A phase step (round six, `cost_phaseH`)

Proved

IntMul.Kappa.R6.IL3H.cost_phaseH

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

complexity-theoryinteger-multiplicationkappa-improvement

A phase step (body and wrapper).

The following holds (Lean statement, in the scope of module Solutions.R6IL3Cost):

theorem IntMul.Kappa.R6.IL3H.cost_phaseH {W m Bn t S0 r Z H x : ℕ} (hW : 1 ≤ W) (hS : 1 ≤ S0) (hBn : 1 ≤ Bn) (v : ℤ)
    (hv : zsz v ≤ Z) (εE : Ex HS) (hsz : εE.size ≤ 12 * m + 27) (hx : x + 2 * W ≤ H) :
    cWrapQ W (nN Bn t S0 * r) Z 0 0 (cPhStep Z v εE Bn t S0 r) x ≤
      MphH W m * ((W * nN Bn t S0) * Yu r Z H)

Context. Round six, item language (I3): cost bounds of the steps of a node

KFWCost2 with the literal W_c, A0, mC replaced by variables: with P = (W n) · Yu r Z H (n records per role, W roles, at most A atoms per gate, m slots):

cost_gateH, cost_permH, cost_phaseH, cost_fixGH, cost_rowH: every local step costs ≤ MlocH W A m · P (plus the row body);
cost_childH: the child wrapper adds ≤ MlocH W A m · P to the (constant) child cost, for gsSz c + 6 ≤ A.

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

Formal statement

theorem IntMul.Kappa.R6.IL3H.cost_phaseH {W m Bn t S0 r Z H x : ℕ} (hW : 1 ≤ W) (hS : 1 ≤ S0) (hBn : 1 ≤ Bn) (v : ℤ)
    (hv : zsz v ≤ Z) (εE : Ex HS) (hsz : εE.size ≤ 12 * m + 27) (hx : x + 2 * W ≤ H) :
    cWrapQ W (nN Bn t S0 * r) Z 0 0 (cPhStep Z v εE Bn t S0 r) x ≤
      MphH W m * ((W * nN Bn t S0) * Yu r Z H) := 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.cost_phaseH`, Solutions/R6IL3Cost.lean (lines 131–179) of the round-six development.

View graph