The swap of a compact field (round six, `swW_RS`)

Proved

IntMul.Kappa.R6.IL2C.swW_RS

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

complexity-theoryinteger-multiplicationkappa-improvement

The swap of a compact field (stride s = Mid QC, on register r) with Bk.

The following holds (Lean statement, in the scope of module Solutions.R6IL2CPhase):

theorem IntMul.Kappa.R6.IL2C.swW_RS {Ksw : ℝ} (H : IntMul.Kappa.R6.IL2C.SwOK tab l12 τ Ksw) (hnG : 1 ≤ n * G) {r : Fin HS} {s P Mid : ℕ}
    (hr : r = kSU ∨ r = kST)
    (hreg : ∀ t, CEnv t l K ρ G n q Br fl → ∃ L, t.G r = .int (s : ℤ) :: L)
    (hs : s = Mid * l.QC) (hM : l.M = P * l.QC * s) (hP0 : 0 < P) (hMid : 0 < Mid) :
    RecStep I tab (IntMul.Kappa.R6.IL2C.swW l12 r) (fun t => CEnv t l K ρ G n q Br fl) l.M Br (swP l.QC s 1)
      (IntMul.Kappa.R6.IL2C.cW Ksw τ l n G Br)

Context. Round six, compact controls (I2C): the compact phase as record steps

The ten table rotations (rotYA, rotYB on Y; rotS, rotC, rotL ±1 on Bk) and the twelve Lemma 12 swaps (swT, swU: the compact fields T, U with Bk) of the compact program, with all parameters read from the registers of CEnv, as RecSteps:

phaseC_step: one phase is l.stepP phInv;
psiC_step: phase, load, phase, unload is l.stepP Psi;
compactC_step: followed by the correction it is l.stepP TlS (exactly).

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_Defs21

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL2C
open NLogN NLogN.KFL NLogN.KIL NLogN.KFD NLogN.KFW NLogN.KFX IntMul NLogN.RS NLogN.KFW.RS AModel AModelL12
variable {I : List Sym} {tab : Fin 1 → IStmt HS 1} {l12 : IStmt HS 1} {τ : ℝ}
  {l : Lay} {K ρ G n L q Br : ℕ} {fl : ℤ}
variable (hP : PhOK l K ρ G n L Br)
include hP

Formal statement

theorem IntMul.Kappa.R6.IL2C.swW_RS {Ksw : ℝ} (H : IntMul.Kappa.R6.IL2C.SwOK tab l12 τ Ksw) (hnG : 1 ≤ n * G) {r : Fin HS} {s P Mid : ℕ}
    (hr : r = kSU ∨ r = kST)
    (hreg : ∀ t, CEnv t l K ρ G n q Br fl → ∃ L, t.G r = .int (s : ℤ) :: L)
    (hs : s = Mid * l.QC) (hM : l.M = P * l.QC * s) (hP0 : 0 < P) (hMid : 0 < Mid) :
    RecStep I tab (IntMul.Kappa.R6.IL2C.swW l12 r) (fun t => CEnv t l K ρ G n q Br fl) l.M Br (swP l.QC s 1)
      (IntMul.Kappa.R6.IL2C.cW Ksw τ l n G Br) := 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 `IntMul.Kappa.R6.IL2C.swW_RS`, Solutions/R6IL2CPhase.lean (lines 446–492) of the round-six development.

View graph