Prove2Me

The branch of one command, on the state after the two pops (`branch_runs`)

Proved
J6Seg.Slab.branch_runs

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

complexity-theoryinteger-multiplicationkappa-improvement

The branch of one command, on the state after the two pops.

The following holds (Lean statement, in the scope of module Solutions.J6SegILSlab3):

theorem J6Seg.Slab.branch_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) (op a : ℤ) (C CU OP ARG ACC : List ℤ)
    (inp tmp : List U) (hlen : op ≠ 0 → m ≤ inp.length) (hok : ∀ u ∈ inp, OkU u)
    (ha : 0 < op → OkA a) :
    IRuns I iprog
      (.ite rOP (fun c => c == .neg) (rptX rM lA lC lD dr)
        (.ite rOP (fun c => c == .zero) (rptX rM lA lC lD (.op (.const rACC 0))) (emitP rd it bk)))
      (encK s (lin inp tmp (skS G C CU (op :: OP) (a :: ARG) ACC)))
      (encK s (lin (if op < 0 then inp.drop m else inp) tmp
        (skS G C CU (op :: OP) (a :: ARG) ((runC m g [(op, a)] inp).reverse ++ ACC))))
      (50 + (m + 1) * (cdr + crd + cit + cbk + 12002))

Context. B-line IL, B1 (streaming slab pass) §3: one command, the command loop, one block

The skeleton contents are kept in the form skS G C CU OP ARG ACC (control rC, used commands rCU, current op rOP, current argument rARG, output rACC, over G).

  • cmdP: pop (op, arg) from rC to rOP, rARG; branch on the sign of op: drop a slab (op < 0), emit m zeros (op = 0), emit the slab mapped by g arg, reading the units to the temps and moving them back (op > 0); push op, arg to rCU. cmd_runs.
  • cmdLoop: rptX rNC cmdP — the whole command list; cmdLoop_runs: the output is runC m g cs inp (reversed onto rACC), the input loses nDrop cs slabs.
  • restLoop: rptX rNC (mv rCU rC; mv rCU rC) restores the control list. restLoop_runs.
  • blockP = cmdLoop; restLoop: block_runs.

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}
variable {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.branch_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) (op a : ℤ) (C CU OP ARG ACC : List ℤ)
    (inp tmp : List U) (hlen : op ≠ 0 → m ≤ inp.length) (hok : ∀ u ∈ inp, OkU u)
    (ha : 0 < op → OkA a) :
    IRuns I iprog
      (.ite rOP (fun c => c == .neg) (rptX rM lA lC lD dr)
        (.ite rOP (fun c => c == .zero) (rptX rM lA lC lD (.op (.const rACC 0))) (emitP rd it bk)))
      (encK s (lin inp tmp (skS G C CU (op :: OP) (a :: ARG) ACC)))
      (encK s (lin (if op < 0 then inp.drop m else inp) tmp
        (skS G C CU (op :: OP) (a :: ARG) ((runC m g [(op, a)] inp).reverse ++ ACC))))
      (50 + (m + 1) * (cdr + crd + cit + cbk + 12002)) := 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.branch_runs`, Solutions/J6SegILSlab3.lean (lines 148–207) of the round-six development.

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