Generalized Prop. 11 (round six, bit networks) (round six, `prop11H`)

Proved

IntMul.Kappa.R6.Bit.prop11H

by xuanji · Oct 19, 2026 · Mathlib 0df444a (Lean v4.33.1)

complexity-theoryinteger-multiplicationkappa-improvement

Generalized Prop. 11 (round six, bit networks).

The following holds (Lean statement, in the scope of module Solutions.R6BitProp11):

theorem IntMul.Kappa.R6.Bit.prop11H (C : QHNet R W T m) {q : ℕ} (hq : 1 ≤ q) (hm : 2 ≤ m)
    (hOK : ∀ f, C.OK (q ^ f)) {τ : ℝ} (hτ0 : 0 < τ) (hτ1 : τ ≤ 1)
    (hmom : histMoment C.hist m τ < W) :
    (∀ (e P L : ℕ) (A B Cs : Shape) (xs : List R), W ^ depH m e ∣ L →
      xs.length = vol (P :: L :: hdShape A B Cs e q) →
      progH C q e P L A B Cs xs =
        pull (P :: L :: hdShape A B Cs e q) (P :: L :: hdShape A B Cs e q)
          (blockSwap (A.length + 2) (A.length + B.length + e + 2) e) 0 xs) ∧
    ∃ K : ℝ, ∀ (e : ℕ) (V : ℝ), 0 ≤ V → costH C e V ≤ K * V * (e : ℝ) ^ τ

Context. Round six, bit side: the generalized Proposition 11 (prop11H)

For a round-six bit network C : QHNet R W T m (general Bruhat frame changes, run children of arbitrary widths 0 < r < m, gates, zero temporaries) satisfying C.OK (q ^ f) at every field size q ^ f, with child-width histogram C.hist of moment Σ n_w (w/m)^τ < W:

denotation: for every width e (any e = m f + r, leftover digits swapped singly), the program progH C q e interchanges the two runs of e radix-q digits of P :: L :: hdShape A B Cs e q whenever W ^ depH m e ∣ L;
cost: costH C e V ≤ K V e^τ for all e and V ≥ 0.

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_Defs4

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.Kappa IntMul.Kappa.R6 IntMul.Kappa.R6.Bit
open AModel
variable {R : Type*} [CommRing R] {W T m : ℕ} [NeZero W]

Formal statement

theorem IntMul.Kappa.R6.Bit.prop11H (C : QHNet R W T m) {q : ℕ} (hq : 1 ≤ q) (hm : 2 ≤ m)
    (hOK : ∀ f, C.OK (q ^ f)) {τ : ℝ} (hτ0 : 0 < τ) (hτ1 : τ ≤ 1)
    (hmom : histMoment C.hist m τ < W) :
    (∀ (e P L : ℕ) (A B Cs : Shape) (xs : List R), W ^ depH m e ∣ L →
      xs.length = vol (P :: L :: hdShape A B Cs e q) →
      progH C q e P L A B Cs xs =
        pull (P :: L :: hdShape A B Cs e q) (P :: L :: hdShape A B Cs e q)
          (blockSwap (A.length + 2) (A.length + B.length + e + 2) e) 0 xs) ∧
    ∃ K : ℝ, ∀ (e : ℕ) (V : ℝ), 0 ≤ V → costH C e V ≤ K * V * (e : ℝ) ^ τ := 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.Bit.prop11H`, Solutions/R6BitProp11.lean (lines 23–33) of the round-six development.

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