HvdH machine lemma Rsp.dStage_okc (The `bapL` slice program)

Proved

NLogN.Rsp.dStage_okc

by xuanji · Oct 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

harvey-van-der-hoeven-tminteger-multiplicationturing-machines

The following holds (Lean statement, in the scope of module NLogN.HRspBap):

theorem NLogN.Rsp.dStage_okc (M : MulSpec) (mm Φ B : ℕ) (hM : MulBnd M mm Φ) (s t kb p α : ℕ) (hs : 0 < s)
    (L3 : List (ℤ × ℤ)) (hL3 : L3.length = s) (τ : IState HS) (RU : List Item)
    (hU : τ.G CU = gints L3 ++ RU) (hP : BParams τ p s t α kb)
    (hu : ∀ z ∈ L3, zsz z.1 ≤ mm ∧ zsz z.2 ≤ mm) (hWD : ∀ W ∈ dTab p s t α, ∀ y ∈ W, zsz y ≤ mm)
    (hAcc : 2 * 2 ^ (2 * mm) ≤ 2 ^ B) (hmB : 2 * mm ≤ B) (hS : zsz ((s + 1 : ℕ) : ℤ) ≤ B)
    (hpB : p + 1 ≤ B) :
    ((stageP Ad).ren σd).ok M τ ∧
      ((stageP Ad).ren σd).cost M τ ≤ stCost s 1 s mm Φ B 1 ∧
      ∀ z ∈ dapL p s t α L3, zsz z.1 ≤ B ∧ zsz z.2 ≤ B

Context. The lemma belongs to the part The bapL slice program: bapP = P̃_s⁻¹ ∘ D̃' ∘ J̃' ∘ C̃ ∘ P̃_t on the slice stack CU:

P̃_t, P̃_s⁻¹: permP 2 relocated (stack 0 ↔ CU, key stack 3 ↔ KS);
C̃: a gresP stage with one all-ones weight row (tables relocated BW ↔ W1, BD ↔ D1);
J̃': JP (the E parameters / tables are the native ones);
D̃': a gresP stage with the diagonal table (relocated BW ↔ W3).

Formalization Note. This is a lemma of the Lean development NLogN, a multitape-Turing-machine implementation of the Harvey–van der Hoeven $O(n\log n)$ integer multiplication algorithm, used in the proof of NLogN.FIN.theorem_1_1_nlogn : IntMul.MulTimeBound fun n => n * Real.log n. All names refer to the definition bundles IntMul_HvdH_M_Defs1–IntMul_HvdH_M_Defs8 (namespace NLogN), built on the platform model Def_IntMul_MultitapeModel and the published reference Def_IntMul_HvdH_MachineRef. The machine layer: IState HS is a state of 64 item stacks; RM programs have a semantics p.run s, a definedness predicate p.ok M s and a cost p.cost M s relative to a multiplication spec M : MulSpec; zsz z is the bit length of $|z|$ .

Preamble

-- Harvey–van der Hoeven multitape-TM development (NLogN), generated by skeleton subtraction.

import Definitions.Def_IntMul_HvdH_M_Defs8

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

set_option linter.unusedSimpArgs false
open NLogN
open NLogN.Rsp
open IntMul NLogN RS

Formal statement

theorem NLogN.Rsp.dStage_okc (M : MulSpec) (mm Φ B : ℕ) (hM : MulBnd M mm Φ) (s t kb p α : ℕ) (hs : 0 < s)
    (L3 : List (ℤ × ℤ)) (hL3 : L3.length = s) (τ : IState HS) (RU : List Item)
    (hU : τ.G CU = gints L3 ++ RU) (hP : BParams τ p s t α kb)
    (hu : ∀ z ∈ L3, zsz z.1 ≤ mm ∧ zsz z.2 ≤ mm) (hWD : ∀ W ∈ dTab p s t α, ∀ y ∈ W, zsz y ≤ mm)
    (hAcc : 2 * 2 ^ (2 * mm) ≤ 2 ^ B) (hmB : 2 * mm ≤ B) (hS : zsz ((s + 1 : ℕ) : ℤ) ≤ B)
    (hpB : p + 1 ≤ B) :
    ((stageP Ad).ren σd).ok M τ ∧
      ((stageP Ad).ren σd).cost M τ ≤ stCost s 1 s mm Φ B 1 ∧
      ∀ z ∈ dapL p s t α L3, zsz z.1 ≤ B ∧ zsz z.2 ≤ B := by sorry

Source

D. Harvey, J. van der Hoeven, Integer multiplication in time O(n log n), Ann. of Math. 193 (2021), 563–617 (Theorem 1.1: multitape Turing machine multiplication in O(n log n)). Lemma `NLogN.Rsp.dStage_okc` of the Lean development NLogN/HRspBap.lean (lines 400–423), written for the prove2.me integer-multiplication campaign.

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