Prove2Me

The three extra hypotheses of `P14NeedH'` hold at round-six parameters for large `n` (round six, `p14_hyps_r6`)

Proved
IntMul.Kappa.R6.IL3.p14_hyps_r6

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

The three extra hypotheses of P14NeedH' hold at round-six parameters for large n. (1) is stated for every N < 2^(6b) (the caller's vol·r + w + Ew D is below that).

The following holds (Lean statement, in the scope of module Solutions.R6IL3Hyps):

theorem IntMul.Kappa.R6.IL3.p14_hyps_r6 (P : Prm) (hP : P.Ok) (K₀ : ℕ) : ∃ n₁ : ℕ, ∀ n, n₁ ≤ n →
    Good P n ∧ 6 ≤ KP6 P n ∧
    (∀ N : ℕ, N < 2 ^ (6 * bP6 n) → zsz (N : ℤ) ≤ pP6 P n) ∧
    (pP6 P n + K₀) ^ 5 ≤ 2 ^ KP6 P n ∧
    ((K₀ * dP6 P n * (zsz (pP6 P n : ℤ) + 1) : ℕ) : ℝ) ≤
      (KP6 P n : ℝ) * (dP6 P n : ℝ) ^ P.lamp

Context. Round six, item language (I3): the extra hypotheses of P14NeedH' at round-six parameters

p14_hyps_r6: for P.Ok, any constant K₀ and all large n, the round-six values K = KP6 = ℓ - 1, D = dP6 = ⌊b^ε⌋, p = pP6 = 6 b^(1+x) satisfy

  1. zsz N ≤ p for every N < 2^(6b) (and n ≤ 2^b, V ≤ 48 n, so every local size of the caller is far below);
  2. (p + K₀)^5 ≤ 2^K;
  3. K₀ D (zsz p + 1) ≤ K D^lamp (this is ε (2 - lamp) < 1, Prm.Ok.eps_two_sub);

and also 6 ≤ K.

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_Kappa_R6_Defs14

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open Real
open IntMul.Kappa.R6 NLogN
Formal statement
theorem IntMul.Kappa.R6.IL3.p14_hyps_r6 (P : Prm) (hP : P.Ok) (K₀ : ℕ) : ∃ n₁ : ℕ, ∀ n, n₁ ≤ n →
    Good P n ∧ 6 ≤ KP6 P n ∧
    (∀ N : ℕ, N < 2 ^ (6 * bP6 n) → zsz (N : ℤ) ≤ pP6 P n) ∧
    (pP6 P n + K₀) ^ 5 ≤ 2 ^ KP6 P n ∧
    ((K₀ * dP6 P n * (zsz (pP6 P n : ℤ) + 1) : ℕ) : ℝ) ≤
      (KP6 P n : ℝ) * (dP6 P n : ℝ) ^ P.lamp := by sorry
Source
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 `IntMul.Kappa.R6.IL3.p14_hyps_r6`, Solutions/R6IL3Hyps.lean (lines 112–239) of the round-six development.

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