Round-six lemma `rdN`

Proved

J6A.DP.Rn.rdN

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.J6AIL11):

theorem J6A.DP.Rn.rdN {X : Fin HS} (hXr : X ∉ rg) (hX5 : X.val < 56) (hXT : X ≠ sTmp)
    (hXC : X ≠ cRdC) {c : Cs} {k pre post s t p α d W : ℕ}
    (hc : c X = J6A.PA.dL k pre post s t p α d)
    (hL : ∀ x ∈ J6A.PA.dL k pre post s t p α d, |x| < 2 ^ W) (hW : 8 ≤ 2 ^ W)
    (hN : 2 * (pre * (t * post)) ≤ 2 ^ W) :
    Rn (J6A.DP.rdN X) (fun _ => 0) c
      (fun v => v 20 = ((2 * (pre * (t * post)) : ℕ) : ℤ) ∧ BV W v) c
      (cRdK 1 W + (cRdK 2 W + (cRdK 4 W + J6Seg.PJ.cT nE W)))

Context. A lines, IL side 11: the copy step dupL 32

Sp.dupKE: dupK with a general count expression;
cpBody T T2 C r: from T = G.reverse, push G onto stack 0 (via T2) and restore T; rpt k b: k copies of b in sequence;
dupP: read pre, post, t of the descriptor on dA into the register file, compute 2·pre·t·post, move the array to T, push it back 32 times, drop T, leave the register file.

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_Defs64

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6A J6A.DP
open IntMul NLogN NLogN.KIL NLogN.RS J6Seg.PJ IntMul.HvdH J6A.PA

Formal statement

theorem J6A.DP.Rn.rdN {X : Fin HS} (hXr : X ∉ rg) (hX5 : X.val < 56) (hXT : X ≠ sTmp)
    (hXC : X ≠ cRdC) {c : Cs} {k pre post s t p α d W : ℕ}
    (hc : c X = J6A.PA.dL k pre post s t p α d)
    (hL : ∀ x ∈ J6A.PA.dL k pre post s t p α d, |x| < 2 ^ W) (hW : 8 ≤ 2 ^ W)
    (hN : 2 * (pre * (t * post)) ≤ 2 ^ W) :
    Rn (J6A.DP.rdN X) (fun _ => 0) c
      (fun v => v 20 = ((2 * (pre * (t * post)) : ℕ) : ℤ) ∧ BV W v) c
      (cRdK 1 W + (cRdK 2 W + (cRdK 4 W + J6Seg.PJ.cT nE W))) := 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 `J6A.DP.Rn.rdN`, Solutions/J6AIL11.lean (lines 189–233) of the round-six development.

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