Round-six lemma `tabK2_bnd`

Proved

J6Seg.KR.tabK2_bnd

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

complexity-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.J6SegAxBud3):

theorem J6Seg.KR.tabK2_bnd (hl : 1 ≤ lam) (hlQ : lam < 2 ^ Q) (hb : 1 ≤ b)
    (hT : ∀ u < lam, |(TT Q R s t α).di u| ≤ 2 ^ (Q + R + 2) ∧
      |(TT Q R s t α).gi u| ≤ 2 ^ (Q + R + 2) ∧ |(TT Q R s t α).gam u| ≤ 2 ^ (Q + R + 2) ∧
      |(TT Q R s t α).dd u| ≤ 2 ^ (Q + R + 2) ∧
      ∀ c, |(TT Q R s t α).am c u| ≤ 2 ^ (Q + R + 2) ∧ |(TT Q R s t α).ap c u| ≤ 2 ^ (Q + R + 2))
    (hBH : Q + R + b + pk + 4 ≤ BH) (q : ℕ) :
    ∀ v ∈ tabK2 Q pk b lam (diL Q R s t lam α) (giL Q R s t lam α) (gamL Q R s t lam α)
      (ddL Q R s t lam α) (descsOf s t lam (TT Q R s t α) q), zsz v ≤ BH

Context. B-line axis budget 3: the size facts of the chunk tables (generic)

hdr2_bnd: the block headers blkHdr2 m b (1 ≤ m ≤ λ < 2^Q) have entries of size ≤ Q + b + 1;
tabK2_bnd: every entry of the chunk table has size ≤ BH once Q + R + b + pk + 4 ≤ BH and the tables T are bounded by 2^(Q+R+2);
tabK2_len: the chunk table has length ≤ 8 + 10 λ;
tabsT_len: |tabsT tab nc| ≤ nc (L + 1);
kerC_leN: kerC as a polynomial.

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_Defs66

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.KR
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFV
variable {Q R s t lam α pk b BH : ℕ}

Formal statement

theorem J6Seg.KR.tabK2_bnd (hl : 1 ≤ lam) (hlQ : lam < 2 ^ Q) (hb : 1 ≤ b)
    (hT : ∀ u < lam, |(TT Q R s t α).di u| ≤ 2 ^ (Q + R + 2) ∧
      |(TT Q R s t α).gi u| ≤ 2 ^ (Q + R + 2) ∧ |(TT Q R s t α).gam u| ≤ 2 ^ (Q + R + 2) ∧
      |(TT Q R s t α).dd u| ≤ 2 ^ (Q + R + 2) ∧
      ∀ c, |(TT Q R s t α).am c u| ≤ 2 ^ (Q + R + 2) ∧ |(TT Q R s t α).ap c u| ≤ 2 ^ (Q + R + 2))
    (hBH : Q + R + b + pk + 4 ≤ BH) (q : ℕ) :
    ∀ v ∈ tabK2 Q pk b lam (diL Q R s t lam α) (giL Q R s t lam α) (gamL Q R s t lam α)
      (ddL Q R s t lam α) (descsOf s t lam (TT Q R s t α) q), zsz v ≤ BH := 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.KR.tabK2_bnd`, Solutions/J6SegAxBud3.lean (lines 65–119) of the round-six development.

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