Round-six lemma `fsI_runs`

Proved

J6A.FS.fsI_runs

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.J6AIL13):

theorem J6A.FS.fsI_runs (s : IState HS) (F : Fin HS → List ℤ) (dF : Fin HS) (hQ : ∀ i, F (Q dF i) = [])
    (w f a y : ℤ) :
    IRuns I iprog (.seq (.op (.dup (Q dF 3) dF)) (.seq (.op (.dup (Q dF 2) dF))
      (.seq (.op (.dup (Q dF 1) dF)) (.seq (.op (.dup (Q dF 0) dF))
      (.seq (.op (.drop (Q dF 0))) (.seq (.op (.drop (Q dF 1))) (.seq (.op (.drop (Q dF 2)))
        (.op (.drop (Q dF 3))))))))))
      (encK s (fsG F dF w f a y)) (encK s (update F dF (w :: f :: a :: y :: F dF)))
      (cFsI w f a y)

Context. A lines, IL side 13: the fine-descriptor set-ups

For a fine descriptor descFine w f a y with w = p + 2, f a literal and a, y expressions in the registers of parAX X:

fsR X ea ey fl Q: enter the register file, parAX X, push w, f, a, y on four temporaries Q 0 … Q 3 (outside rg, away from dF), leave the register file;
fsUp: then copy y, a, f, w onto dF and drop the temporaries; fsDn: drop four from dF.

fsUp_runs / fsDn_runs.

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_Defs64

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6A J6A.FS
open IntMul NLogN NLogN.KIL NLogN.RS J6Seg.PJ IntMul.HvdH Function J6A.PA IntMul.Kappa.R6
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}

Formal statement

theorem J6A.FS.fsI_runs (s : IState HS) (F : Fin HS → List ℤ) (dF : Fin HS) (hQ : ∀ i, F (Q dF i) = [])
    (w f a y : ℤ) :
    IRuns I iprog (.seq (.op (.dup (Q dF 3) dF)) (.seq (.op (.dup (Q dF 2) dF))
      (.seq (.op (.dup (Q dF 1) dF)) (.seq (.op (.dup (Q dF 0) dF))
      (.seq (.op (.drop (Q dF 0))) (.seq (.op (.drop (Q dF 1))) (.seq (.op (.drop (Q dF 2)))
        (.op (.drop (Q dF 3))))))))))
      (encK s (fsG F dF w f a y)) (encK s (update F dF (w :: f :: a :: y :: F dF)))
      (cFsI w f a y) := 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 `J6A.FS.fsI_runs`, Solutions/J6AIL13.lean (lines 162–221) of the round-six development.

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