Proposition 18 at round six from the uniform Proposition 14 (`P14UH'`) (round six, `p18Body6_of`)

Proved

NLogN.KFV.R6P.p18Body6_of

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Proposition 18 at round six from the uniform Proposition 14 (P14UH').

The following holds (Lean statement, in the scope of module Solutions.R6P18Main):

theorem NLogN.KFV.R6P.p18Body6_of (P : Prm) (hP : P.Ok) (tab : Fin 1 → IStmt HS 1) (Ew : ℕ → ℕ)
    (hEw : EwOK6 P Ew) (HU : IntMul.Kappa.R6.IL3.P14UH' tab Ew P.lamp) : P18Body6 P tab

Context. Round six, Proposition 18: the assembly (P18IL6)

p18Body6_of: from the uniform Proposition 14 in its explicit-hypothesis form (IL3.P14UH' tab Ew P.lamp, with a guard EwOK6 P Ew) the body P18Body6 P tab.

The table is IL.kTableP tab (ext6 nU U e su td q) (R6P18Tab): the Proposition 14 procedures, a copy of the frozen Lemma 12 procedure with the frozen uniform swap KFW.swGU q on it (so the frozen layout programs and their correctness proofs are reused unchanged, with the frozen swap exponent τS), and the frozen body body18. At round six K = ℓ - 1, so the swap is never called and its exponent does not enter the cost (R6P18Lay); the cost is cost6 (R6P18Cost), the side conditions par6 (R6P18Par), the chunk rounds chunkU_of6 (R6P18Chunk), setup and teardown the frozen S18M.setup18Need, the error the frozen KFM.err18Need (2 N (3ℓ+1) 2^{-p} ≤ 2^{⌊p/2⌋} 2^{-p} since N ≤ 2^{(d-1)ℓ}).

p18IL6: P18IL6 P for every admissible P (through IL3.P14UH.toH').

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
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_SApprox
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
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_KappaFramework
import Definitions.Def_IntMul_KappaJainData
import Definitions.Def_IntMul_Kappa_R6_Defs23
import Definitions.Def_IntMul_Kappa_R6_Defs16

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFV
open IntMul NLogN.RS NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFH NLogN.KFC NLogN.KFT Synth AModel AModelL12 GlueLayout GlueP14 GlueError
open IntMul.Kappa.R6

Formal statement

theorem NLogN.KFV.R6P.p18Body6_of (P : Prm) (hP : P.Ok) (tab : Fin 1 → IStmt HS 1) (Ew : ℕ → ℕ)
    (hEw : EwOK6 P Ew) (HU : IntMul.Kappa.R6.IL3.P14UH' tab Ew P.lamp) : P18Body6 P tab := 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.KFV.R6P.p18Body6_of`, Solutions/R6P18Main.lean (lines 31–216) of the round-six development.

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