Round-six lemma `touch_sub`

Proved

NLogN.KFW.RS.touch_sub

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.KFWSub):

theorem NLogN.KFW.RS.touch_sub (σ : Fin HS → Option ℤ) : ∀ (p : NLogN.RS), ∀ i ∈ touch p,
    i ∈ touch (RS.sub σ p) ∨ (σ i).isSome

Context. KFW: literals read from registers (the uniformity meta-theorem for register programs)

A uniform register program U reads its run-time parameters from registers (tops of stacks). RS.sub σ U replaces each register i with σ i = some v by the literal v; the result is the (non-uniform) program with baked-in literals, for which a spec is usually already proved.

FSp.of_sub: if FSp (U.sub σ) s t c, the registers of σ hold their values in s (Agree σ s) and U never writes them, then FSp U s t (400 c). Also fok_of_sub.

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.KFW.RS NLogN.RS NLogN.KFD

Formal statement

theorem NLogN.KFW.RS.touch_sub (σ : Fin HS → Option ℤ) : ∀ (p : NLogN.RS), ∀ i ∈ touch p,
    i ∈ touch (RS.sub σ p) ∨ (σ i).isSome := 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.RS.touch_sub`, Solutions/KFWSub.lean (lines 339–384) of the round-six development.

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