Scalars of the whole program (round six, `sc_prog`)

Proved

IntMul.Kappa.R6.Bit.NB.sc_prog

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

complexity-theoryinteger-multiplicationkappa-improvement

Scalars of the whole program.

The following holds (Lean statement, in the scope of module Solutions.R6BitNBNetC):

theorem IntMul.Kappa.R6.Bit.NB.sc_prog :
    (∀ a < 1771, ∀ s < 1771,
      (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι2 a (wY s)) = φ0 ch N (ι2 a (wX s)) ∧
      (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι2 a (wX s)) = φ0 ch N (ι2 a (wY s))) ∧
    (∀ b < 1771, ∀ q < 40077, (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι1 b (wS q)) = φ0 ch N (ι1 b (wS q))) ∧
    (∀ a < 1771, ∀ q < 40077, (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι2 a (wS q)) = φ0 ch N (ι2 a (wS q)))

Context. Round six, bit network: the scalar circuit and formal correctness (FormalOK)

The module-valued scalar circuit (formal values over ZMod 2) of the whole program: stage one adds X(T, b) into Y(T, b), stage two adds Y(a, S) into X(a, S), the correction adds X into τ and τ into Y. With the labels of R6BitNBNetB and formal_labels, every role ρ w ends with (swapMat, w).

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_Defs52

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.Bit IntMul.Kappa.R6.Bit.NB
open Matrix R6Chk R6Chk.Bit R6Chk.Bit.Loc AModel AModelFac
variable {N : ℕ}
variable (ch : Ch)

Formal statement

theorem IntMul.Kappa.R6.Bit.NB.sc_prog :
    (∀ a < 1771, ∀ s < 1771,
      (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι2 a (wY s)) = φ0 ch N (ι2 a (wX s)) ∧
      (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι2 a (wX s)) = φ0 ch N (ι2 a (wY s))) ∧
    (∀ b < 1771, ∀ q < 40077, (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι1 b (wS q)) = φ0 ch N (ι1 b (wS q))) ∧
    (∀ a < 1771, ∀ q < 40077, (IntMul.Kappa.R6.Bit.NB.prog ch).foldl scalarStepM (φ0 ch N) (ι2 a (wS q)) = φ0 ch N (ι2 a (wS q))) := 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.Bit.NB.sc_prog`, Solutions/R6BitNBNetC.lean (lines 247–295) of the round-six development.

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