HvdH machine lemma ST.tailP_spec (The recursive step, part 5: the tail (`crtOut`, `evalDigits`, result) )

Proved

NLogN.ST.tailP_spec

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.HSt5):

theorem NLogN.ST.tailP_spec (M : MulSpec) (q : Params) (sl : List ℕ) (h : CR.CrtOK sl) (t : IState HS)
    (w : List ℤ) (S Mo kb b : ℕ) (R28 R0 L1 L2 R3 LT L33 L31 : List Item)
    (hS : sl.prod = S) (hw : w.length = S) (hM : Mo ≤ S) (hkb : S ≤ 2 ^ kb)
    (h28 : t.G 28 = gints (chirpL q) ++ R28) (h9 : t.G 9 = .int (q.tdims.prod : ℤ) :: LT)
    (h0 : t.G 0 = ints w ++ R0) (h1 : t.G 1 = .int S :: L1) (h2 : t.G 2 = .int kb :: L2)
    (h3 : t.G 3 = natInts (CR.keyL sl) ++ R3) (h33 : t.G 33 = .int Mo :: L33)
    (h31 : t.G 31 = .int (2 ^ b) :: L31) (B : ℕ)
    (hc : ∀ x ∈ crtOut sl Mo w, 0 ≤ x ∧ x < 2 ^ B) (hb1 : 1 ≤ b) (hbB : b + 1 ≤ B)
    (hpB : q.p + 1 ≤ B) (hTB : 2 * q.tdims.prod < 2 ^ B) (hSB : S < 2 ^ B) (hwB : Bnd B w)
    (hkbB : kb < 2 ^ B) (BE : ℕ) (hEB : zsz (evalDigits b (crtOut sl Mo w)) ≤ BE) :
    Spec M tailP t (((((((t.upd 28 R28).upd 0 R0).upd 3 R3).upd 1 L1).upd 2 L2)).upd Top.XR
      (.int (evalDigits b (crtOut sl Mo w)) :: t.G Top.XR))
      (tailCost q.tdims.prod S Mo kb b B + (4000 * (BE + 2) + (2 * BE + 2)))

Context. The lemma belongs to the part The recursive step, part 5: the tail (crtOut, evalDigits, result) and the final pops.

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
set_option linter.unusedVariables false
set_option linter.unreachableTactic false
set_option linter.unusedTactic false
open NLogN
open NLogN.ST
open IntMul NLogN IntMul.HvdH IntMul.HvdH.Ref
open NLogN.FW (Spec)
set_option maxHeartbeats 8000000

Formal statement

theorem NLogN.ST.tailP_spec (M : MulSpec) (q : Params) (sl : List ℕ) (h : CR.CrtOK sl) (t : IState HS)
    (w : List ℤ) (S Mo kb b : ℕ) (R28 R0 L1 L2 R3 LT L33 L31 : List Item)
    (hS : sl.prod = S) (hw : w.length = S) (hM : Mo ≤ S) (hkb : S ≤ 2 ^ kb)
    (h28 : t.G 28 = gints (chirpL q) ++ R28) (h9 : t.G 9 = .int (q.tdims.prod : ℤ) :: LT)
    (h0 : t.G 0 = ints w ++ R0) (h1 : t.G 1 = .int S :: L1) (h2 : t.G 2 = .int kb :: L2)
    (h3 : t.G 3 = natInts (CR.keyL sl) ++ R3) (h33 : t.G 33 = .int Mo :: L33)
    (h31 : t.G 31 = .int (2 ^ b) :: L31) (B : ℕ)
    (hc : ∀ x ∈ crtOut sl Mo w, 0 ≤ x ∧ x < 2 ^ B) (hb1 : 1 ≤ b) (hbB : b + 1 ≤ B)
    (hpB : q.p + 1 ≤ B) (hTB : 2 * q.tdims.prod < 2 ^ B) (hSB : S < 2 ^ B) (hwB : Bnd B w)
    (hkbB : kb < 2 ^ B) (BE : ℕ) (hEB : zsz (evalDigits b (crtOut sl Mo w)) ≤ BE) :
    Spec M tailP t (((((((t.upd 28 R28).upd 0 R0).upd 3 R3).upd 1 L1).upd 2 L2)).upd Top.XR
      (.int (evalDigits b (crtOut sl Mo w)) :: t.G Top.XR))
      (tailCost q.tdims.prod S Mo kb b B + (4000 * (BE + 2) + (2 * BE + 2))) := 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.ST.tailP_spec` of the Lean development NLogN/HSt5.lean (lines 31–158), written for the prove2.me integer-multiplication campaign.

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