The register-file core of the chunk set-up (round six, `eCcore`)

Proved

J6Seg.PJ.Rn.eCcore

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

complexity-theoryinteger-multiplicationkappa-improvement

The register-file core of the chunk set-up.

The following holds (Lean statement, in the scope of module Solutions.J6SegILPreJC4):

theorem J6Seg.PJ.Rn.eCcore {pkA pkC bA bC kA kC wA wC k pre post s t p α d W M : ℕ} {Z : Fin HS}
    (hZ : Z ∉ rg) (hZ5 : Z.val < 56) (hZT : Z ≠ sTmp) (hZC : Z ≠ cRdC)
    (hZx : Z ≠ J6Seg.PJ.sSt ∧ Z ≠ sX ∧ Z ≠ sOut ∧ Z ≠ sLn)
    {c : Cs} (hcZ : c Z = dsc k pre post s t p α d) (hcL : c cLC = [])
    (h41 : c sOut = []) (h42 : c sLn = []) (h44 : c J6Seg.PJ.sSt = []) (h45 : c sX = [])
    (h13 : c 13 = []) (h51 : c 51 = []) (h53 : c 53 = []) (h54 : c 54 = []) (hcu : c PreT.Cu = [])
    (hW : parBig kA kC wA wC k pre post s t p α d < 2 ^ W)
    (bpk : pkA * segPF p d s + pkC ≤ 2 ^ W) (bb : bA * segPF p d s + bC ≤ 2 ^ W)
    (bl : pkA + pkC + bA + bC + 2 ≤ 2 ^ W)
    (hPA : segPF p d s ≤ W) (hA2 : 2 ≤ W)
    (TB : PreT.TabB (segPF p d s) (p / 7) s t α (lamP d) W)
    (hpkM : pkA * segPF p d s + pkC ≤ M) (hbM : bA * segPF p d s + bC ≤ M)
    (hncA : t / lamP d * lamP d ≤ 2 ^ W) (hncA' : t / lamP d * (9 + 10 * lamP d) ≤ 2 ^ W) :
    Rn (J6Seg.PJ.eCcore pkA pkC bA bC kA kC wA wC Z) (fun _ => 0) c (BV W)
      (update (update (update c sX (cVals (wA * segPF p d s + wC) (Nat.log 2 (lamF d)) pre
        (t / lamP d) post)) sOut
        (tabsT (PreT.tabQ2 (segPF p d s) (p / 7) s t (lamP d) α (pkA * segPF p d s + pkC)
          (bA * segPF p d s + bC)) (t / lamP d))) sLn
        [((tabsT (PreT.tabQ2 (segPF p d s) (p / 7) s t (lamP d) α (pkA * segPF p d s + pkC)
          (bA * segPF p d s + bC)) (t / lamP d)).length : ℤ)])
      (cCcore pkA pkC bA bC kA kC wA wC W d (NF d s) (segPF p d s) (kA * segPF p d s + kC)
        (lamP d) (p / 7) M (t / lamP d))

Context. B5 chunk side (eC), part 4: the register-file core

parP; cPre; cMid; chunkStoreP2 41 42; cA3 from the descriptor on Z: the table stream tabsT (tabQ2 …) nc on 41, its length on 42, the register values ρ 0 … ρ 8 on 45 (Rn.eCcore).

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_Defs71

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.PJ
open IntMul NLogN NLogN.KIL NLogN.RS Function

Formal statement

theorem J6Seg.PJ.Rn.eCcore {pkA pkC bA bC kA kC wA wC k pre post s t p α d W M : ℕ} {Z : Fin HS}
    (hZ : Z ∉ rg) (hZ5 : Z.val < 56) (hZT : Z ≠ sTmp) (hZC : Z ≠ cRdC)
    (hZx : Z ≠ J6Seg.PJ.sSt ∧ Z ≠ sX ∧ Z ≠ sOut ∧ Z ≠ sLn)
    {c : Cs} (hcZ : c Z = dsc k pre post s t p α d) (hcL : c cLC = [])
    (h41 : c sOut = []) (h42 : c sLn = []) (h44 : c J6Seg.PJ.sSt = []) (h45 : c sX = [])
    (h13 : c 13 = []) (h51 : c 51 = []) (h53 : c 53 = []) (h54 : c 54 = []) (hcu : c PreT.Cu = [])
    (hW : parBig kA kC wA wC k pre post s t p α d < 2 ^ W)
    (bpk : pkA * segPF p d s + pkC ≤ 2 ^ W) (bb : bA * segPF p d s + bC ≤ 2 ^ W)
    (bl : pkA + pkC + bA + bC + 2 ≤ 2 ^ W)
    (hPA : segPF p d s ≤ W) (hA2 : 2 ≤ W)
    (TB : PreT.TabB (segPF p d s) (p / 7) s t α (lamP d) W)
    (hpkM : pkA * segPF p d s + pkC ≤ M) (hbM : bA * segPF p d s + bC ≤ M)
    (hncA : t / lamP d * lamP d ≤ 2 ^ W) (hncA' : t / lamP d * (9 + 10 * lamP d) ≤ 2 ^ W) :
    Rn (J6Seg.PJ.eCcore pkA pkC bA bC kA kC wA wC Z) (fun _ => 0) c (BV W)
      (update (update (update c sX (cVals (wA * segPF p d s + wC) (Nat.log 2 (lamF d)) pre
        (t / lamP d) post)) sOut
        (tabsT (PreT.tabQ2 (segPF p d s) (p / 7) s t (lamP d) α (pkA * segPF p d s + pkC)
          (bA * segPF p d s + bC)) (t / lamP d))) sLn
        [((tabsT (PreT.tabQ2 (segPF p d s) (p / 7) s t (lamP d) α (pkA * segPF p d s + pkC)
          (bA * segPF p d s + bC)) (t / lamP d)).length : ℤ)])
      (cCcore pkA pkC bA bC kA kC wA wC W d (NF d s) (segPF p d s) (kA * segPF p d s + kC)
        (lamP d) (p / 7) M (t / lamP d)) := 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.PJ.Rn.eCcore`, Solutions/J6SegILPreJC4.lean (lines 41–164) of the round-six development.

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