Round-six lemma `p3B_fsp`

Proved

NLogN.KFL.p3B_fsp

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.KFLCorr3):

theorem NLogN.KFL.p3B_fsp {κ γ : ℕ → ℕ} (hγκ : ∀ m, γ (κ m) = m) {recs : ℕ → List ℤ} {M A B Bb : ℕ}
    (hB2 : 2 ≤ Bb) (hlen : ∀ m, (recs m).length = B) (hbits : ∀ m, NLogN.Bnd 1 (recs m))
    (zB : zsz (B : ℤ) ≤ Bb) (zK : ∀ d < M, zsz ((d / 2 ^ A : ℕ) : ℤ) ≤ Bb)
    (s : IState HS) (G0 : Fin HS → List ℤ) {k : ℕ} (hk : k < M) :
    FSp (NLogN.KFL.p3B B) (encK s (P3 κ γ recs M A G0 (k + 1))) (encK s (P3 κ γ recs M A G0 k)) (cP3 B Bb)

Context. KFL §14: the correction, pass 3 (merge) and the whole correction

Pass 3 rebuilds stack 0 from the bottom: for d = M-1, …, 0 it pops the flag of d and moves the top record of sKB (a moved destination: the sorted entries are in decreasing destination order) or of sT (an unmoved record) to stack 0.

corrRS = pass 1 + radix sort + pass 3; corrRS_fsp: on flR recs M it produces flR (recs ∘ corrR) M.

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 Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFL
open IntMul NLogN.RS NLogN.KIL NLogN.KFD Butterfly

Formal statement

theorem NLogN.KFL.p3B_fsp {κ γ : ℕ → ℕ} (hγκ : ∀ m, γ (κ m) = m) {recs : ℕ → List ℤ} {M A B Bb : ℕ}
    (hB2 : 2 ≤ Bb) (hlen : ∀ m, (recs m).length = B) (hbits : ∀ m, NLogN.Bnd 1 (recs m))
    (zB : zsz (B : ℤ) ≤ Bb) (zK : ∀ d < M, zsz ((d / 2 ^ A : ℕ) : ℤ) ≤ Bb)
    (s : IState HS) (G0 : Fin HS → List ℤ) {k : ℕ} (hk : k < M) :
    FSp (NLogN.KFL.p3B B) (encK s (P3 κ γ recs M A G0 (k + 1))) (encK s (P3 κ γ recs M A G0 k)) (cP3 B Bb) := 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.KFL.p3B_fsp`, Solutions/KFLCorr3.lean (lines 44–134) of the round-six development.

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