The simulation theorem (round six, `sim`)

Proved

NLogN.KIL.RSim.sim

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

complexity-theoryinteger-multiplicationkappa-improvement

The simulation theorem.

The following holds (Lean statement, in the scope of module Solutions.KIL4Sim):

theorem NLogN.KIL.RSim.sim {π : PMap} (hπ : PLow π) {p p' : RS} (h : RSim π p p') :
    (∀ i v, π i = some v → i ∉ p.writes) → ∀ s, PV π s → p.ok s →
      p'.ok s ∧ p'.run s = p.run s ∧ p'.cost s ≤ 200 * p.cost s

Context. KIL4: literal-to-register simulation of register programs

The KIL1–KIL3 programs carry the sizes of a configuration (field sizes, volumes, the modulus, pass counts, …) as literals Ex.lit v. A uniform program reads them from parameter stacks instead. RSim π p p' says that p' is p with some literal occurrences .lit v replaced by register reads .reg i, where the parameter map π says π i = some v.

RSim.sim: on a state where every parameter stack i holds v on top (PV π s) and is not written by p, the program p' is executable whenever p is, ends in the same state, and costs at most 200 × the cost of p (a register read costs at most 200 × a literal of the same value). Hence every KIL specification Sp p s t c transfers to Sp p' s t (200 * c) (Sp.sim).

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.KIL
open IntMul NLogN.RS

Formal statement

theorem NLogN.KIL.RSim.sim {π : PMap} (hπ : PLow π) {p p' : RS} (h : RSim π p p') :
    (∀ i v, π i = some v → i ∉ p.writes) → ∀ s, PV π s → p.ok s →
      p'.ok s ∧ p'.run s = p.run s ∧ p'.cost s ≤ 200 * p.cost s := 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.KIL.RSim.sim`, Solutions/KIL4Sim.lean (lines 249–356) of the round-six development.

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