The `J` budget on one axis (round six, `jbudget_at`)
ProvedJ6Seg.Side.jbudget_atcomplexity-theoryinteger-multiplicationkappa-improvement
The J budget on one axis.
The following holds (Lean statement, in the scope of module Solutions.J6SegJCost6):
theorem J6Seg.Side.jbudget_at (hc : Val.ValCond6 ap) (hD : dP6 P n = D + 1) :
JBudgetAt P ap n src D k (KJc P ap * cU6 P n 2 / (D + 1))
Context. J cost 6: jBudget6_all
With W2 = 3 kA + kC, W3 = W + W2 + bcC + 16, C1 = Cfm (W2+10)(W2+x+14)², the J pass costs
≤ KJ · cU6 P n 2 / (D+1), KJ = 4·10^5 (487500 C1 + 3·10^6 (3 W2 + 12) + 3·10^9 (10+bcC)²) + 6·10^11 W3.
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.Side
open Finset
open IntMul NLogN NLogN.KIL IntMul.Kappa.R6 J6Seg
variable {P : Prm} {n : ℕ} {src : ℕ → ℕ} {D k : ℕ} (ap : AxPlan6) (h : AxJ P n src D k)
include h
Formal statement
theorem J6Seg.Side.jbudget_at (hc : Val.ValCond6 ap) (hD : dP6 P n = D + 1) :
JBudgetAt P ap n src D k (KJc P ap * cU6 P n 2 / (D + 1)) := 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.Side.jbudget_at`, Solutions/J6SegJCost6.lean (lines 172–235) of the round-six development.