Round-six lemma `sp_copy3`

Proved

IntMul.Kappa.R6.sp_copy3

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.R6SDesc):

theorem IntMul.Kappa.R6.sp_copy3 (s : IState HS) (f : Fin HS → List ℤ) (Lz l37 l25 : List ℤ) (Aa Bb : ℕ)
    (h51 : f 51 = []) (h25 : f 25 = (Lz.length : ℤ) :: l25) (h37 : f 37 = Lz ++ l37)
    (hb : ∀ z ∈ Lz, |z| ≤ (Aa : ℤ)) (hNA : Lz.length ≤ Aa) (hAa : 2 ≤ Aa)
    (hB : Aa ^ 6 < 2 ^ Bb) :
    Sp cpR (encK s f) (encK s (update (update (update (update f 37 l37) 26 (Lz.reverse ++ f 26))
      27 (Lz.reverse ++ f 27)) 28 (Lz.reverse ++ f 28))) ((Lz.length * 8 + 1) * kU Bb)

Context. Round-six setup, part 6: the descriptor loops

cpR: pops N values off 37 and pushes each on 26, 27, 28 (so each receives the reversed list).
btR: pushes the bits [d' ≤ j] for j = N-1, …, 0 on 26, 27, 28 (so each receives [b_0, …, b_{N-1}] on top).

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_Defs69

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open IntMul NLogN NLogN.KIL NLogN.KFM NLogN.RS Finset Function

Formal statement

theorem IntMul.Kappa.R6.sp_copy3 (s : IState HS) (f : Fin HS → List ℤ) (Lz l37 l25 : List ℤ) (Aa Bb : ℕ)
    (h51 : f 51 = []) (h25 : f 25 = (Lz.length : ℤ) :: l25) (h37 : f 37 = Lz ++ l37)
    (hb : ∀ z ∈ Lz, |z| ≤ (Aa : ℤ)) (hNA : Lz.length ≤ Aa) (hAa : 2 ≤ Aa)
    (hB : Aa ^ 6 < 2 ^ Bb) :
    Sp cpR (encK s f) (encK s (update (update (update (update f 37 l37) 26 (Lz.reverse ++ f 26))
      27 (Lz.reverse ++ f 27)) 28 (Lz.reverse ++ f 28))) ((Lz.length * 8 + 1) * kU Bb) := 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 `IntMul.Kappa.R6.sp_copy3`, Solutions/R6SDesc.lean (lines 32–100) of the round-six development.

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