Round-six lemma `setup_of`

Proved

J6Seg.Src.setup_of

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.J6SegILSrc19):

theorem J6Seg.Src.setup_of {Pp : ℕ} {I : List Sym} {tbl : Fin Pp → IStmt HS Pp} {dA Z Z1 Cm : Fin HS}
    (hP : ParkOK dA Z Z1 Cm) (hZ : ZOK Z) {body : IStmt HS Pp} {dn : RS}
    {k pre post sv t p α d W : ℕ} (hW : parBig 0 0 0 0 k pre post sv t p α d < 2 ^ W) (hW3 : 3 ≤ W)
    (cs : List (ℤ × ℤ)) (n' : ℕ) (K1 K2 : List ℤ) {cB cD : ℕ}
    (hbody : ∀ s0 v c, ParV 0 0 0 0 pre post sv t p α d v → BV W v →
      (∀ j, j ≠ 0 → j ≠ sZ → j ≠ Z → c j = []) →
      IRuns I tbl body (encK s0 (RF v c)) (encK s0 (regOut (RF v c) cs pre post n' K1 K2)) cB)
    (hdn : ∀ s0 F, (∀ i, F (ρS i) = []) → F Cm = [] →
      Sp dn (encK s0 (regOut F cs pre post n' K1 K2)) (encK s0 F) cD)
    (C : ℝ) (hCu : ((frameC W d sv p + cB : ℕ) : ℝ) ≤ C) (hCd : (cD : ℝ) ≤ C) :
    SetupS I tbl (upF dA Z Z1 Cm body) dn.code ρS dA (dsc k pre post sv t p α d)
      (slabV cs pre post n' K1 K2) C

Context. B-line IL, source-table producers, part 19: SetupS for the four slab passes

setup_of: a body taking RF v c to regOut (RF v c) cs … (for every v with ParV and every c empty off 0, sZ, Z) and a tear-down taking regOut F cs … back to F give SetupS … (upF dA Z Z1 Cm body) dn ρS dA desc (slabV cs …) C. setup1/3/5/7: the four passes (body1/3/5/7, dn0/dn7).

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_Defs70

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.Src
open IntMul NLogN NLogN.KIL Function NLogN.RS J6Seg
open J6Seg.PreT hiding rg RF RF_in RF_out upd_RF_in upd_RF_out cE InRg
open J6Seg.Slab (srcTab flatC ctlB ctlCost)
open J6Seg.PJ (rg RF RF_in RF_out ParV BV parBig cPar dsc)
open J6Seg.Ax (ρS sZ slabV SetupS)

Formal statement

theorem J6Seg.Src.setup_of {Pp : ℕ} {I : List Sym} {tbl : Fin Pp → IStmt HS Pp} {dA Z Z1 Cm : Fin HS}
    (hP : ParkOK dA Z Z1 Cm) (hZ : ZOK Z) {body : IStmt HS Pp} {dn : RS}
    {k pre post sv t p α d W : ℕ} (hW : parBig 0 0 0 0 k pre post sv t p α d < 2 ^ W) (hW3 : 3 ≤ W)
    (cs : List (ℤ × ℤ)) (n' : ℕ) (K1 K2 : List ℤ) {cB cD : ℕ}
    (hbody : ∀ s0 v c, ParV 0 0 0 0 pre post sv t p α d v → BV W v →
      (∀ j, j ≠ 0 → j ≠ sZ → j ≠ Z → c j = []) →
      IRuns I tbl body (encK s0 (RF v c)) (encK s0 (regOut (RF v c) cs pre post n' K1 K2)) cB)
    (hdn : ∀ s0 F, (∀ i, F (ρS i) = []) → F Cm = [] →
      Sp dn (encK s0 (regOut F cs pre post n' K1 K2)) (encK s0 F) cD)
    (C : ℝ) (hCu : ((frameC W d sv p + cB : ℕ) : ℝ) ≤ C) (hCd : (cD : ℝ) ≤ C) :
    SetupS I tbl (upF dA Z Z1 Cm body) dn.code ρS dA (dsc k pre post sv t p α d)
      (slabV cs pre post n' K1 K2) C := 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.Src.setup_of`, Solutions/J6SegILSrc19.lean (lines 53–96) of the round-six development.

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