Round-six lemma `chunk_facts`

Proved

J6Seg.ChB.chunk_facts

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.J6SegAxBud4):

theorem J6Seg.ChB.chunk_facts (hT : J6Seg.Ax.TabAt P n src k) (hc : AxCh.ChunkPlan6 ap) (hcb : ChunkPlanB ap) :
    ChunkFacts P ap n src D k (cBI P n src k ap) (cBH P n src k ap) (cBk P n src k ap)
      (cLc P n) (cCk P n src k ap)

Context. B-line axis budget 4: the chunk half of AxBudgetAt (statement, witnesses, size facts)

ChunkPlanB ap: 2 pkA + 1 ≤ bA, 2 pkC + 3 ≤ bC (so b ≥ 2 pk + Q + 3; ap6 satisfies it);
ChunkFacts … BI BH Bk Lc Ck: the chunk conjuncts of AxBudgetAt (same order, no cost);
ChunkBudgetAt, ChunkBudget6: ChunkFacts ∧ kerC … ≤ Cc;
the witnesses cBI … cCk and chunk_facts: ChunkFacts at the witnesses on every axis with AxNum, TabAt, ChunkPlan6, ChunkPlanB.

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.ChB
open IntMul NLogN NLogN.KIL NLogN.KFB IntMul.Kappa.R6 J6Seg J6Seg.KR
variable {P : Prm} {n : ℕ} {src : ℕ → ℕ} {D k : ℕ} (ap : AxPlan6) (h : Side.AxNum P n src D k)
include h

Formal statement

theorem J6Seg.ChB.chunk_facts (hT : J6Seg.Ax.TabAt P n src k) (hc : AxCh.ChunkPlan6 ap) (hcb : ChunkPlanB ap) :
    ChunkFacts P ap n src D k (cBI P n src k ap) (cBH P n src k ap) (cBk P n src k ap)
      (cLc P n) (cCk P n src k ap) := 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.ChB.chunk_facts`, Solutions/J6SegAxBud4.lean (lines 141–291) of the round-six development.

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