The middle-part preconditions (round six, `midPre`)

Proved

IntMul.Kappa.R6.IL3H.Hyp14.midPre

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

complexity-theoryinteger-multiplicationkappa-improvement

The middle-part preconditions.

The following holds (Lean statement, in the scope of module Solutions.R6IL3Top14):

theorem IntMul.Kappa.R6.IL3H.Hyp14.midPre (hc : CfgOK c) {Pp : ℕ} {Isym : List Sym} {tbl : Fin Pp → IStmt HS Pp}
    {jL : Fin Pp} {rowC : IStmt HS Pp} {CB : ℕ → ℝ} {ρ : ℕ} (hρ : ρ < K)
    (LV : ∀ E, LvH c Isym tbl jL rowC K r (IntMul.Kappa.R6.IL3H.Zl c P K D S r w) (3 * Z) (w + EwH c D)
      (Gv czH (IntMul.Kappa.R6.IL3H.Zl c P K D S r w)) CB E)
    (hlt : (chV c (cWc c) P K D S r w).q0 + (chV c (cWc c) P K D S r w).qB < D) :
    MidPre c Isym tbl jL rowC CB K S r (chV c (cWc c) P K D S r w).H P D ρ
      (chV c (cWc c) P K D S r w).q0 (chV c (cWc c) P K D S r w).Bp
      (chV c (cWc c) P K D S r w).qB (chV c (cWc c) P K D S r w).Zl Z (3 * Z)
      (chV c (cWc c) P K D S r w).G

Context. Round six, item language (I3): the middle-part preconditions of the chain

Under Hyp14 and the level claims of lvAllH, Hyp14.midPre gives MidPre for the chain values chV c (cWc c) P K D S r w (with Zr = 3 Z), and Hyp14.hGm the small-value bound G + m ≤ N used by the core stage.

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_Defs23

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.IL3H IntMul.Kappa.R6.IL3H.Hyp14
open IntMul AModel 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} {P K D S r p w Z : ℕ} {lamp : ℝ} (h : Hyp14 c P K D S r p w Z lamp)
include h

Formal statement

theorem IntMul.Kappa.R6.IL3H.Hyp14.midPre (hc : CfgOK c) {Pp : ℕ} {Isym : List Sym} {tbl : Fin Pp → IStmt HS Pp}
    {jL : Fin Pp} {rowC : IStmt HS Pp} {CB : ℕ → ℝ} {ρ : ℕ} (hρ : ρ < K)
    (LV : ∀ E, LvH c Isym tbl jL rowC K r (IntMul.Kappa.R6.IL3H.Zl c P K D S r w) (3 * Z) (w + EwH c D)
      (Gv czH (IntMul.Kappa.R6.IL3H.Zl c P K D S r w)) CB E)
    (hlt : (chV c (cWc c) P K D S r w).q0 + (chV c (cWc c) P K D S r w).qB < D) :
    MidPre c Isym tbl jL rowC CB K S r (chV c (cWc c) P K D S r w).H P D ρ
      (chV c (cWc c) P K D S r w).q0 (chV c (cWc c) P K D S r w).Bp
      (chV c (cWc c) P K D S r w).qB (chV c (cWc c) P K D S r w).Zl Z (3 * Z)
      (chV c (cWc c) P K D S r w).G := 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.Hyp14.midPre`, Solutions/R6IL3Top14.lean (lines 38–90) of the round-six development.

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