`segInvS 0 = 0` (round six, `segInvS_zero`)

Proved

J6Seg.segInvS_zero

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

complexity-theoryinteger-multiplicationkappa-improvement

segInvS 0 = 0 under the hypotheses of segInvS_eq.

The following holds (Lean statement, in the scope of module Solutions.J6SegLn10):

theorem J6Seg.segInvS_zero {E Dm Bu : ℝ} (hs : 0 < s) (hst : s ≤ t) (hα : 2 ≤ α) (hx : 3 ≤ xθ s t α)
    (hB : BlocksOK s t B) (hlam : ∀ i < s, B.len i ≤ lam)
    (hT : TabOK P s t α B T E Dm) (hE : 0 ≤ E) (hBu : 0 ≤ Bu)
    (hP : 1000 * segG B lam E Dm Bu ≤ 2 ^ P) :
    segInvS P lam T B (fun _ => 0) = fun _ => 0

Context. Segmented inverse (S2), machine side 10: the solver maps zero to zero

B₀ must send an all-zero line to an all-zero line (the padded entries of the tensor stay exactly zero). The only non-obvious step is the rounded division fdiv P 0 d = 0, true for d ≥ 0: every pivot of the elimination is positive in the near-identity regime.

PivPos, geSolveS_zero: nonnegative pivots give geSolveS … 0 = 0;
pivPos_of: the hypotheses of geSolveS_eq give nonnegative pivots (same induction);
segInvS_zero (hypotheses of segInvS_eq), segInvFullS_zero (parameter hypotheses of segInvFullS_err).

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_Defs61

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 s t α lam : ℕ} {B : Blocks} {T : Tables}

Formal statement

theorem J6Seg.segInvS_zero {E Dm Bu : ℝ} (hs : 0 < s) (hst : s ≤ t) (hα : 2 ≤ α) (hx : 3 ≤ xθ s t α)
    (hB : BlocksOK s t B) (hlam : ∀ i < s, B.len i ≤ lam)
    (hT : TabOK P s t α B T E Dm) (hE : 0 ≤ E) (hBu : 0 ≤ Bu)
    (hP : 1000 * segG B lam E Dm Bu ≤ 2 ^ P) :
    segInvS P lam T B (fun _ => 0) = fun _ => 0 := 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.segInvS_zero`, Solutions/J6SegLn10.lean (lines 195–275) of the round-six development.

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