Round-six lemma `umt3_chk`

Proved

R6Chk.Bit.umt3_chk

by xuanji · Oct 19, 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.R6ChkBitBase):

theorem R6Chk.Bit.umt3_chk : umtOK UMT3 3 = true

Context. Bit round six (h = 23): support trees and triple tables

BT: a binary tree of numbers; btGet t d i follows the low d bits of i (most significant first). Node supports (bitmasks over the 1771 triples of Fin 23) are stored 16 per leaf in 1792-bit fields (R6BitDataSup*), read by fieldOf.
TRIPL: row i (32-bit field) is the bitmask of the i-th triple (lexicographic order); STARL: field c (1792 bits) is star c, the set of triples containing c; UMT0..3: field b of UMTj is the set of triples meeting the 6-bit chunk b of points 6j, …, 6j+5 (so um P = triples meeting P is four lookups).
Table facts are proved by one kernel evaluation each (star_spec, umt_spec).

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_Defs9

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open R6Chk R6Chk.Bit
open R6Chk

Formal statement

theorem R6Chk.Bit.umt3_chk : umtOK UMT3 3 = 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 `R6Chk.Bit.umt3_chk`, Solutions/R6ChkBitBase.lean (lines 109–109) of the round-six development.

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