Round-six lemma `replP_runs`

Proved

NLogN.KFZ.replP_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.KFZPass):

theorem NLogN.KFZ.replP_runs (H W : Fin HS) (hHU : H ≠ pU) (hWU : W ≠ pU) (hHW : H ≠ W) (hH : NC H)
    (hW : NC W) (hHo : H ≠ oA ∧ H ≠ oC ∧ H ≠ oD) (hWo : W ≠ oA ∧ W ≠ oC ∧ W ≠ oD)
    (hHr : H ≠ rN2) (hWr : W ≠ rN2) (hWr1 : W ≠ KFM.rN) (τ : List GI) (B : ℕ) (hB : GB B τ)
    (N : ℕ) (s : IState HS) (RH L1 L2 : List Item) (h0 : s.G H = ints (gflat τ) ++ RH)
    (hN1 : s.G KFM.rN = .int (N : ℤ) :: L1) (hN2 : s.G rN2 = .int ((2 * τ.length : ℕ) : ℤ) :: L2) :
    IRuns I iprog (replP H W) s
      (s.upd W (ints (gflat (List.replicate N τ).flatten) ++ s.G W))
      (cRepl N (2 * τ.length) B)

Context. KFZ Pass: list moves, conjugate copies and record replication (uniform counts)

All counts are run-time data (rptX, the count copied from a register). Every spec is frame-preserving: only the named stacks change.

mvX_runs: move L from X to Y (reversed); mv2_runs: move through pU (order kept); dropX_runs: drop L.
cnjCp_runs: push the conjugate of the Gaussian list on H onto W (order kept), H unchanged; counts 2N on nX2, N on nX.
replP_runs: push N copies of the Gaussian list on H onto W; counts N on rN, 2|τ| on rN2.

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

Formal statement

theorem NLogN.KFZ.replP_runs (H W : Fin HS) (hHU : H ≠ pU) (hWU : W ≠ pU) (hHW : H ≠ W) (hH : NC H)
    (hW : NC W) (hHo : H ≠ oA ∧ H ≠ oC ∧ H ≠ oD) (hWo : W ≠ oA ∧ W ≠ oC ∧ W ≠ oD)
    (hHr : H ≠ rN2) (hWr : W ≠ rN2) (hWr1 : W ≠ KFM.rN) (τ : List GI) (B : ℕ) (hB : GB B τ)
    (N : ℕ) (s : IState HS) (RH L1 L2 : List Item) (h0 : s.G H = ints (gflat τ) ++ RH)
    (hN1 : s.G KFM.rN = .int (N : ℤ) :: L1) (hN2 : s.G rN2 = .int ((2 * τ.length : ℕ) : ℤ) :: L2) :
    IRuns I iprog (replP H W) s
      (s.upd W (ints (gflat (List.replicate N τ).flatten) ++ s.G W))
      (cRepl N (2 * τ.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.KFZ.replP_runs`, Solutions/KFZPass.lean (lines 207–257) of the round-six development.

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