Prove2Me

Round-six lemma `chkTV`

Proved
IntMul.Kappa.R6.Cx.NetData.chkTV

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.R6CxNetChkI):

theorem IntMul.Kappa.R6.Cx.NetData.chkTV : allN (fun t => Nat.beq (pcR (dat.tv t)) 3 && lineOK t) NV = true

Context. Kernel checks of the boundary states and of the triple data

  • chkW: the snapshot trees are well formed;
  • chkI1 / chkI2: the initial snapshots hold init1 / init2;
  • chkF1 / chkF2: the final codes of the x, y, a wires;
  • chkTV: every triple mask has three bits and its line code is psp a b {c} with {a,b,c} the triple;
  • chkG: every gate has an input.

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_Defs25

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.Cx IntMul.Kappa.R6.Cx.NetData
open IntMul.Kappa.R6.Cx.NetBT IntMul.Kappa.R6.Cx.NetLoc IntMul.Kappa.R6.Cx.NetSpec IntMul.Kappa.R6.Cx.NetFast IntMul.Kappa.R6.Cx.NetRaw IntMul.Kappa.R6.Cx.NetMask
Formal statement
theorem IntMul.Kappa.R6.Cx.NetData.chkTV : allN (fun t => Nat.beq (pcR (dat.tv t)) 3 && lineOK t) NV = true := 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.Cx.NetData.chkTV`, Solutions/R6CxNetChkI.lean (lines 73–75) of the round-six development.

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