The pass (round six, `pass_runs`)

Proved

J6Seg.Slab.pass_runs

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

complexity-theoryinteger-multiplicationkappa-improvement

The pass.

The following holds (Lean statement, in the scope of module Solutions.J6SegILSlab4):

theorem J6Seg.Slab.pass_runs (H : UnitsOK (I := I) (iprog := iprog) lin OkU OkA KG g dr rd it bk cdr crd cit cbk)
    (s : IState HS) (G : Fin HS → List ℤ) (hK : ∀ C CU OP ARG ACC, KG (skS G C CU OP ARG ACC)) (hA : G lA = []) (hC : G lC = []) (hD : G lD = [])
    {m : ℕ} {LM : List ℤ} (hM : G rM = (m : ℤ) :: LM) (cs : List (ℤ × ℤ)) {LN : List ℤ}
    (hN : G rNC = (cs.length : ℤ) :: LN) {Ba : ℕ} (hcs : CmdsOK OkA Ba cs) (n : ℕ)
    (hV : VC n cs 0) (hND : nDrop cs = n) (pre : ℕ) {LB : List ℤ}
    (hB : G rNB = (pre : ℤ) :: LB) (xs : List U) (hxs : xs.length = pre * (n * m))
    (hok : ∀ u ∈ xs, OkU u) {LT : List ℤ}
    (hT : G rTOT = ((passOut m g cs n pre xs).length : ℤ) :: LT) {Bo : ℕ}
    (hBo : ∀ y ∈ passOut m g cs n pre xs, zsz y ≤ Bo) (RC CU OP ARG : List ℤ) :
    IRuns I iprog (J6Seg.Slab.passP dr rd it bk)
      (encK s (lin xs [] (skS G (flatC cs ++ RC) CU OP ARG [])))
      (encK s (skS (update G 0 (passOut m g cs n pre xs ++ G 0)) (flatC cs ++ RC) CU OP ARG []))
      (passC pre cs.length m Ba (cdr + crd + cit + cbk) (passOut m g cs n pre xs).length Bo)

Context. B-line IL, B1 (streaming slab pass) §4: the whole pass (generic lanes)

passP = rptX rNB blockP; rptX rTOT (mv rACC 0): the blocks, then the output (collected reversed on rACC) is moved onto stack 0. pass_runs: from the lane state with the input xs (pre blocks of n slabs of m units) the output passOut … ++ G 0 lands on stack 0; all skeleton registers are restored (rACC empty again). passOut_eq: for the control list ctlOf n n' src argOf of a monotone source, passOut is slabPass (block form).

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_Defs64

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open J6Seg J6Seg.Slab
set_option maxRecDepth 20000
open IntMul NLogN NLogN.KIL Function
open NLogN.KFUU (lA lC lD)
open NLogN.KFT (cf3 rptX)
variable {U : Type*} {P : ℕ} {I : List Sym} {iprog : Fin P → IStmt HS P}
  {lin : List U → List U → (Fin HS → List ℤ) → (Fin HS → List ℤ)} {OkU : U → Prop}
  {OkA : ℤ → Prop} {KG : (Fin HS → List ℤ) → Prop} {g : ℤ → U → ℤ} {dr rd it bk : IStmt HS P}
  {cdr crd cit cbk : ℕ}

Formal statement

theorem J6Seg.Slab.pass_runs (H : UnitsOK (I := I) (iprog := iprog) lin OkU OkA KG g dr rd it bk cdr crd cit cbk)
    (s : IState HS) (G : Fin HS → List ℤ) (hK : ∀ C CU OP ARG ACC, KG (skS G C CU OP ARG ACC)) (hA : G lA = []) (hC : G lC = []) (hD : G lD = [])
    {m : ℕ} {LM : List ℤ} (hM : G rM = (m : ℤ) :: LM) (cs : List (ℤ × ℤ)) {LN : List ℤ}
    (hN : G rNC = (cs.length : ℤ) :: LN) {Ba : ℕ} (hcs : CmdsOK OkA Ba cs) (n : ℕ)
    (hV : VC n cs 0) (hND : nDrop cs = n) (pre : ℕ) {LB : List ℤ}
    (hB : G rNB = (pre : ℤ) :: LB) (xs : List U) (hxs : xs.length = pre * (n * m))
    (hok : ∀ u ∈ xs, OkU u) {LT : List ℤ}
    (hT : G rTOT = ((passOut m g cs n pre xs).length : ℤ) :: LT) {Bo : ℕ}
    (hBo : ∀ y ∈ passOut m g cs n pre xs, zsz y ≤ Bo) (RC CU OP ARG : List ℤ) :
    IRuns I iprog (J6Seg.Slab.passP dr rd it bk)
      (encK s (lin xs [] (skS G (flatC cs ++ RC) CU OP ARG [])))
      (encK s (skS (update G 0 (passOut m g cs n pre xs ++ G 0)) (flatC cs ++ RC) CU OP ARG []))
      (passC pre cs.length m Ba (cdr + crd + cit + cbk) (passOut m g cs n pre xs).length Bo) := 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.Slab.pass_runs`, Solutions/J6SegILSlab4.lean (lines 94–137) of the round-six development.

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