Round-six lemma `MS_of`

Proved

IntMul.Kappa.R6.MS_of

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

complexity-theoryinteger-multiplicationkappa-improvement

S ≥ (1 - 1/(4d)) T-type bound: 2 M ≤ S.

The following holds (Lean statement, in the scope of module Solutions.R6MFacts):

theorem IntMul.Kappa.R6.MS_of (P : Prm) (n : ℕ) (src : ℕ → ℕ) (hs : SrcOK6 P n src) (hd : 1 ≤ dP6 P n)
    (hl : 1 ≤ ℓP6 P n) (hL : 3 ≤ LP6 P n) (hb : 1 ≤ b'P6 P n) :
    2 * M6 P n ≤ S6 P n src

Context. Round-six machine: QOk for large n

prod_tF6: ∏_{j<d} t_j = T.
eps_room: eventually 4 b^ε ≤ b (so ℓ ≥ 2).
qok_of: for P.Ok, there is n₁ such that QOk ⟨P, n, src⟩ holds for every n ≥ n₁ and every source list with SrcOK6 P n src.

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_MultitapeModel
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
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_MachineRef
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData
import Definitions.Def_IntMul_Kappa_R6_Defs14

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.KFB NLogN.KFM IntMul.OpenAI.RT AModel

Formal statement

theorem IntMul.Kappa.R6.MS_of (P : Prm) (n : ℕ) (src : ℕ → ℕ) (hs : SrcOK6 P n src) (hd : 1 ≤ dP6 P n)
    (hl : 1 ≤ ℓP6 P n) (hL : 3 ≤ LP6 P n) (hb : 1 ≤ b'P6 P n) :
    2 * M6 P n ≤ S6 P n src := 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.MS_of`, Solutions/R6MFacts.lean (lines 60–139) of the round-six development.

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