Round-six lemma `dupMv_spec`

Proved

J6Seg.JK.dupMv_spec

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.J6SegILJKer7):

theorem J6Seg.JK.dupMv_spec {X Y T C : Fin HS} (hXY : X ≠ Y) (hXT : X ≠ T) (hYT : Y ≠ T) (hXC : X ≠ C)
    (hYC : Y ≠ C) (hTC : T ≠ C) (hX5 : X.val < 56) (hY5 : Y.val < 56) (hT5 : T.val < 56)
    (hC5 : C.val < 56) (hC : G C = []) {L l : List ℤ} (hX : G X = L ++ l) {n : Ex HS}
    (hnc : n.Clean) (hno : n.ok (encK s G)) (hnl : n.Lin) {B : ℕ} (hnb : n.Bnd (encK s G) B)
    (hn : n.eval (encK s G) = L.length) (hL : Bnd B L) :
    Sp (dupMv X Y T C n) (encK s G)
      (encK s (update (update (update G Y (L.reverse ++ G Y)) T (L.reverse ++ G T)) X l))
      ((n.size + 1) * 2000 * (B + 2) + L.length * (20000 * (B + 2)) + 32)

Context. B-line IL, the J kernel (B4), part 7: list moves on managed states

Plain register programs (no products, linear cost) used around the kernel loops:

rdT X: one table entry from dTJ onto X, recorded on the consumed-table stack TT;
rdH X: one header entry from dTJ onto X (not recorded);
popL X C n: drop n entries of X;
dupMv X Y T C n: copy the top n entries of X (reversed) onto Y and onto T, removing them from X.

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_Defs66

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.JK
open IntMul NLogN NLogN.KIL Function Finset
variable {s : IState HS} {G : Fin HS → List ℤ}

Formal statement

theorem J6Seg.JK.dupMv_spec {X Y T C : Fin HS} (hXY : X ≠ Y) (hXT : X ≠ T) (hYT : Y ≠ T) (hXC : X ≠ C)
    (hYC : Y ≠ C) (hTC : T ≠ C) (hX5 : X.val < 56) (hY5 : Y.val < 56) (hT5 : T.val < 56)
    (hC5 : C.val < 56) (hC : G C = []) {L l : List ℤ} (hX : G X = L ++ l) {n : Ex HS}
    (hnc : n.Clean) (hno : n.ok (encK s G)) (hnl : n.Lin) {B : ℕ} (hnb : n.Bnd (encK s G) B)
    (hn : n.eval (encK s G) = L.length) (hL : Bnd B L) :
    Sp (dupMv X Y T C n) (encK s G)
      (encK s (update (update (update G Y (L.reverse ++ G Y)) T (L.reverse ++ G T)) X l))
      ((n.size + 1) * 2000 * (B + 2) + L.length * (20000 * (B + 2)) + 32) := 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.JK.dupMv_spec`, Solutions/J6SegILJKer7.lean (lines 96–155) of the round-six development.

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