The node frames (round six, `sImplQ_nodeFr`)

Proved

IntMul.Kappa.R6.IL3H.sImplQ_nodeFr

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

complexity-theoryinteger-multiplicationkappa-improvement

The node frames around a statement under the node assignment.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Node2):

theorem IntMul.Kappa.R6.IL3H.sImplQ_nodeFr {ι : Type*} [Fintype ι] [DecidableEq ι] {lo hi : ℕ} {st : ι → ℕ} {n : ℕ}
    {T : Matrix (ι → Fin 2) (ι → Fin 2) ℂ} {U : IStmt HS Pp} {e' dw : ℕ} {cost : ℕ → ℕ}
    (hL : LevRH c σ K t S0 r H E e B) (hm : 1 ≤ c.m) (hW1 : 1 ≤ c.W)
    (hA : 1 ≤ c.Aw) (hp : 1 ≤ c.p) (hS : 1 ≤ S0) (hr : 1 ≤ r) (hB1 : 1 ≤ B)
    (hBn : 1 ≤ B / c.W)
    (hδ : cLH c + ((chWs c).map fun s => c.G (s * (E / c.m))).sum + E % c.m ≤ c.G E)
    (hcw : ∀ s ∈ chWs c, s < c.m) (M Bz : ℕ)
    (hM : c.Aw * (E + 1) ^ c.p + (E + 1) ^ c.p + E + 2 * c.m + cLH c + B + c.W + S0 + r + K +
      S0 * r + K * E + 1 ≤ M)
    (hMB : zsz (M : ℤ) ≤ Bz)
    (hzA : zsz ((2 * (c.Wr * (nN (B / c.W) t S0 * r)) + 16 * (B * 2 ^ t * S0 * r) : ℕ) : ℤ) ≤ Bz)
    (H : SImplQ I iprog (fun s => Agree (σNodeH σ c K t S0 r E B) s) lo hi r st n T U e' dw cost) :
    SImplQ I iprog (fun s => Agree σ s) lo hi r st n T
      (withPushI (fr (nodeFr1 c)) (withPushI (nodeFr2 c) U)) e' dw
      (fun w => cFrame 6 5 Bz + (cNF2 c Bz (δv c E) + cost w +
        (6 * (2 * (Bz + δv c E + 1) + 2) + 1)) + (6 * (2 * Bz + 2) + 1))

Context. Round six, item language (I3): the node frames

sumGsNE_val/regs/OK, δNE_val/regs/OK: the pad-scale expression;
nodeVL1, nodeVL2, σNodeH: the node registers over a level frame;
frOK_nodeFr1, pushSeq_nodeFr2 (the 2^δ push at linear cost);
nodeRH_of: the node frame over a level frame on E ≥ 2m bits is a NodeRH;
sImplQ_nodeFr: the two node frames around a statement under Agree σNodeH.

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_R6_Defs22

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL3H
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly IntMul.Kappa.R6.Cx
variable {Pp : ℕ} {I : List Sym} {iprog : Fin Pp → IStmt HS Pp}
variable {c : IntMul.Kappa.R6.IL3H.Cfg} {σ : Fin HS → Option ℤ} {K t S0 r H E e B : ℕ}

Formal statement

theorem IntMul.Kappa.R6.IL3H.sImplQ_nodeFr {ι : Type*} [Fintype ι] [DecidableEq ι] {lo hi : ℕ} {st : ι → ℕ} {n : ℕ}
    {T : Matrix (ι → Fin 2) (ι → Fin 2) ℂ} {U : IStmt HS Pp} {e' dw : ℕ} {cost : ℕ → ℕ}
    (hL : LevRH c σ K t S0 r H E e B) (hm : 1 ≤ c.m) (hW1 : 1 ≤ c.W)
    (hA : 1 ≤ c.Aw) (hp : 1 ≤ c.p) (hS : 1 ≤ S0) (hr : 1 ≤ r) (hB1 : 1 ≤ B)
    (hBn : 1 ≤ B / c.W)
    (hδ : cLH c + ((chWs c).map fun s => c.G (s * (E / c.m))).sum + E % c.m ≤ c.G E)
    (hcw : ∀ s ∈ chWs c, s < c.m) (M Bz : ℕ)
    (hM : c.Aw * (E + 1) ^ c.p + (E + 1) ^ c.p + E + 2 * c.m + cLH c + B + c.W + S0 + r + K +
      S0 * r + K * E + 1 ≤ M)
    (hMB : zsz (M : ℤ) ≤ Bz)
    (hzA : zsz ((2 * (c.Wr * (nN (B / c.W) t S0 * r)) + 16 * (B * 2 ^ t * S0 * r) : ℕ) : ℤ) ≤ Bz)
    (H : SImplQ I iprog (fun s => Agree (σNodeH σ c K t S0 r E B) s) lo hi r st n T U e' dw cost) :
    SImplQ I iprog (fun s => Agree σ s) lo hi r st n T
      (withPushI (fr (nodeFr1 c)) (withPushI (nodeFr2 c) U)) e' dw
      (fun w => cFrame 6 5 Bz + (cNF2 c Bz (δv c E) + cost w +
        (6 * (2 * (Bz + δv c E + 1) + 2) + 1)) + (6 * (2 * Bz + 2) + 1)) := 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 `IntMul.Kappa.R6.IL3H.sImplQ_nodeFr`, Solutions/R6IL3Node2.lean (lines 385–445) of the round-six development.

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