A child step (round six, `stepH_child`)

Proved

IntMul.Kappa.R6.IL3H.stepH_child

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

complexity-theoryinteger-multiplicationkappa-improvement

A child step (normalized: slots < s, forward) from an implementation of the child level.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Child):

theorem IntMul.Kappa.R6.IL3H.stepH_child (hN : NodeRH c σ K t S0 r H E e B) (hQ : ∀ s, Q s → Agree σ s) (hhi : hi ≤ H)
    (hA : 1 ≤ c.Aw) (hp : 1 ≤ c.p) (hm : 1 ≤ c.m) (hW : 1 ≤ c.Wr)
    (hr : 1 ≤ r) (hS : 1 ≤ S0) (hBn : 1 ≤ B / c.W) (Z : ℕ)
    (hZ : zsz ((2 * (c.Wr * (nN (B / c.W) t S0 * r)) : ℕ) : ℤ) ≤ Z) (M Bz : ℕ)
    (hM : c.Aw * (E + 1) ^ c.p + (E + 1) ^ c.p + E + 2 * c.m + B + 1 ≤ M)
    (hMB : zsz (M : ℤ) ≤ Bz) (s : ℕ) (hs : s < c.m)
    {jL : Fin Pp} {rowC : IStmt HS Pp} (hj : tbl jL = levCH c jL rowC)
    (w : Fin c.Wr) (DWc : ℕ) (cIH : ℕ → ℕ)
    (IH : SImplQ Isym tbl (fun s' => Agree (σChildH σ c s (E / c.m) (B / c.W)) s') 0 H r
      (fun i : Fin (s * (E / c.m)) => 2 ^ (e - K * i.val) * S0) (nN (B / c.W) t S0)
      (tens Cm) (levCH c jL rowC) (c.G (s * (E / c.m))) DWc cIH) :
    RImplQ Isym tbl Q lo hi r (nN (B / c.W) t S0) (stH c.m K e (E / c.m) S0)
      ((HStep.child w (pref c.m s) ∅ : HStep (Fin c.Wr) c.m).den (chEx (κ := Fin (E / c.m))))
      (withPushI (childFrH c s) (wrapQ c.Wr w.val wL wP (.call jL)))
      (c.G (s * (E / c.m))) (DWc + (c.G (s * (E / c.m)) + 2) * c.Wr)
      (fun x => cChF c Bz (c.G (s * (E / c.m))) + cWrapQ c.Wr (nN (B / c.W) t S0 * r) Z
        (c.G (s * (E / c.m))) DWc (fun x => cIH x + 2) x +
        (4 * (2 * (Bz + c.G (s * (E / c.m)) + 1) + 2) + 1))

Context. Round six, item language (I3): the child step of a node

child w (pref m s) ∅ (normalized: the slots < s, forward) on role w: the frame childFrH (wP = 2^{Gs (sF)}, vX = Gs (sF), vB = B/W, vk = s F) around wrapQ … (call jL). The slots < s of role w are the top s F bits e - K i, i < s F, so the call runs the level procedure on s F bits at the same e.

rImplQ_push: a frame around a role-stream statement (copy of sImplQ_push);
gsNE_OK, powNE_OK: side conditions of the scale expressions;
levRH_child, stepH_child.

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} {Q : IState HS → Prop}
  {lo hi : ℕ}
variable {c : IntMul.Kappa.R6.IL3H.Cfg} {σ : Fin HS → Option ℤ} {K t S0 r H E e B : ℕ}

Formal statement

theorem IntMul.Kappa.R6.IL3H.stepH_child (hN : NodeRH c σ K t S0 r H E e B) (hQ : ∀ s, Q s → Agree σ s) (hhi : hi ≤ H)
    (hA : 1 ≤ c.Aw) (hp : 1 ≤ c.p) (hm : 1 ≤ c.m) (hW : 1 ≤ c.Wr)
    (hr : 1 ≤ r) (hS : 1 ≤ S0) (hBn : 1 ≤ B / c.W) (Z : ℕ)
    (hZ : zsz ((2 * (c.Wr * (nN (B / c.W) t S0 * r)) : ℕ) : ℤ) ≤ Z) (M Bz : ℕ)
    (hM : c.Aw * (E + 1) ^ c.p + (E + 1) ^ c.p + E + 2 * c.m + B + 1 ≤ M)
    (hMB : zsz (M : ℤ) ≤ Bz) (s : ℕ) (hs : s < c.m)
    {jL : Fin Pp} {rowC : IStmt HS Pp} (hj : tbl jL = levCH c jL rowC)
    (w : Fin c.Wr) (DWc : ℕ) (cIH : ℕ → ℕ)
    (IH : SImplQ Isym tbl (fun s' => Agree (σChildH σ c s (E / c.m) (B / c.W)) s') 0 H r
      (fun i : Fin (s * (E / c.m)) => 2 ^ (e - K * i.val) * S0) (nN (B / c.W) t S0)
      (tens Cm) (levCH c jL rowC) (c.G (s * (E / c.m))) DWc cIH) :
    RImplQ Isym tbl Q lo hi r (nN (B / c.W) t S0) (stH c.m K e (E / c.m) S0)
      ((HStep.child w (pref c.m s) ∅ : HStep (Fin c.Wr) c.m).den (chEx (κ := Fin (E / c.m))))
      (withPushI (childFrH c s) (wrapQ c.Wr w.val wL wP (.call jL)))
      (c.G (s * (E / c.m))) (DWc + (c.G (s * (E / c.m)) + 2) * c.Wr)
      (fun x => cChF c Bz (c.G (s * (E / c.m))) + cWrapQ c.Wr (nN (B / c.W) t S0 * r) Z
        (c.G (s * (E / c.m))) DWc (fun x => cIH x + 2) x +
        (4 * (2 * (Bz + c.G (s * (E / c.m)) + 1) + 2) + 1)) := 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.stepH_child`, Solutions/R6IL3Child.lean (lines 338–406) of the round-six development.

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