Round-six lemma `hdr_sp`

Proved

J6Seg.hdr_sp

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.J6SegILSweep2):

theorem J6Seg.hdr_sp (s : IState HS) (F : Fin HS → List ℤ) (z : List ℤ) (p α h c : ℕ)
    (bs sv : List ℤ) (B : ℕ) (hB : 1 ≤ B) (h01 : ∀ b ∈ bs, b = 0 ∨ b = 1)
    (hc : bs.sum = (c : ℤ)) (hzp : zsz (p : ℤ) ≤ B) (hzα : zsz (α : ℤ) ≤ B)
    (hBe : h * bs.length + bs.length + 1 ≤ B) (hhB : h + 1 ≤ B)
    (hzT : zsz ((2 : ℤ) ^ (h * bs.length + c)) ≤ B) :
    Sp (hdrA σ dS)
      (encK s (SS σ dS F z ((p : ℤ) :: (α : ℤ) :: (h : ℤ) :: (bs.length : ℤ) :: (bs ++ sv))
        (fun _ : Fin 19 => ([] : List ℤ))))
      (encK s (SS σ dS F z sv (vH p α h bs.length c bs))) (cHdr B bs.length)

Context. B-line IL, B6 §2: reading the line descriptor

hdrA σ dS (on dS = descLine6 = p :: α :: h :: d :: (bits ++ svals)): pop p, α, h, d into σ 0..3, move the d bits onto σ 12 while counting the ones in σ 4, move them on to σ 13 (in order), then σ 5 := T = 2^{h d + #ones}, σ 6 := 1 (pre), σ 8 := 0 (k).

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_Defs67

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg
open IntMul NLogN NLogN.KIL NLogN.KFB IntMul.Kappa.R6 Function
variable {σ : Fin 19 → Fin HS} {dS dA : Fin HS} (hW : SwOK σ dS dA)
include hW

Formal statement

theorem J6Seg.hdr_sp (s : IState HS) (F : Fin HS → List ℤ) (z : List ℤ) (p α h c : ℕ)
    (bs sv : List ℤ) (B : ℕ) (hB : 1 ≤ B) (h01 : ∀ b ∈ bs, b = 0 ∨ b = 1)
    (hc : bs.sum = (c : ℤ)) (hzp : zsz (p : ℤ) ≤ B) (hzα : zsz (α : ℤ) ≤ B)
    (hBe : h * bs.length + bs.length + 1 ≤ B) (hhB : h + 1 ≤ B)
    (hzT : zsz ((2 : ℤ) ^ (h * bs.length + c)) ≤ B) :
    Sp (hdrA σ dS)
      (encK s (SS σ dS F z ((p : ℤ) :: (α : ℤ) :: (h : ℤ) :: (bs.length : ℤ) :: (bs ++ sv))
        (fun _ : Fin 19 => ([] : List ℤ))))
      (encK s (SS σ dS F z sv (vH p α h bs.length c bs))) (cHdr B bs.length) := 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 `J6Seg.hdr_sp`, Solutions/J6SegILSweep2.lean (lines 191–285) of the round-six development.

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