One chunk (new layout) (round six, `chB2_sp`)

Proved

J6Seg.PreT.chB2_sp

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

complexity-theoryinteger-multiplicationkappa-improvement

One chunk (new layout).

The following holds (Lean statement, in the scope of module Solutions.J6SegILPre24):

theorem J6Seg.PreT.chB2_sp (s : IState HS) (u : Fin HS → ℤ) (c : Fin HS → List ℤ) {O L : Fin HS}
    (hO : O ∉ rg) (hO5 : O.val < 56) (hO13 : O ≠ 13) (hO50 : O ≠ 50) (hOC : O ≠ Cu)
    (hL : L ∉ rg) (hL5 : L.val < 56) (hOL : O ≠ L) (hL13 : L ≠ 13)
    (hc13 : c 13 = []) (hc53 : c 53 = []) (hc54 : c 54 = []) (hcu : c Cu = [])
    {P R sN tN lam aN A pk b M q : ℕ} {ℓ : ℤ} {lr : List ℤ}
    (hi : StIn2 u P R sN tN lam aN A) (hPA : P ≤ A) (TB : J6Seg.PreT.TabB P R sN tN aN lam A)
    (h26 : u 26 = pk) (h28 : u 28 = b) (hpkM : pk ≤ M) (hbM : b ≤ M) (h4l : 4 * lam ≤ 2 ^ A)
    (hl : 1 ≤ lam) (hsA : sN ≤ 2 ^ A) (hqA : q * lam ≤ 2 ^ A) (hq : q ≤ 2 ^ A)
    (hcL : c L = ℓ :: lr) (hℓ : |ℓ| ≤ 2 ^ A) :
    Sp (chB2 O L) (encK s (J6Seg.PreT.RF (CS u q) c))
      (encK s (J6Seg.PreT.RF (CS u ((q : ℤ) - 1)) (Function.update (Function.update c O
        ((((tabQ2 P R sN tN lam aN pk b q).length : ℤ) :: tabQ2 P R sN tN lam aN pk b q) ++ c O)) L
        ((ℓ + (1 + ((tabQ2 P R sN tN lam aN pk b q).length : ℤ))) :: lr))))
      (cCh2 P R A lam M)

Context. B-line IL, pre-tables (B5′, tabK2 layout), part 24: one chunk

chB2 O L = partA2 O; partB O; partC O L. From CS u q it pushes |tab q| :: tab q with tab q = tabK2 P pk b λ di gi gam dd (descsOf … q) onto O and adds 1 + |tab q| on L. Part C is the old program, re-costed with 2^pk, 2^b at width M + 2 (pk, b ≤ 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.chB2_sp (s : IState HS) (u : Fin HS → ℤ) (c : Fin HS → List ℤ) {O L : Fin HS}
    (hO : O ∉ rg) (hO5 : O.val < 56) (hO13 : O ≠ 13) (hO50 : O ≠ 50) (hOC : O ≠ Cu)
    (hL : L ∉ rg) (hL5 : L.val < 56) (hOL : O ≠ L) (hL13 : L ≠ 13)
    (hc13 : c 13 = []) (hc53 : c 53 = []) (hc54 : c 54 = []) (hcu : c Cu = [])
    {P R sN tN lam aN A pk b M q : ℕ} {ℓ : ℤ} {lr : List ℤ}
    (hi : StIn2 u P R sN tN lam aN A) (hPA : P ≤ A) (TB : J6Seg.PreT.TabB P R sN tN aN lam A)
    (h26 : u 26 = pk) (h28 : u 28 = b) (hpkM : pk ≤ M) (hbM : b ≤ M) (h4l : 4 * lam ≤ 2 ^ A)
    (hl : 1 ≤ lam) (hsA : sN ≤ 2 ^ A) (hqA : q * lam ≤ 2 ^ A) (hq : q ≤ 2 ^ A)
    (hcL : c L = ℓ :: lr) (hℓ : |ℓ| ≤ 2 ^ A) :
    Sp (chB2 O L) (encK s (J6Seg.PreT.RF (CS u q) c))
      (encK s (J6Seg.PreT.RF (CS u ((q : ℤ) - 1)) (Function.update (Function.update c O
        ((((tabQ2 P R sN tN lam aN pk b q).length : ℤ) :: tabQ2 P R sN tN lam aN pk b q) ++ c O)) L
        ((ℓ + (1 + ((tabQ2 P R sN tN lam aN pk b q).length : ℤ))) :: lr))))
      (cCh2 P R A lam M) := 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.chB2_sp`, Solutions/J6SegILPre24.lean (lines 155–248) of the round-six development.

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