`AxisUA6` from the `A`-line components (round six, `axisUA6_of`)

Proved

J6Seg.Ax.axisUA6_of

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

complexity-theoryinteger-multiplicationkappa-improvement

AxisUA6 from the A-line components.

The following holds (Lean statement, in the scope of module Solutions.J6SegAxA7):

theorem J6Seg.Ax.axisUA6_of (P : Prm) (hP : P.Ok) (hStep : AStepNeed6 P) (hKer : AKerNeedA6 P)
    (tab : Fin 1 → IStmt HS 1) (hFU : FineU6 tab) (hPre : PreA6 P 8 16 17 40 tab) :
    AxisUA6 P tab

Context. A-line axis program (AxisUA6), part 7: assembly

AKerNeedA6 P (the numeric leaf of the A axis): eventually, for every axis, kernel parameters satisfying AKCond (J6SegAxA5; pk = 8p + 16, b = 17p + 40, input bound p + 2, 2^f = 8H) and the nine non-set-up cost terms of axACost (six fine exposures, two slab passes, the kernel run) each ≤ Cb · cU6 P n A / (D + 1).

axisUA6_of: AxisUA6 P tab from FineU6 tab, PreA6 P 8 16 17 40 tab (B4, the kernel register set-up), AStepNeed6 P (A worker, the copy step and the descriptor set-ups) and AKerNeedA6 P. The descriptor stack is dA = 31 (30 when the fine stack is 31), the kernel registers ρK dF (10 … 29 without dF), the kernel aKprog2 and the axis body are the machine's own two procedures.

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_Defs70

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.Ax
open IntMul NLogN NLogN.KIL NLogN.KFB IntMul.Kappa.R6 Function
open J6Seg.Slab

Formal statement

theorem J6Seg.Ax.axisUA6_of (P : Prm) (hP : P.Ok) (hStep : AStepNeed6 P) (hKer : AKerNeedA6 P)
    (tab : Fin 1 → IStmt HS 1) (hFU : FineU6 tab) (hPre : PreA6 P 8 16 17 40 tab) :
    AxisUA6 P tab := 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 `J6Seg.Ax.axisUA6_of`, Solutions/J6SegAxA7.lean (lines 89–204) of the round-six development.

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