The total cost of one context (round six, `total_le6`)

Proved

IntMul.Kappa.R6.total_le6

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

complexity-theoryinteger-multiplicationkappa-improvement

The total cost of one context.

The following holds (Lean statement, in the scope of module Solutions.R6MTotal):

theorem IntMul.Kappa.R6.total_le6 {E : ℕ} {ix : Ix6 (7 + E)} {Isym : List Sym} {tab : Fin 1 → IStmt HS 1}
    {extM : Fin E → IStmt HS (7 + E)} {Q : Q6} {D : ℕ} [∀ i, NeZero (Q.sB D i)]
    [∀ i, NeZero (Q.tB D i)] (C : BigCtx6 ix Isym tab extM Q D) (hP : Q.P.Ok)
    {CΦ CL CC CS : ℝ} (h1 : 0 ≤ CΦ) (h2 : 0 ≤ CL) (h3 : 0 ≤ CC) (h4 : 0 ≤ CS)
    (AΦ AL AC AS : ℕ)
    (e1 : C.CΦ = CΦ) (e2 : C.AΦ = AΦ) (e3 : C.BΦ = dzB Q.dPh) (e4 : C.CL = CL) (e5 : C.AL = AL)
    (e6 : C.BL = dzB Q.dLn) (e7 : C.CC = CC) (e8 : C.AC = AC) (e9 : C.BC = dzB Q.dCh)
    (c0 : ℕ) (hc0 : (c0 : ℝ) ≤ CS * cU6 Q.P Q.n AS) :
    (c0 : ℝ) + kMul6 C + kFin6 C + 2 ≤
      Ctot6 Q.P CΦ CL CC CS * cU6 Q.P Q.n (AΦ + AL + AC + AS + 2)

Context. Round-six machine: the total cost of one context

cU6_mono, ZU6_le: the unit ZU6 is within Kz6 · cU6 A (A ≥ 2).
total_le6: the setup (within C_S · cU6 A_S) plus kMul6 + kFin6 + 2 of a context is at most Ctot6 · cU6 A' with A' = A_Φ + A_L + A_C + A_S + 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_R6_Defs17

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6
open Finset IntMul NLogN NLogN.KIL NLogN.KFB NLogN.KFT NLogN.KFU NLogN.RS IntMul.OpenAI.Glue IntMul.OpenAI.RCost NLogN.KFM

Formal statement

theorem IntMul.Kappa.R6.total_le6 {E : ℕ} {ix : Ix6 (7 + E)} {Isym : List Sym} {tab : Fin 1 → IStmt HS 1}
    {extM : Fin E → IStmt HS (7 + E)} {Q : Q6} {D : ℕ} [∀ i, NeZero (Q.sB D i)]
    [∀ i, NeZero (Q.tB D i)] (C : BigCtx6 ix Isym tab extM Q D) (hP : Q.P.Ok)
    {CΦ CL CC CS : ℝ} (h1 : 0 ≤ CΦ) (h2 : 0 ≤ CL) (h3 : 0 ≤ CC) (h4 : 0 ≤ CS)
    (AΦ AL AC AS : ℕ)
    (e1 : C.CΦ = CΦ) (e2 : C.AΦ = AΦ) (e3 : C.BΦ = dzB Q.dPh) (e4 : C.CL = CL) (e5 : C.AL = AL)
    (e6 : C.BL = dzB Q.dLn) (e7 : C.CC = CC) (e8 : C.AC = AC) (e9 : C.BC = dzB Q.dCh)
    (c0 : ℕ) (hc0 : (c0 : ℝ) ≤ CS * cU6 Q.P Q.n AS) :
    (c0 : ℝ) + kMul6 C + kFin6 C + 2 ≤
      Ctot6 Q.P CΦ CL CC CS * cU6 Q.P Q.n (AΦ + AL + AC + AS + 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.Kappa.R6.total_le6`, Solutions/R6MTotal.lean (lines 154–267) of the round-six development.

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