The forward values (round six, `bt_bound`)

Proved

J6Seg.bt_bound

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

complexity-theoryinteger-multiplicationkappa-improvement

The forward values.

The following holds (Lean statement, in the scope of module Solutions.J6SegJPre5):

theorem J6Seg.bt_bound (h : ElimOK P S L A η) (b : ℕ → ℤ) {Zm : ℝ}
    (hz : ∀ k, |(geSolveS P S L A b k : ℝ)| ≤ Zm) {r : ℕ} (hr : r < L.length) :
    |(btV P S L A b r : ℝ)| ≤ 2 * Zm + 3 / 4 + L.length / 2

Context. J kernel precondition (JPre), part 5: value bounds

fmul_abs_le, dotF_abs_le: |fmul x y| ≤ |x||y|/2^P + 1/2, and the dot product bound;
z_bound: under ElimOK, η ≤ 1/4, (|L|+1)²/2^P ≤ 1/8 and |b| ≤ M, every value of the sparse solve satisfies |z| ≤ 4M + 2^P (from geSolve_bound and geSolveS_eq);
bt_bound: the forward values satisfy |bt_r| ≤ 2 Z + 3/4 + |L|/2 when |z| ≤ Z (from the closed back substitution and fdiv_err);
cpl_abs_le, ccOf_abs_le: the coupling entries are ≤ 2^P + 1, the window coefficients ≤ (2w+1)(2^P+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_Defs67

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open Finset Real
open IntMul.HvdH IntMul.HvdH.JPf J6QLU
variable {P : ℕ} {S : ℕ → ℕ → Prop} [DecidableRel S] {L : List ℕ} {A : ℕ → ℕ → ℤ} {η : ℝ}

Formal statement

theorem J6Seg.bt_bound (h : ElimOK P S L A η) (b : ℕ → ℤ) {Zm : ℝ}
    (hz : ∀ k, |(geSolveS P S L A b k : ℝ)| ≤ Zm) {r : ℕ} (hr : r < L.length) :
    |(btV P S L A b r : ℝ)| ≤ 2 * Zm + 3 / 4 + L.length / 2 := 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.bt_bound`, Solutions/J6SegJPre5.lean (lines 81–147) of the round-six development.

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