The square system `N = C T D` is `I + E` with `E` the map of Lemma 4.11 (`N_eq`)

Proved

IntMul.HvdH.P47.N_eq

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

complexity-theoryinteger-multiplicationkappa-improvement

The square system N = C T D is I + E with E the map of Lemma 4.11.

The following holds (Lean statement, in the scope of module Solutions.KFMBaseP47):

theorem IntMul.HvdH.P47.N_eq (hst : s < t) (α : ℕ) (hx : 1 ≤ xθ s t α) (w : ZMod s → ℂ) (l : ZMod s) :
    rowSel s t (resT s t α (diagD s t α 1 w)) l =
      w l + Eexact s t α (fun i => w (i : ZMod s)) l.val

Context. Definitions (HvdH §2.4, §4.1)

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Mathlib
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.HvdH IntMul.HvdH.P47
open Complex Real
open IntMul.HvdH IntMul.HvdH.Res
variable {s t : ℕ} [NeZero s] [NeZero t]

Formal statement

theorem IntMul.HvdH.P47.N_eq (hst : s < t) (α : ℕ) (hx : 1 ≤ xθ s t α) (w : ZMod s → ℂ) (l : ZMod s) :
    rowSel s t (resT s t α (diagD s t α 1 w)) l =
      w l + Eexact s t α (fun i => w (i : ZMod s)) l.val := 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.P47.N_eq`, Solutions/KFMBaseP47.lean (lines 1133–1177) of the round-six development.

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