Round-six lemma `pops_end`

Proved

NLogN.KFZ.pops_end

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.KFZFull):

theorem NLogN.KFZ.pops_end (s : IState HS) (f : Fin HS → List ℤ) (hf : EmptyEx f) {p h d σ K B : ℕ}
    (hB : RB p h d σ K B) (z : List ℤ) :
    Sp (popsL regL) (encK s (NLogN.KFZ.St f p h d σ K [] z [] [] [] [] []))
      (encK s (Function.update f 0 z)) (5 * (2 * B + 2) + 1)

Context. KFZ Full: the whole chirp procedure body

body j d18 = setupP; T1; …; T7; wP setC (PP j d18); popsL regL.

body_runs: from encK s f (stack 0 = gflat xs ++ l, the chirp descriptor on cS, all other managed stacks empty) the body ends in encK s (update f 0 (gflat ys ++ l)) with ys = Q(Q(H θ̃⁻) ã), given the run of the Prop 18 call on its state.

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
import Definitions.Def_IntMul_HvdH_Numeric
import Definitions.Def_IntMul_HvdH_ExpApprox
import Definitions.Def_IntMul_HvdH_Twiddle
import Definitions.Def_IntMul_HvdH_RealExp
import Definitions.Def_IntMul_HvdH_SApprox
import Definitions.Def_IntMul_HvdH_MachineRef
import Definitions.Def_IntMul_HvdH_DiagApprox
import Definitions.Def_IntMul_HvdH_EApprox
import Definitions.Def_IntMul_HvdH_JApprox
import Definitions.Def_IntMul_HvdH_BApprox
import Definitions.Def_IntMul_HvdH_FPApprox

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN NLogN.KFZ
open IntMul NLogN.KIL NLogN.KFB NLogN.KFM IntMul.OpenAI.Glue

Formal statement

theorem NLogN.KFZ.pops_end (s : IState HS) (f : Fin HS → List ℤ) (hf : EmptyEx f) {p h d σ K B : ℕ}
    (hB : RB p h d σ K B) (z : List ℤ) :
    Sp (popsL regL) (encK s (NLogN.KFZ.St f p h d σ K [] z [] [] [] [] []))
      (encK s (Function.update f 0 z)) (5 * (2 * B + 2) + 1) := 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.KFZ.pops_end`, Solutions/KFZFull.lean (lines 87–130) of the round-six development.

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