The middle level size (round six, `VbH_mid`)

Proved

IntMul.Kappa.R6.IL3H.Hyp14.VbH_mid

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

complexity-theoryinteger-multiplicationkappa-improvement

The middle level size.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Top13):

theorem IntMul.Kappa.R6.IL3H.Hyp14.VbH_mid (hc : CfgOK c) {ρ : ℕ} (hρ : ρ < K) :
    zsz ((VbH c K (K * (D - (chV c (cWc c) P K D S r w).q0 - (chV c (cWc c) P K D S r w).qB))
      (S * 2 ^ (K * (chV c (cWc c) P K D S r w).qB)) r (chV c (cWc c) P K D S r w).H
      (D - (chV c (cWc c) P K D S r w).q0 - (chV c (cWc c) P K D S r w).qB)
      (ρ + K * (D - (chV c (cWc c) P K D S r w).q0 - (chV c (cWc c) P K D S r w).qB - 1))
      (chV c (cWc c) P K D S r w).Bp : ℕ) : ℤ) ≤ 3 * Z

Context. Round six, item language (I3): the size bounds and the level claims of the chain

Under Hyp14, with Zr = 3 Z and CB = CBc c Kc G Cc θ (the closed form of cbRec_of with the row factor Kc (G+2) + 1 of RowUCC):

VbH at the top level and at the middle level, and MbChain, are below 2^{3Z};
lvAllH: every level claim LvH … E holds;
midPre: the middle-part preconditions MidPre.

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_Defs23

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL3H IntMul.Kappa.R6.IL3H.Hyp14
open IntMul AModel NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly IntMul.Kappa.R6.Cx
variable {c : IntMul.Kappa.R6.IL3H.Cfg} {P K D S r p w Z : ℕ} {lamp : ℝ} (h : Hyp14 c P K D S r p w Z lamp)
include h

Formal statement

theorem IntMul.Kappa.R6.IL3H.Hyp14.VbH_mid (hc : CfgOK c) {ρ : ℕ} (hρ : ρ < K) :
    zsz ((VbH c K (K * (D - (chV c (cWc c) P K D S r w).q0 - (chV c (cWc c) P K D S r w).qB))
      (S * 2 ^ (K * (chV c (cWc c) P K D S r w).qB)) r (chV c (cWc c) P K D S r w).H
      (D - (chV c (cWc c) P K D S r w).q0 - (chV c (cWc c) P K D S r w).qB)
      (ρ + K * (D - (chV c (cWc c) P K D S r w).q0 - (chV c (cWc c) P K D S r w).qB - 1))
      (chV c (cWc c) P K D S r w).Bp : ℕ) : ℤ) ≤ 3 * Z := 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.Hyp14.VbH_mid`, Solutions/R6IL3Top13.lean (lines 120–199) of the round-six development.

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