Schur lemma (report L9) (round six, `schur_row`)

Proved

J6Seg.schur_row

by xuanji · Oct 19, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Schur lemma (report L9). If every row of a - I on j :: L has absolute sum ≤ η ≤ 1/2, then so does every row of the Schur complement a_ik - a_ij a_jk / a_jj on L.

The following holds (Lean statement, in the scope of module Solutions.J6SegFix):

theorem J6Seg.schur_row {a : ℕ → ℕ → ℝ} {j : ℕ} {L : List ℕ} (hj : j ∉ L) {η : ℝ} (hη : η ≤ 1 / 2)
    (hrow : ∀ i ∈ j :: L, |a i j - kd i j| + ∑ k ∈ L.toFinset, |a i k - kd i k| ≤ η) :
    ∀ i ∈ L, ∑ k ∈ L.toFinset, |a i k - a i j * a j k / a j j - kd i k| ≤ η

Context. Segmented inverse (S2): fixed-point primitives and elimination without pivoting

Fixed-point numbers of precision P are integers a standing for a / 2^P.

rq a b = ⌊a/b + 1/2⌋ (rounded quotient), rs P a = rq a (2^P), fmul P x y = rs P (x*y) and fdiv P x y = rq (x 2^P) y (errors ≤ 1/2 ulp).
geSolve P L A b: Gaussian elimination without pivoting in the order of the index list L, written as forward elimination of the head pivot followed by back substitution.
schur_row (report L9): if every row of A - I has absolute sum ≤ η ≤ 1/2, the same holds for the Schur complement after eliminating one pivot; hence for every Schur complement, and no pivoting is ever needed (any order, any sparsity pattern, cyclic borders included).
near_id: ‖A - I‖_∞ ≤ η < 1 and ‖A z - r‖ ≤ ρ give ‖z‖ ≤ (‖r‖ + ρ)/(1 - η).
geSolve_resid: the computed solution has residual ≤ (Z+1)(|L|+1)²/2^P; geSolve_bound: and |z| ≤ 4B + 1.

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_Defs3

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset

Formal statement

theorem J6Seg.schur_row {a : ℕ → ℕ → ℝ} {j : ℕ} {L : List ℕ} (hj : j ∉ L) {η : ℝ} (hη : η ≤ 1 / 2)
    (hrow : ∀ i ∈ j :: L, |a i j - kd i j| + ∑ k ∈ L.toFinset, |a i k - kd i k| ≤ η) :
    ∀ i ∈ L, ∑ k ∈ L.toFinset, |a i k - a i j * a j k / a j j - kd i k| ≤ η := 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 `J6Seg.schur_row`, Solutions/J6SegFix.lean (lines 98–138) of the round-six development.

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