The register part of a pad (round six, `cost_padH`)

Proved

IntMul.Kappa.R6.IL3H.cost_padH

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

complexity-theoryinteger-multiplicationkappa-improvement

The register part of a pad.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Cost2):

theorem IntMul.Kappa.R6.IL3H.cost_padH {L0 Q Z w dw r H Nn : ℕ} (hL : L0 ≤ 2 * (Nn * r)) (hQ : Q ≤ 2 * (Nn * r))
    (hw : w + dw ≤ H) (hZ : zsz (L0 : ℤ) ≤ Z) (hN : 1 ≤ Nn) :
    400 * (cPadC L0 Q Z w dw (fun _ => 0) + 0) ≤ 10 ^ 8 * (Nn * Yu r Z H)

Context. Round six, item language (I3): cost bounds of the pads

With Y = Yu r Z H:

cost_padH: the register part of a pad (padCU) on ≤ 2 Nn r entries costs ≤ 10^8 Nn Y;
cost_scaleH: the scale pad on N records costs ≤ 10^16 N Y.

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 Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_KappaFramework

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly

Formal statement

theorem IntMul.Kappa.R6.IL3H.cost_padH {L0 Q Z w dw r H Nn : ℕ} (hL : L0 ≤ 2 * (Nn * r)) (hQ : Q ≤ 2 * (Nn * r))
    (hw : w + dw ≤ H) (hZ : zsz (L0 : ℤ) ≤ Z) (hN : 1 ≤ Nn) :
    400 * (cPadC L0 Q Z w dw (fun _ => 0) + 0) ≤ 10 ^ 8 * (Nn * Yu r Z H) := 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.Kappa.R6.IL3H.cost_padH`, Solutions/R6IL3Cost2.lean (lines 27–70) of the round-six development.

View graph