Round-six lemma `expApprox_bound`

Proved

IntMul.HvdH.ExpPf.expApprox_bound

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.KFMBaseExp):

theorem IntMul.HvdH.ExpPf.expApprox_bound (N : ℕ) (x y : ℤ) (hw : ‖toC (x, y) / 2 ^ N‖ ≤ 1) :
    ‖toC (expApprox N (N + 4) x y) - 2 ^ N * Complex.exp (toC (x, y) / 2 ^ N)‖
      ≤ 4 * (N + 4) + 2

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_Kappa_R6_Defs3

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.HvdH IntMul.HvdH.ExpPf
open Finset Complex
open IntMul.HvdH

Formal statement

theorem IntMul.HvdH.ExpPf.expApprox_bound (N : ℕ) (x y : ℤ) (hw : ‖toC (x, y) / 2 ^ N‖ ≤ 1) :
    ‖toC (expApprox N (N + 4) x y) - 2 ^ N * Complex.exp (toC (x, y) / 2 ^ N)‖
      ≤ 4 * (N + 4) + 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 `IntMul.HvdH.ExpPf.expApprox_bound`, Solutions/KFMBaseExp.lean (lines 130–175) of the round-six development.

View graph