Round-six lemma `grpBody_sound`

Proved

R6Chk.grpBody_sound

by xuanji · Oct 19, 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.R6ChkCx):

theorem R6Chk.grpBody_sound (hbad : bad = BADf n) (hn : n ≤ 24) {kV cnt B H D E s5 : ℕ}
    (hB : B < 2 ^ (32 * kV)) (hH : H < 2 ^ (32 * (n - kV))) (hD : D < 2 ^ (32 * kV))
    (hE : E < 2 ^ (32 * (n - kV))) (h : grpBody n bad kV cnt B H D E s5 = true) :
    finrank (ZMod 2) (spanP n B kV) = kV ∧ NondegF (spanP n B kV) ∧
      ∀ p ∈ entsDec kV cnt s5, PairOK n p.1 p.2 B kV

Context. Reflective checker for nested nondegenerate F₂ label pairs (complex round six)

Data: a list of groups. A group is one natural number read as 32-bit fields (LSB first): kV, cnt, B (kV rows), H (n-kV), D (kV), E (n-kV), then cnt entries kU, U (kU), EU (kU), R (kV-kU). Rows are vectors of F₂^n (n ≤ 24).

grpChk n bad G : Bool — two packed products for the label V = span B (H·[B;E]ᵀ = [0 | I], D·[H;B]ᵀ = [0 | I]) and two per entry (U·[H;EU]ᵀ = [0 | I], R·[H;U;R]ᵀ = [0 | 0 | I]), plus row range checks.
grpChk_sound: if grpChk n (BADf n) G = true then V is nondegenerate of dimension kV, and for every entry (U, kU), PairOK n U kU V kV: span U ≤ span V, dim span U = kU, span U nondegenerate, and span U = span V or some w ∈ span V has w·w = 1, w ⊥ U (i.e. NetComplexP8.Up (span U) (span V) with Fd = kV - kU; see R6ChkCxBridge).

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_Defs9

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open R6Chk
open Matrix Matrix.Module Module
variable {n bad : ℕ}

Formal statement

theorem R6Chk.grpBody_sound (hbad : bad = BADf n) (hn : n ≤ 24) {kV cnt B H D E s5 : ℕ}
    (hB : B < 2 ^ (32 * kV)) (hH : H < 2 ^ (32 * (n - kV))) (hD : D < 2 ^ (32 * kV))
    (hE : E < 2 ^ (32 * (n - kV))) (h : grpBody n bad kV cnt B H D E s5 = true) :
    finrank (ZMod 2) (spanP n B kV) = kV ∧ NondegF (spanP n B kV) ∧
      ∀ p ∈ entsDec kV cnt s5, PairOK n p.1 p.2 B kV := 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 `R6Chk.grpBody_sound`, Solutions/R6ChkCx.lean (lines 247–295) of the round-six development.

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