Round-six lemma `pushSeq_child`

Proved

IntMul.Kappa.R6.IL3H.pushSeq_child

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

complexity-theoryinteger-multiplicationkappa-improvement

An auxiliary lemma of the round-six development, published as a separate node because its proof is long or shared.

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

theorem IntMul.Kappa.R6.IL3H.pushSeq_child (hN : NodeRH c σ K t S0 r H E e B) (hA : 1 ≤ c.Aw) (hp : 1 ≤ c.p)
    (hm : 1 ≤ c.m) (s : ℕ) (hs : s < c.m) (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) :
    PushSeq σ (childFrH c s) (childVLH c s (E / c.m) (B / c.W))
      (cChF c Bz (c.G (s * (E / c.m))))

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 {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.pushSeq_child (hN : NodeRH c σ K t S0 r H E e B) (hA : 1 ≤ c.Aw) (hp : 1 ≤ c.p)
    (hm : 1 ≤ c.m) (s : ℕ) (hs : s < c.m) (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) :
    PushSeq σ (childFrH c s) (childVLH c s (E / c.m) (B / c.W))
      (cChF c Bz (c.G (s * (E / c.m)))) := 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.pushSeq_child`, Solutions/R6IL3Child.lean (lines 230–308) of the round-six development.

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