Round-six lemma `dupK`

Proved

NLogN.KFD.FSp.dupK

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.KFDOps):

theorem NLogN.KFD.FSp.dupK {X Y Z C : Fin HS} (hXY : X ≠ Y) (hXZ : X ≠ Z) (hYZ : Y ≠ Z) (hXC : X ≠ C)
    (hYC : Y ≠ C) (hZC : Z ≠ C) (hX5 : X.val < 56) (hY5 : Y.val < 56) (hZ5 : Z.val < 56)
    (hC5 : C.val < 56) (s : IState HS) (f : Fin HS → List ℤ) (L l : List ℤ)
    (hfX : f X = L ++ l) (hfC : f C = []) (w Bb : ℕ) (hL : Bnd w L)
    (hn : zsz (L.length : ℤ) ≤ Bb) :
    FSp (dupK X Y Z C L.length) (encK s f)
      (encK s (Function.update (Function.update (Function.update f X l) Y (L.reverse ++ f Y)) Z
        (L.reverse ++ f Z))) (cLp L.length (9000 * (w + 2)) Bb)

Context. KFD: fine element loops

Fine (FSp) versions of the element loops of KIL3Ops (dupK, dropK, zeroK, zipK), of mvK with a register count, and of the conditional / c-way branch (sel). The per-element cost depends only on a data bound w (bits: w = 1); the count enters only through the loop overhead cRep N = O(N + zsz N).

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.KFD
open IntMul NLogN.RS
open NLogN.KIL

Formal statement

theorem NLogN.KFD.FSp.dupK {X Y Z C : Fin HS} (hXY : X ≠ Y) (hXZ : X ≠ Z) (hYZ : Y ≠ Z) (hXC : X ≠ C)
    (hYC : Y ≠ C) (hZC : Z ≠ C) (hX5 : X.val < 56) (hY5 : Y.val < 56) (hZ5 : Z.val < 56)
    (hC5 : C.val < 56) (s : IState HS) (f : Fin HS → List ℤ) (L l : List ℤ)
    (hfX : f X = L ++ l) (hfC : f C = []) (w Bb : ℕ) (hL : Bnd w L)
    (hn : zsz (L.length : ℤ) ≤ Bb) :
    FSp (dupK X Y Z C L.length) (encK s f)
      (encK s (Function.update (Function.update (Function.update f X l) Y (L.reverse ++ f Y)) Z
        (L.reverse ++ f Z))) (cLp L.length (9000 * (w + 2)) Bb) := 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.KFD.FSp.dupK`, Solutions/KFDOps.lean (lines 108–168) of the round-six development.

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