Round-six lemma `cAxU_le_prime`

Proved

NLogN.KFZ.cAxU_le_prime

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.KFZCost):

theorem NLogN.KFZ.cAxU_le_prime (hP : dlV p h (D + 1) + 3 ≤ P) (s t : ℕ) (hst : s ≤ t) (ht : t ≤ 2 ^ (h + 1)) :
    cAxU (dlV p h (D + 1)) s t (KFM.fmB (2 * (dlV p h (D + 1) + 2))) ≤
      10 ^ 27 * ((P + 31) ^ 5 + (2 ^ (h + 1) + 1) * (FM P + P + 1))

Context. KFZ Cost: a closed-form bound for the body cost

cBody_le: with a master width P (bounding Δ + 3, B, Bd, b, the descriptor length and 2h + 6), N = ∏ t_i, r = 2^{h+1}, d r ≤ N:

cBody ≤ 10^30 (N + 1)(fmB(8P) + P + 1) + 10^30 (P + 31)^6.

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

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFZ
open IntMul NLogN.KIL NLogN.KFB NLogN.KFM
variable {p h D σ B Bd b len P N : ℕ} {sv t : Fin (D + 1) → ℕ}

Formal statement

theorem NLogN.KFZ.cAxU_le_prime (hP : dlV p h (D + 1) + 3 ≤ P) (s t : ℕ) (hst : s ≤ t) (ht : t ≤ 2 ^ (h + 1)) :
    cAxU (dlV p h (D + 1)) s t (KFM.fmB (2 * (dlV p h (D + 1) + 2))) ≤
      10 ^ 27 * ((P + 31) ^ 5 + (2 ^ (h + 1) + 1) * (FM P + P + 1)) := 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.KFZ.cAxU_le'`, Solutions/KFZCost.lean (lines 104–145) of the round-six development.

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