Round-six lemma `branchOn`

Proved

NLogN.Runs.branchOn

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 NLogN.IComp):

theorem NLogN.Runs.branchOn {i : Fin S} {k : Cls → Stmt (S + 4) P} {s : IState S} {σ' : St (S + 4)}
    {c : ℕ} (h : Runs I prog (k (s.cl i)) (encI s) σ' c) :
    Runs I prog (NLogN.branchOn i k) (encI s) σ' (c + 20)

Context. Control flow of the item language and the compiler correctness theorem

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_K_Defs33
import Definitions.Def_IntMul_Kappa_R6_Defs1
import Definitions.Def_IntMul_MultitapeModel
import Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN
open IntMul
variable {S P : ℕ}
variable {I : List Sym} {prog : Fin P → Stmt (S + 4) P}

Formal statement

theorem NLogN.Runs.branchOn {i : Fin S} {k : Cls → Stmt (S + 4) P} {s : IState S} {σ' : St (S + 4)}
    {c : ℕ} (h : Runs I prog (k (s.cl i)) (encI s) σ' c) :
    Runs I prog (NLogN.branchOn i k) (encI s) σ' (c + 20) := 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 `NLogN.Runs.branchOn`, Solutions/IComp.lean (lines 99–146) of the round-six development.

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