Compilation is correct (round six, `good`)

Proved

NLogN.KIL.NE.good

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

complexity-theoryinteger-multiplicationkappa-improvement

Compilation is correct.

The following holds (Lean statement, in the scope of module Solutions.KIL6NE):

theorem NLogN.KIL.NE.good {t : IState HS} {v : Fin HS → ℕ} {M B : ℕ} (hM : zsz (M : ℤ) ≤ B) :
    ∀ e : NE, Holds t v e → e.OK v M →
      e.toEx.ok t ∧ e.toEx.Bnd t B ∧ e.toEx.eval t = (e.val v : ℤ)

Context. KIL6: natural-number expressions

NE is a small language of natural-number expressions over registers (+, - (truncation excluded by hypothesis), *, /, %, if a < b then x else y), compiled to Ex HS. NE.good: on a state whose registers hold the naturals v i, if every intermediate value is ≤ M and zsz M ≤ B, then the compiled expression is clean, well defined, B-bounded and evaluates to the natural value. This turns every descriptor computation into arithmetic.

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 NLogN.KIL.NE
open IntMul NLogN.RS AModel AModelL12

Formal statement

theorem NLogN.KIL.NE.good {t : IState HS} {v : Fin HS → ℕ} {M B : ℕ} (hM : zsz (M : ℤ) ≤ B) :
    ∀ e : NE, Holds t v e → e.OK v M →
      e.toEx.ok t ∧ e.toEx.Bnd t B ∧ e.toEx.eval t = (e.val v : ℤ) := 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.NE.good`, Solutions/KIL6NE.lean (lines 115–191) of the round-six development.

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