budget_at:ChunkBudgetAt P ap n src D k (Kch · cU6 P n 2 / (D+1))on every axis withAxNumandTabAt;chunkBudget6_all P hP ap (hc : ChunkPlan6 ap) (hcb : ChunkPlanB ap) : ChunkBudget6 P ap.
Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).
Preamble
-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.
import Definitions.Def_IntMul_Kappa_R6_Defs70
set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000
open J6Seg J6Seg.ChB
open IntMul NLogN NLogN.KIL NLogN.KFB IntMul.Kappa.R6 J6Seg J6Seg.KR
variable {P : Prm} {n : ℕ} {src : ℕ → ℕ} {D k : ℕ} (ap : AxPlan6) (h : Side.AxNum P n src D k)
include h
Formal statement
theorem J6Seg.ChB.cY_le :
(cY P n src k ap : ℝ) ≤ Kch P ap * ((pP6 P n : ℝ) * (Real.log (IntMul.lg n) + 1) ^ 2) := by sorrySource
S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `J6Seg.ChB.cY_le`, Solutions/J6SegAxBud7.lean (lines 25–171) of the round-six development.