Moving a reversed list of pairs back (round six, `pairMove_runs`)

Proved

J6Seg.Slab.pairMove_runs

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

complexity-theoryinteger-multiplicationkappa-improvement

Moving a reversed list of pairs back, rptX X (mv i j; mv i j).

The following holds (Lean statement, in the scope of module Solutions.J6SegILCtl3):

theorem J6Seg.Slab.pairMove_runs {i j X : Fin HS} (hij : i ≠ j) (hXA : X ≠ lA)
    (Φ : List ℤ → List ℤ → Fin HS → List ℤ) (hi : ∀ A B, Φ A B i = A) (hj : ∀ A B, Φ A B j = B)
    (hup : ∀ A B v w, update (update (Φ A B) i v) j w = Φ v w)
    (hA : ∀ A B, Φ A B lA = []) (hC : ∀ A B, Φ A B lC = []) (hD : ∀ A B, Φ A B lD = [])
    (L : ℕ) {LM : List ℤ} (hX : ∀ A B, Φ A B X = (L : ℤ) :: LM) (fc A0 B0 : List ℤ)
    (hfc : fc.length = 2 * L) {Z : ℕ} (hZ : ∀ y ∈ fc, zsz y ≤ Z) (s : IState HS) :
    IRuns I iprog (rptX X lA lC lD (.seq (.op (.mv i j)) (.op (.mv i j))))
      (encK s (Φ (fc.reverse ++ A0) B0)) (encK s (Φ A0 (fc ++ B0)))
      (L * (2 * cU Z) + 3000 * (L + 1))

Context. B-line IL, B1 follow-up: building the control list, §3 list facts and the pair transfer

pos src argOf j: the input position after the first j output slabs; pos_succ, pos_mono, pos_le_n, sum_tele (the drops of the entries telescope to pos).
length_ctlOf: |ctlOf n n' src argOf| = n + n' for a monotone source.
srcTab_drop: the table seen from entry j.
pairMove_runs: rptX X (mv i j; mv i j) moves a reversed list of 2L entries back in order.
cS_upd_G, cS_self: the builder state over other registers.

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
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option maxRecDepth 20000
open IntMul NLogN NLogN.KIL Function
open NLogN.KFUU (lA lC lD cU)
open NLogN.KFT (cf3 rptX)
variable {n n' : ℕ} {src : ℕ → Option ℕ} (argOf : ℕ → ℤ)
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem J6Seg.Slab.pairMove_runs {i j X : Fin HS} (hij : i ≠ j) (hXA : X ≠ lA)
    (Φ : List ℤ → List ℤ → Fin HS → List ℤ) (hi : ∀ A B, Φ A B i = A) (hj : ∀ A B, Φ A B j = B)
    (hup : ∀ A B v w, update (update (Φ A B) i v) j w = Φ v w)
    (hA : ∀ A B, Φ A B lA = []) (hC : ∀ A B, Φ A B lC = []) (hD : ∀ A B, Φ A B lD = [])
    (L : ℕ) {LM : List ℤ} (hX : ∀ A B, Φ A B X = (L : ℤ) :: LM) (fc A0 B0 : List ℤ)
    (hfc : fc.length = 2 * L) {Z : ℕ} (hZ : ∀ y ∈ fc, zsz y ≤ Z) (s : IState HS) :
    IRuns I iprog (rptX X lA lC lD (.seq (.op (.mv i j)) (.op (.mv i j))))
      (encK s (Φ (fc.reverse ++ A0) B0)) (encK s (Φ A0 (fc ++ B0)))
      (L * (2 * cU Z) + 3000 * (L + 1)) := 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.Slab.pairMove_runs`, Solutions/J6SegILCtl3.lean (lines 124–166) of the round-six development.

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