Round-six lemma `stA_spX`

Proved

J6Seg.PreT.stA_spX

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.J6SegILPreJK7):

theorem J6Seg.PreT.stA_spX (s : IState HS) (v : Fin HS → ℤ) (c : Fin HS → List ℤ) {O : Fin HS}
    (hO : O ∉ rg) (hO5 : O.val < 56) (hOC : O ≠ Cu) (hc : c Cu = []) {P R sN tN lam aN A : ℕ}
    (hi : StIn2 v P R sN tN lam aN A) (hPA : P ≤ A) {cN nN : ℕ} (h1 : v 1 = cN) (h2 : v 2 = nN)
    (hn : nN ≤ lam) {eJ : Ex HS} (hiJ : InRg eJ) (hpJ : Poly eJ = true) (hdJ : J6Seg.PreT.dg eJ ≤ 40)
    (hoJ : ∀ (u : Fin HS → ℤ) (cu : Fin HS → List ℤ), eJ.ok (encK s (J6Seg.PreT.RF u cu))) (jv : ℕ → ℕ)
    (hJ : ∀ (k : ℕ) (u : Fin HS → ℤ) (cu : Fin HS → List ℤ), u 1 = cN → u 22 = sN → u 23 = tN →
      u 25 = aN → u 12 = k → eJ.eval (encK s (J6Seg.PreT.RF u cu)) = jv k)
    (bX : ∀ k < nN, |expNeg P sN (jv k)| ≤ 2 ^ A)
    (bF : ∀ k < nN, |(-1) ^ k * J6Seg.fmul P (expNeg P sN (jv k)) (giI P sN tN aN k)| ≤ 2 ^ A)
    (bq : ∀ k < lam, |expNeg P sN (2 * aN ^ 2 * tN * (k + 1))| ≤ 2 ^ A)
    (bG : ∀ k ≤ lam, |gamI P sN tN aN k| ≤ 2 ^ A) :
    (∀ j, j ≠ 3 → j ≠ 4 → j ≠ 5 → j ≠ 11 → j ≠ 12 → stAV v P sN tN aN jv nN j = v j) ∧
    VB (stAV v P sN tN aN jv nN) A ∧
    Sp (stA O eJ) (encK s (J6Seg.PreT.RF v c)) (encK s (J6Seg.PreT.RF (stAV v P sN tN aN jv nN) (Function.update c O
      (rv (fun (k : ℕ) => (-1) ^ k * J6Seg.fmul P (expNeg P sN (jv k)) (giI P sN tN aN k)) nN ++ c O))))
      (cStA A lam eJ)

Context. B5 J side, part K7: ap / am lists as Rn steps

stA_spX: stA_sp (J6SegILPre9) with the final registers given explicitly (stAV), so the block-table loops apP, amP become Rn steps (Rn.apP, Rn.amP).

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_Defs71

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.stA_spX (s : IState HS) (v : Fin HS → ℤ) (c : Fin HS → List ℤ) {O : Fin HS}
    (hO : O ∉ rg) (hO5 : O.val < 56) (hOC : O ≠ Cu) (hc : c Cu = []) {P R sN tN lam aN A : ℕ}
    (hi : StIn2 v P R sN tN lam aN A) (hPA : P ≤ A) {cN nN : ℕ} (h1 : v 1 = cN) (h2 : v 2 = nN)
    (hn : nN ≤ lam) {eJ : Ex HS} (hiJ : InRg eJ) (hpJ : Poly eJ = true) (hdJ : J6Seg.PreT.dg eJ ≤ 40)
    (hoJ : ∀ (u : Fin HS → ℤ) (cu : Fin HS → List ℤ), eJ.ok (encK s (J6Seg.PreT.RF u cu))) (jv : ℕ → ℕ)
    (hJ : ∀ (k : ℕ) (u : Fin HS → ℤ) (cu : Fin HS → List ℤ), u 1 = cN → u 22 = sN → u 23 = tN →
      u 25 = aN → u 12 = k → eJ.eval (encK s (J6Seg.PreT.RF u cu)) = jv k)
    (bX : ∀ k < nN, |expNeg P sN (jv k)| ≤ 2 ^ A)
    (bF : ∀ k < nN, |(-1) ^ k * J6Seg.fmul P (expNeg P sN (jv k)) (giI P sN tN aN k)| ≤ 2 ^ A)
    (bq : ∀ k < lam, |expNeg P sN (2 * aN ^ 2 * tN * (k + 1))| ≤ 2 ^ A)
    (bG : ∀ k ≤ lam, |gamI P sN tN aN k| ≤ 2 ^ A) :
    (∀ j, j ≠ 3 → j ≠ 4 → j ≠ 5 → j ≠ 11 → j ≠ 12 → stAV v P sN tN aN jv nN j = v j) ∧
    VB (stAV v P sN tN aN jv nN) A ∧
    Sp (stA O eJ) (encK s (J6Seg.PreT.RF v c)) (encK s (J6Seg.PreT.RF (stAV v P sN tN aN jv nN) (Function.update c O
      (rv (fun (k : ℕ) => (-1) ^ k * J6Seg.fmul P (expNeg P sN (jv k)) (giI P sN tN aN k)) nN ++ c O))))
      (cStA A lam eJ) := 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.stA_spX`, Solutions/J6SegILPreJK7.lean (lines 26–115) of the round-six development.

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