Round-six lemma `conjPU_runs`

Proved

NLogN.KFM.conjPU_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.KFMMv):

theorem NLogN.KFM.conjPU_runs (B : ℕ) (s : IState HS) (zs : List GI) (R0 : List Item)
    (h0 : s.G 0 = ints (gflat zs) ++ R0) (hB : GB B zs) (LX LX2 : List Item)
    (hX : s.G nX = .int (zs.length : ℤ) :: LX)
    (hX2 : s.G nX2 = .int ((2 * zs.length : ℕ) : ℤ) :: LX2) :
    IRuns I iprog conjPU s (s.upd 0 (ints (gflat (zs.map gconj)) ++ R0)) (cConj zs.length B)

Context. KFM: counted moves and the conjugation pass

mvNU Cn X Y: move N items (count on Cn) from X to Y (mvNU_runs: the moved block arrives reversed). Two of them move a block in order (mv2_runs).
conjPU: zs ↦ zs.map gconj on stack 0 (N Gaussian items, counts N on nX and 2N on nX2), as KFM's itemPU with a two-item body (conjPU_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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Definitions.Def_IntMul_MultitapeModel
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_MachineRef
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

open NLogN NLogN.KFM
open IntMul NLogN.KFC NLogN.KFB NLogN.KFT NLogN.KIL
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}

Formal statement

theorem NLogN.KFM.conjPU_runs (B : ℕ) (s : IState HS) (zs : List GI) (R0 : List Item)
    (h0 : s.G 0 = ints (gflat zs) ++ R0) (hB : GB B zs) (LX LX2 : List Item)
    (hX : s.G nX = .int (zs.length : ℤ) :: LX)
    (hX2 : s.G nX2 = .int ((2 * zs.length : ℕ) : ℤ) :: LX2) :
    IRuns I iprog conjPU s (s.upd 0 (ints (gflat (zs.map gconj)) ++ R0)) (cConj zs.length B) := 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 `NLogN.KFM.conjPU_runs`, Solutions/KFMMv.lean (lines 114–178) of the round-six development.

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