All table entries `≤ 2^(P+R+2)` (round six, `tab_bnd`)

Proved

J6Seg.TabBnd.tab_bnd

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

complexity-theoryinteger-multiplicationkappa-improvement

All table entries ≤ 2^(P+R+2).

The following holds (Lean statement, in the scope of module Solutions.J6SegTabBnd1):

theorem J6Seg.TabBnd.tab_bnd (hs : 0 < s) (hst : s ≤ t) (hα : 2 ≤ α) (hP : 1 ≤ P) (hl : 1 ≤ lam)
    (hPl : 200 * (3 * (lam : ℝ) + 1) ≤ 2 ^ P)
    (hR : ddR (α : ℝ) ((t : ℝ) / s) (lam - 1) ≤ 2 ^ R) :
    ∀ u < lam, |(mkTables P R s t α).di u| ≤ 2 ^ (P + R + 2) ∧
      |(mkTables P R s t α).gi u| ≤ 2 ^ (P + R + 2) ∧
      |(mkTables P R s t α).gam u| ≤ 2 ^ (P + R + 2) ∧
      |(mkTables P R s t α).dd u| ≤ 2 ^ (P + R + 2) ∧
      ∀ c, |(mkTables P R s t α).am c u| ≤ 2 ^ (P + R + 2) ∧
        |(mkTables P R s t α).ap c u| ≤ 2 ^ (P + R + 2)

Context. Chunk-table sizes 1: every entry of mkTables is ≤ 2^(P+R+2)

From mkTables_ok (on a one-block partition with st = c, len = λ): each entry is within E = 3λ+3 ≤ 2^P (or E·D_u for dd) of 2^P times a real value of modulus ≤ 1.0001 (diR, gamR ≤ 1, giR ≤ 1.0001, |a^±| ≤ giR) or ≤ D_{λ-1} ≤ 2^R (ddR).

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_Defs61

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset Real
open IntMul.HvdH IntMul.HvdH.JPf J6QLU J6Seg
variable {P R s t α lam : ℕ}

Formal statement

theorem J6Seg.TabBnd.tab_bnd (hs : 0 < s) (hst : s ≤ t) (hα : 2 ≤ α) (hP : 1 ≤ P) (hl : 1 ≤ lam)
    (hPl : 200 * (3 * (lam : ℝ) + 1) ≤ 2 ^ P)
    (hR : ddR (α : ℝ) ((t : ℝ) / s) (lam - 1) ≤ 2 ^ R) :
    ∀ u < lam, |(mkTables P R s t α).di u| ≤ 2 ^ (P + R + 2) ∧
      |(mkTables P R s t α).gi u| ≤ 2 ^ (P + R + 2) ∧
      |(mkTables P R s t α).gam u| ≤ 2 ^ (P + R + 2) ∧
      |(mkTables P R s t α).dd u| ≤ 2 ^ (P + R + 2) ∧
      ∀ c, |(mkTables P R s t α).am c u| ≤ 2 ^ (P + R + 2) ∧
        |(mkTables P R s t α).ap c u| ≤ 2 ^ (P + R + 2) := 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 `J6Seg.TabBnd.tab_bnd`, Solutions/J6SegTabBnd1.lean (lines 35–121) of the round-six development.

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