A generic loop (round six, `loopQS`)

Proved

IntMul.Kappa.R6.IL3H.loopQS

by xuanji · Oct 20, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

A generic loop (test of size ≤ sz).

The following holds (Lean statement, in the scope of module Solutions.R6IL3Top1):

theorem IntMul.Kappa.R6.IL3H.loopQS {sz : ℕ} {j : Fin Pp} {cond : NE} {step : List (Fin HS × NE)} {cont : IStmt HS Pp}
    (hj : tbl j = testC 16 cond (withPushI (fr step) (.call j)) cont)
    (Inv : ℕ → (Fin HS → Option ℤ) → Prop) {Mb Bz : ℕ} (hMb : zsz (Mb : ℤ) ≤ Bz)
    (hcR : cond.RegsLt) (hcS : cond.size ≤ sz) (hsN : NoRdN [] step) (hs0 : ∀ x ∈ step, x.1 ≠ 0)
    {f g : Fin HS → List ℤ} (hf : Clean0 f) (hg : Clean0 g) {cc : ℕ}
    (hbase : ∀ σ, Inv 0 σ → σ 0 = none ∧ NatR σ cond ∧ cond.OK (vOf σ) Mb ∧
      cond.val (vOf σ) ≠ 0 ∧
      RunQ Isym tbl (σPush σ [((16 : Fin HS), ((cond.val (vOf σ) : ℕ) : ℤ))]) cont f g cc)
    (hstep : ∀ n σ, Inv (n + 1) σ → σ 0 = none ∧ NatR σ cond ∧ cond.OK (vOf σ) Mb ∧
      cond.val (vOf σ) = 0 ∧ FrOK (σPush σ [((16 : Fin HS), ((0 : ℕ) : ℤ))]) Mb 12 step ∧
      Inv n (σPush (σPush σ [((16 : Fin HS), ((0 : ℕ) : ℤ))])
        (frV (σPush σ [((16 : Fin HS), ((0 : ℕ) : ℤ))]) step))) :
    ∀ n σ, Inv n σ → RunQ Isym tbl σ (.call j) f g (cc + (n + 1) * cLoopS sz step.length Bz)

Context. Round six, item language (I3): the entry loops of the uniform Proposition 14

loopQ: a generic continuation-passing loop (testC 16 cond (push step; call j) cont) from an invariant indexed by the number of remaining iterations.
gloopC cz: with Z on vG = 24, finds G = clog₂ (4 (Z + cz) (Z + cz + 2)) on 11 (gloopQ; the compact-controls field width).
dloopC c: with D on 22, finds d = dep m D on 12 and W^d on 13 (dloopQ).

Formalization Note. Part of the Lean formalization of the round-six κ-improvement (integer-multiplication-kappa campaign, target IntMul.Kappa.jain_round6_framework). The statement is copied verbatim from the Lean development; the platform node carries the same declaration name. Definitions it mentions live in the IntMul_Kappa_R6_Defs* bundles (on top of the published κ bundles IntMul_Kappa_K_Defs*).

Preamble

-- Round-six κ development (integer-multiplication-kappa), generated by skeleton subtraction.

import Definitions.Def_IntMul_Kappa_R6_Defs22

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL3H
open IntMul NLogN NLogN.KIL NLogN.KFD NLogN.KFB NLogN.KFG NLogN.KFH NLogN.KFN NLogN.KFW Complex P8Lift Butterfly IntMul.Kappa.R6.Cx
variable {Pp : ℕ} {Isym : List Sym} {tbl : Fin Pp → IStmt HS Pp}

Formal statement

theorem IntMul.Kappa.R6.IL3H.loopQS {sz : ℕ} {j : Fin Pp} {cond : NE} {step : List (Fin HS × NE)} {cont : IStmt HS Pp}
    (hj : tbl j = testC 16 cond (withPushI (fr step) (.call j)) cont)
    (Inv : ℕ → (Fin HS → Option ℤ) → Prop) {Mb Bz : ℕ} (hMb : zsz (Mb : ℤ) ≤ Bz)
    (hcR : cond.RegsLt) (hcS : cond.size ≤ sz) (hsN : NoRdN [] step) (hs0 : ∀ x ∈ step, x.1 ≠ 0)
    {f g : Fin HS → List ℤ} (hf : Clean0 f) (hg : Clean0 g) {cc : ℕ}
    (hbase : ∀ σ, Inv 0 σ → σ 0 = none ∧ NatR σ cond ∧ cond.OK (vOf σ) Mb ∧
      cond.val (vOf σ) ≠ 0 ∧
      RunQ Isym tbl (σPush σ [((16 : Fin HS), ((cond.val (vOf σ) : ℕ) : ℤ))]) cont f g cc)
    (hstep : ∀ n σ, Inv (n + 1) σ → σ 0 = none ∧ NatR σ cond ∧ cond.OK (vOf σ) Mb ∧
      cond.val (vOf σ) = 0 ∧ FrOK (σPush σ [((16 : Fin HS), ((0 : ℕ) : ℤ))]) Mb 12 step ∧
      Inv n (σPush (σPush σ [((16 : Fin HS), ((0 : ℕ) : ℤ))])
        (frV (σPush σ [((16 : Fin HS), ((0 : ℕ) : ℤ))]) step))) :
    ∀ n σ, Inv n σ → RunQ Isym tbl σ (.call j) f g (cc + (n + 1) * cLoopS sz step.length Bz) := by sorry

Source

S. Jain, round-six networks for the integer-multiplication κ-improvement campaign (https://github.com/Swapnil-jain/integer-mult-kappa, commit 5868dc7) and D. Colkitt, integer-mult-bounds (https://github.com/CrocSwap/integer-mult-bounds); target `IntMul.Kappa.jain_round6_framework`. Lean declaration `IntMul.Kappa.R6.IL3H.loopQS`, Solutions/R6IL3Top1.lean (lines 40–85) of the round-six development.

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