Prove2Me

The node frame over a level frame (round six, `nodeRH_of`)

Proved
IntMul.Kappa.R6.IL3H.nodeRH_of

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

complexity-theoryinteger-multiplicationkappa-improvement

The node frame over a level frame.

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

theorem IntMul.Kappa.R6.IL3H.nodeRH_of (hL : LevRH c σ K t S0 r H E e B) :
    NodeRH c (σNodeH σ c K t S0 r E B) K t S0 r H E e B

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 {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.nodeRH_of (hL : LevRH c σ K t S0 r H E e B) :
    NodeRH c (σNodeH σ c K t S0 r E B) K t S0 r H E e 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 `IntMul.Kappa.R6.IL3H.nodeRH_of`, Solutions/R6IL3Node2.lean (lines 130–171) of the round-six development.

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