Carries out

Proved

NLogN.KFM.carIU_spec

by xuanji · Oct 14, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Carries out.

The following holds (Lean statement, in the scope of module Solutions.KFMU3):

theorem NLogN.KFM.carIU_spec (b B : ℕ) (hb : 1 ≤ b) (zs : List GI) (hre : ∀ z ∈ zs, 0 ≤ z.1)
    (hB : NLogN.KFB.GB B zs) (s : IState HS) (R0 L1 L2 : List Item) (h0 : s.G 0 = ints (gflat zs) ++ R0)
    (hB2 : s.G uB2 = .int ((2 : ℤ) ^ b) :: L1) (hM : s.G uM = .int (zs.length : ℤ) :: L2) :
    IRuns I iprog carIU s (s.upd 0 (.int (carOut b zs : ℤ) :: R0)) (cCarU b B zs.length)

Context. The lemma belongs to the part KFM U3: uniform digits in and carries out.

Formalization Note. This lemma belongs to the item-language (IL) layer of the integer-multiplication-kappa campaign: tape-level programs (register programs over 64 item stacks of the NLogN multitape framework, definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8) that implement the data-movement primitives of the OpenAI preprint Integer multiplication below $n\log n$ (field moves, row split/merge, shifts, the uniform Proposition 11 interchange, Lemma 12 with fine $O(1)$ -per-element costs, Gaussian record primitives, Proposition 14 pieces) with exact list denotations from the array-model bundles IntMul_Kappa_K_Defs* and explicit cost bounds. All names refer to the definition bundles up to IntMul_Kappa_K_Defs22.

Preamble

-- OpenAI-preprint / κ-improvement development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_K_Defs22

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option linter.unusedSimpArgs false
set_option linter.unusedVariables false
set_option linter.unusedSectionVars false
set_option linter.unnecessarySeqFocus false
set_option maxRecDepth 20000
open NLogN
open NLogN.KFM
open IntMul KFC KFB KFT
variable {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}
set_option maxHeartbeats 1000000

Formal statement

theorem NLogN.KFM.carIU_spec (b B : ℕ) (hb : 1 ≤ b) (zs : List GI) (hre : ∀ z ∈ zs, 0 ≤ z.1)
    (hB : NLogN.KFB.GB B zs) (s : IState HS) (R0 L1 L2 : List Item) (h0 : s.G 0 = ints (gflat zs) ++ R0)
    (hB2 : s.G uB2 = .int ((2 : ℤ) ^ b) :: L1) (hM : s.G uM = .int (zs.length : ℤ) :: L2) :
    IRuns I iprog carIU s (s.upd 0 (.int (carOut b zs : ℤ) :: R0)) (cCarU b B zs.length) := by sorry

Source

OpenAI, Integer multiplication below n log n (preprint, 23 Sep 2026), https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026. Lemma `NLogN.KFM.carIU_spec` of the Lean development Solutions/KFMU3.lean (lines 121–211), integer-multiplication-kappa campaign.

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