Round-six lemma `sp_bits3`

Proved

IntMul.Kappa.R6.sp_bits3

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_bits3 (s : IState HS) (f : Fin HS → List ℤ) (N dp : ℕ) (l25 l34 : List ℤ) (Aa Bb : ℕ)
    (h51 : f 51 = []) (h25 : f 25 = (N : ℤ) :: l25) (h34 : f 34 = (dp : ℤ) :: l34)
    (hNA : N ≤ Aa) (hdA : dp ≤ Aa) (hAa : 2 ≤ Aa) (hB : Aa ^ 6 < 2 ^ Bb) :
    Sp btR (encK s f) (encK s (update (update (update f 26 (bitL dp N ++ f 26))
      27 (bitL dp N ++ f 27)) 28 (bitL dp N ++ f 28))) ((N * 31 + 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_bits3 (s : IState HS) (f : Fin HS → List ℤ) (N dp : ℕ) (l25 l34 : List ℤ) (Aa Bb : ℕ)
    (h51 : f 51 = []) (h25 : f 25 = (N : ℤ) :: l25) (h34 : f 34 = (dp : ℤ) :: l34)
    (hNA : N ≤ Aa) (hdA : dp ≤ Aa) (hAa : 2 ≤ Aa) (hB : Aa ^ 6 < 2 ^ Bb) :
    Sp btR (encK s f) (encK s (update (update (update f 26 (bitL dp N ++ f 26))
      27 (bitL dp N ++ f 27)) 28 (bitL dp N ++ f 28))) ((N * 31 + 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_bits3`, Solutions/R6SDesc.lean (lines 105–184) of the round-six development.

View graph