Pass 5 (round six, `pass5`)

Proved

J6Seg.Src.pass5

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

complexity-theoryinteger-multiplicationkappa-improvement

Pass 5.

The following holds (Lean statement, in the scope of module Solutions.J6SegILSrc23):

theorem J6Seg.Src.pass5 {Pp : ℕ} {I : List Sym} {tbl : Fin Pp → IStmt HS Pp} {dA Z Z1 Cm : Fin HS}
    (hP : ParkOK dA Z Z1 Cm) (hZ : ZOK Z) (hCm : CmOK Cm) {k pre post s t p α d T : ℕ}
    (R : Reg k pre post s t p α d T)
    (hG : 2 ^ Nat.clog 2 (Jlist (mkBlocks s t (lamP d) (wF d))).length ≤ 3 * t) :
    SetupS I tbl (upF dA Z Z1 Cm body5) (dn0 Cm).code ρS dA (dsc k pre post s t p α d)
      (slabV (ctlOf (2 ^ Nat.clog 2 (Jlist (mkBlocks s t (lamP d) (wF d))).length) t
        (scSrc s t (lamP d) (wF d)) (fun _ => 0)) pre post t [] []) (B0 p d s t α)

Context. B-line IL, source-table producers, part 23: the four slab set-ups on plain parameters

With the parking registers ParkOK dA Z Z1 Cm, ZOK Z, CmOK Cm, and the numerical regime of an axis (pre · t · post = T, 2T < 2^p, p + 1 ≤ Q ≤ 3p, …), the procedures upF dA Z Z1 Cm body{1,3,5,7} / dn0 Cm, dn7 Cm are SetupS pairs for the slab register files at cost B0 = (Kb + 10^4) (NF d s + p + t + 1) · Ut p α (pass1 … pass7), with A = W = 4p.

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.Src
open IntMul NLogN NLogN.KIL Function NLogN.RS J6Seg
open J6Seg.PreT hiding rg RF RF_in RF_out upd_RF_in upd_RF_out cE InRg
open J6Seg.Slab (srcTab flatC ctlB ctlCost argO)
open J6Seg.PJ (rg RF RF_in RF_out ParV BV parBig cPar dsc)
open J6Seg.Ax (ρS sZ slabV SetupS)

Formal statement

theorem J6Seg.Src.pass5 {Pp : ℕ} {I : List Sym} {tbl : Fin Pp → IStmt HS Pp} {dA Z Z1 Cm : Fin HS}
    (hP : ParkOK dA Z Z1 Cm) (hZ : ZOK Z) (hCm : CmOK Cm) {k pre post s t p α d T : ℕ}
    (R : Reg k pre post s t p α d T)
    (hG : 2 ^ Nat.clog 2 (Jlist (mkBlocks s t (lamP d) (wF d))).length ≤ 3 * t) :
    SetupS I tbl (upF dA Z Z1 Cm body5) (dn0 Cm).code ρS dA (dsc k pre post s t p α d)
      (slabV (ctlOf (2 ^ Nat.clog 2 (Jlist (mkBlocks s t (lamP d) (wF d))).length) t
        (scSrc s t (lamP d) (wF d)) (fun _ => 0)) pre post t [] []) (B0 p d s t α) := 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.Src.pass5`, Solutions/J6SegILSrc23.lean (lines 273–322) of the round-six development.

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