Part A of a chunk (new layout) (round six, `partA2_sp`)

Proved

J6Seg.PreT.partA2_sp

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

complexity-theoryinteger-multiplicationkappa-improvement

Part A of a chunk (new layout).

The following holds (Lean statement, in the scope of module Solutions.J6SegILPre23):

theorem J6Seg.PreT.partA2_sp (s : IState HS) (u : Fin HS → ℤ) (c : Fin HS → List ℤ) {O : Fin HS}
    (hO : O ∉ rg) (hO5 : O.val < 56) (hO13 : O ≠ 13) (hO50 : O ≠ 50)
    (hc13 : c 13 = []) (hc53 : c 53 = []) (hc54 : c 54 = []) (hcu : c Cu = [])
    {P R sN tN lam aN A b q : ℕ}
    (hi : StIn2 u P R sN tN lam aN A) (hPA : P ≤ A) (TB : J6Seg.PreT.TabB P R sN tN aN lam A)
    (h28 : u 28 = b) (h4l : 4 * lam ≤ 2 ^ A)
    (hl : 1 ≤ lam) (hsA : sN ≤ 2 ^ A) (hqA : q * lam ≤ 2 ^ A) (hq : q ≤ 2 ^ A) :
    Sp (partA2 O) (encK s (J6Seg.PreT.RF (CS u q) c))
      (encK s (J6Seg.PreT.RF (RA P R sN tN lam aN u q) (Function.update c O
        (J6Seg.KR.dsEnc2 b (J6Seg.descsOf sN tN lam (mkTables P R sN tN aN) q) ++ c O))))
      (cPA2 A lam)

Context. B-line IL, pre-tables (B5′, tabK2 layout), part 23: the block loop and chunk part A

blkL2, partA2 are blkL, partA with blkB2; the descriptors are dsEnc2 b, all at width BW A (no BH/BD, no M).

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_Defs68

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PreT
open IntMul NLogN NLogN.KIL NLogN.RS IntMul.HvdH

Formal statement

theorem J6Seg.PreT.partA2_sp (s : IState HS) (u : Fin HS → ℤ) (c : Fin HS → List ℤ) {O : Fin HS}
    (hO : O ∉ rg) (hO5 : O.val < 56) (hO13 : O ≠ 13) (hO50 : O ≠ 50)
    (hc13 : c 13 = []) (hc53 : c 53 = []) (hc54 : c 54 = []) (hcu : c Cu = [])
    {P R sN tN lam aN A b q : ℕ}
    (hi : StIn2 u P R sN tN lam aN A) (hPA : P ≤ A) (TB : J6Seg.PreT.TabB P R sN tN aN lam A)
    (h28 : u 28 = b) (h4l : 4 * lam ≤ 2 ^ A)
    (hl : 1 ≤ lam) (hsA : sN ≤ 2 ^ A) (hqA : q * lam ≤ 2 ^ A) (hq : q ≤ 2 ^ A) :
    Sp (partA2 O) (encK s (J6Seg.PreT.RF (CS u q) c))
      (encK s (J6Seg.PreT.RF (RA P R sN tN lam aN u q) (Function.update c O
        (J6Seg.KR.dsEnc2 b (J6Seg.descsOf sN tN lam (mkTables P R sN tN aN) q) ++ c O))))
      (cPA2 A lam) := 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.PreT.partA2_sp`, Solutions/J6SegILPre23.lean (lines 254–361) of the round-six development.

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