Round-six lemma `recStep_sound`

Proved

IntMul.Kappa.R6.Bit.recStep_sound

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.R6BitNBScalA):

theorem IntMul.Kappa.R6.Bit.recStep_sound (r : Rec) (hn : r.1 < 2 ^ 16) (acc : (Bool × ℕ) × List (ℕ × ℕ × ℕ))
    (hg : GateOK supG (acc.2.headD (0, 0, 0))) (h : (recStep acc r).1.1 = true) (z : ℕ → M)
    (hz : IdInv x acc.1.2 z) : IdInv x (recStep acc r).1.2 (xrL (recEl r) z)

Context. Round six, bit network: the producer computes the node values (B A C = I)

Over a module M of characteristic two, for inputs x : ℕ → M:

XOR-run algebra: xrL is additive, is inverted by the reversed run, and acts on read/write separated gate lists by adding the sources;
msum x mask = Σ_{i ∈ mask} x_i;
prod_vals: after copying x_t into the source slots (Cg) and running the producer (Ag), slot q holds msum x (supG (idF IDS3 q)) (the node-id simulation, checked by the kernel in R6BitNBIdB0..2);
reads_eq: the six slots read for target T sum to x_T (R6BitNBIdRd).

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_Defs51

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.Bit
open R6Chk R6Chk.Bit R6Chk.Bit.Loc
variable {M : Type*} [AddCommGroup M] (h2 : ∀ a : M, a + a = 0) (x : ℕ → M)
include h2

Formal statement

theorem IntMul.Kappa.R6.Bit.recStep_sound (r : Rec) (hn : r.1 < 2 ^ 16) (acc : (Bool × ℕ) × List (ℕ × ℕ × ℕ))
    (hg : GateOK supG (acc.2.headD (0, 0, 0))) (h : (recStep acc r).1.1 = true) (z : ℕ → M)
    (hz : IdInv x acc.1.2 z) : IdInv x (recStep acc r).1.2 (xrL (recEl r) z) := 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.recStep_sound`, Solutions/R6BitNBScalA.lean (lines 209–250) of the round-six development.

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