HvdH machine lemma Rsp.gresP_E_run (`Japprox` on a slice: the `E` pass (`gresP` against the `E` tables) it)

Proved

NLogN.Rsp.gresP_E_run

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.HRspJ):

theorem NLogN.Rsp.gresP_E_run (p s t α : ℕ) (σ : IState HS) (U : List (ℤ × ℤ)) (RU : List Item)
    (hs : 0 < s) (hUl : U.length = s) (hE : EParams σ p s t α) (hU : σ.G CU = gints U ++ RU) :
    gresP.run σ = σ.upd CU (gints (eapL p s t α U) ++ RU)

Context. The lemma belongs to the part Japprox on a slice: the E pass (gresP against the E tables) iterated with signed.

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.gresP_E_run (p s t α : ℕ) (σ : IState HS) (U : List (ℤ × ℤ)) (RU : List Item)
    (hs : 0 < s) (hUl : U.length = s) (hE : EParams σ p s t α) (hU : σ.G CU = gints U ++ RU) :
    gresP.run σ = σ.upd CU (gints (eapL p s t α U) ++ RU) := 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.gresP_E_run` of the Lean development NLogN/HRspJ.lean (lines 44–53), written for the prove2.me integer-multiplication campaign.

View graph