Round-six lemma `cL12F_lin`

Proved

NLogN.KFE.cL12F_lin

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

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

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

theorem NLogN.KFE.cL12F_lin {q m W u e Kp r n Vb Vq V' : ℕ} (cO : ℕ → ℕ) (B : ℕ) (hm : 2 ≤ m)
    (hq : 2 ≤ q) (h1 : e ≤ u) (h3 : Vq ≤ q ^ 2 * Vb) (h4 : r ≤ W * Kp) (h5 : n ≤ e)
    (h6 : n ≠ 0 → V' ≤ 2 * Vq) (h7 : Kp ≤ u) :
    cL12F u e r Kp n Vq V' m q cO B ≤ Av q W * ((Vb + 1) * (Kp + 1)) +
      Ao m W * ((B + 2) ^ 2 * (u + 1)) +
      ∑ j ∈ range (Nat.digits m n).length, n / m ^ j % m * cO j

Context. KFE: closed fine cost bound of the Lemma 12 program

cL12F_lin: cL12F ≤ Av·(Vb+1)(Kp+1) + Ao·(Bb+2)²(u+1) + Σ_j digit_j · cO j — the data passes are linear in the bit volume Vb (no (Bb+2) factor); the descriptor work costs O((u+1)(Bb+2)²).
calls_le: the per-call overhead of the oracle, Σ_{j<L} digit_j (S+1)^j ≤ m·n^(c+1) with c = clog_m (S+1): the number of Prop. 11 calls is polynomial in n ≤ u.
cL12F_le: with the fine oracle cost ilCostF and S/W < m^τ, cL12F ≤ K·((Vb + 1)·u^τ + (u+1)^(c+1)·(Bb+2)²), c = clog_m (S+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 Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFE
open IntMul NLogN.RS AModel AModelL12 Finset NLogN.KIL NLogN.KFD

Formal statement

theorem NLogN.KFE.cL12F_lin {q m W u e Kp r n Vb Vq V' : ℕ} (cO : ℕ → ℕ) (B : ℕ) (hm : 2 ≤ m)
    (hq : 2 ≤ q) (h1 : e ≤ u) (h3 : Vq ≤ q ^ 2 * Vb) (h4 : r ≤ W * Kp) (h5 : n ≤ e)
    (h6 : n ≠ 0 → V' ≤ 2 * Vq) (h7 : Kp ≤ u) :
    cL12F u e r Kp n Vq V' m q cO B ≤ Av q W * ((Vb + 1) * (Kp + 1)) +
      Ao m W * ((B + 2) ^ 2 * (u + 1)) +
      ∑ j ∈ range (Nat.digits m n).length, n / m ^ j % m * cO j := 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 `NLogN.KFE.cL12F_lin`, Solutions/KFECost.lean (lines 30–139) of the round-six development.

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