Stack-machine split of an integer item by a ruler 2^m (quotient and remainder)

Proved

NLogN.Runs.splitR

by xuanji · Oct 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

integer-multiplicationschonhage-strassen-tmturing-machines

Let $q,i,j,k,t_1,t_2$ be pairwise distinct stacks of a state $\sigma$ . Assume stack $i$ is $\mathrm{enc}(z)$ followed by $R_i$ and stack $j$ is $\mathrm{enc}(2^m)$ followed by $R_j$ , where $R_i$ and the tops of $k,t_1,t_2$ do not start with a bit symbol. Then splitR q i j k t1 t2 runs from $\sigma$ to the state where

i := \mathrm{enc}\big(\operatorname{sg}(z,\lfloor |z|/2^m\rfloor)\big)\mathbin{+\!\!+}R_i,\qquad k \mathrel{+}= \mathrm{enc}\big(\operatorname{sg}(z,\ |z| \bmod 2^m)\big),

with cost at most $\mathrm{splitCost}(m)+17 = O(m)$ . Here $\operatorname{sg}(z,n)$ is $n$ with the sign of $z$ , so this is the signed (truncating) division of $z$ by $2^m$ ; the ruler on $j$ is kept.

Splitting is used to cut inputs into $M$ -bit pieces, to reduce modulo $2^n+1$ (via $2^n\equiv -1$ ), and to extract carry digits; again the cost depends only on $m$ .

Formalization Note. Machine states are St S records: stacks stk : Fin S → List Sym over the 16-letter alphabet Sym = Fin 16 (head = top), input position ip and output out; σ.set i L replaces stack i. An integer item is stored as (Item.int z).enc: a sign header followed by the minimal binary magnitude, least significant bit nearest the header; bitOf detects bit symbols, and the hypotheses bitOf (…).head? = none say that the following stack content does not start with a bit (so item boundaries are recognisable). Runs I prog p σ σ' c means p runs from σ to exactly σ' with some cost $c'\le c$ ; the procedures prog and input I are arbitrary.

Preamble

-- Schönhage–Strassen multitape-TM development (NLogN), generated by skeleton subtraction.

import Definitions.Def_IntMul_SS_Defs2

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN
variable {S P : ℕ} {I : List Sym} {prog : Fin P → Stmt S P}

Formal statement

theorem NLogN.Runs.splitR {q i j k t1 t2 : Fin S} (hqi : q ≠ i) (hqj : q ≠ j) (hqk : q ≠ k)
    (hqt1 : q ≠ t1) (hqt2 : q ≠ t2) (hij : i ≠ j) (hik : i ≠ k) (hit1 : i ≠ t1) (hit2 : i ≠ t2)
    (hjk : j ≠ k) (hjt1 : j ≠ t1) (hjt2 : j ≠ t2) (hkt1 : k ≠ t1) (hkt2 : k ≠ t2)
    (ht12 : t1 ≠ t2) (σ : St S) (z : ℤ) (m : ℕ) (Ri Rj : List Sym)
    (hi : σ.stk i = (Item.int z).enc ++ Ri) (hj : σ.stk j = (Item.int ((2 : ℤ) ^ m)).enc ++ Rj)
    (hRi : bitOf Ri.head? = none) (hk : bitOf (σ.stk k).head? = none)
    (ht1 : bitOf (σ.stk t1).head? = none) (ht2 : bitOf (σ.stk t2).head? = none) :
    NLogN.Runs I prog (NLogN.splitR q i j k t1 t2 : Stmt S P) σ
      ((σ.set i ((Item.int (sg z (z.natAbs / 2 ^ m))).enc ++ Ri)).set k
        ((Item.int (sg z (z.natAbs % 2 ^ m))).enc ++ σ.stk k)) (splitCost m + 17) := by sorry

Source

A. Schönhage, V. Strassen, Schnelle Multiplikation großer Zahlen, Computing 7 (1971), 281–292 (main theorem: O(n log n log log n) multitape TM multiplication). Lemma `NLogN.Runs.splitR` of the Lean development NLogN/Shift.lean (lines 230–323), written for the prove2.me integer-multiplication campaign.

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