Uniform padding (round six, `sImplQ_padU`)

Proved

NLogN.KFW.sImplQ_padU

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

complexity-theoryinteger-multiplicationkappa-improvement

Uniform padding.

The following holds (Lean statement, in the scope of module Solutions.KFWC4):

theorem NLogN.KFW.sImplQ_padU {ι : Type*} [Fintype ι] [DecidableEq ι] {lo hi r : ℕ} {st : ι → ℕ} {n np : ℕ}
    (hs : SLaws st n) (hnp : n ≤ np) {T : Matrix (ι → Fin 2) (ι → Fin 2) ℂ}
    {σ : Fin HS → Option ℤ} (hlt : ∀ i v, σ i = some v → i.val < 56) (hoff : OffG σ)
    {xL xQ : Fin HS} (hL : σ xL = some ((2 * (n * r) : ℕ) : ℤ))
    (hQ : σ xQ = some ((2 * ((np - n) * r) : ℕ) : ℤ))
    {U : IStmt HS Pp} {e dw : ℕ} {cost : ℕ → ℕ}
    (h : SImplQ Isym iprog (fun s => Agree σ s) lo hi r st np T U e dw cost) (Z : ℕ)
    (hZ : zsz ((2 * ((np - n) * r) : ℕ) : ℤ) ≤ Z) :
    SImplQ Isym iprog (fun s => Agree σ s) lo hi r st n T (padCU xL xQ U) e dw
      (fun w => 400 * (cPadC (2 * (n * r)) (2 * ((np - n) * r)) Z w dw cost + 0) + cost w)

Context. KFW C-layer 4: uniform truncation and uniform padding

truncU xD xN: KFH.truncP with the divisor 2^Δ read from xD and the item count from xN; runQ_truncU: on records at scale p + Δ it gives Q_p at scale p (a fixed run).
padPreU xL xQ, padPostU xL xQ: KFN.padPre/padPost with the lengths read from registers; sImplQ_padU: a uniform program on np ≥ n records implements the same operator on n records, padded (as KFN.sImpl_pad).

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.KFW
open IntMul NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN Complex Butterfly
variable {Pp : ℕ} {Isym : List Sym} {iprog : Fin Pp → IStmt HS Pp}

Formal statement

theorem NLogN.KFW.sImplQ_padU {ι : Type*} [Fintype ι] [DecidableEq ι] {lo hi r : ℕ} {st : ι → ℕ} {n np : ℕ}
    (hs : SLaws st n) (hnp : n ≤ np) {T : Matrix (ι → Fin 2) (ι → Fin 2) ℂ}
    {σ : Fin HS → Option ℤ} (hlt : ∀ i v, σ i = some v → i.val < 56) (hoff : OffG σ)
    {xL xQ : Fin HS} (hL : σ xL = some ((2 * (n * r) : ℕ) : ℤ))
    (hQ : σ xQ = some ((2 * ((np - n) * r) : ℕ) : ℤ))
    {U : IStmt HS Pp} {e dw : ℕ} {cost : ℕ → ℕ}
    (h : SImplQ Isym iprog (fun s => Agree σ s) lo hi r st np T U e dw cost) (Z : ℕ)
    (hZ : zsz ((2 * ((np - n) * r) : ℕ) : ℤ) ≤ Z) :
    SImplQ Isym iprog (fun s => Agree σ s) lo hi r st n T (padCU xL xQ U) e dw
      (fun w => 400 * (cPadC (2 * (n * r)) (2 * ((np - n) * r)) Z w dw cost + 0) + cost w) := 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.KFW.sImplQ_padU`, Solutions/KFWC4.lean (lines 126–245) of the round-six development.

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