Recognize a managed state as `St` (`St_eq`)

Proved

NLogN.KFZ.St_eq

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

complexity-theoryinteger-multiplicationkappa-improvement

Recognize a managed state as St.

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

theorem NLogN.KFZ.St_eq (F0 G : Fin HS → List ℤ) (p h d σ K : ℕ) (td z w tb ta tt ti : List ℤ)
    (e0 : G 0 = z) (e1 : G pW = w) (e2 : G tD = td) (e3 : G qB = tb) (e4 : G NLogN.KFZ.qA = ta)
    (e5 : G NLogN.KFZ.qT = tt) (e6 : G qI = ti) (e7 : G rgP = [(p : ℤ)]) (e8 : G rgH = [(h : ℤ)])
    (e9 : G rgD = [(d : ℤ)]) (e10 : G rgS = [(σ : ℤ)]) (e11 : G rgK = [(K : ℤ)])
    (er : ∀ j, j ≠ 0 → j ≠ pW → j ≠ tD → j ≠ qB → j ≠ NLogN.KFZ.qA → j ≠ NLogN.KFZ.qT → j ≠ qI → j ≠ rgP →
      j ≠ rgH → j ≠ rgD → j ≠ rgS → j ≠ rgK → G j = F0 j) :
    NLogN.KFZ.St F0 p h d σ K td z w tb ta tt ti = G

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.St_eq (F0 G : Fin HS → List ℤ) (p h d σ K : ℕ) (td z w tb ta tt ti : List ℤ)
    (e0 : G 0 = z) (e1 : G pW = w) (e2 : G tD = td) (e3 : G qB = tb) (e4 : G NLogN.KFZ.qA = ta)
    (e5 : G NLogN.KFZ.qT = tt) (e6 : G qI = ti) (e7 : G rgP = [(p : ℤ)]) (e8 : G rgH = [(h : ℤ)])
    (e9 : G rgD = [(d : ℤ)]) (e10 : G rgS = [(σ : ℤ)]) (e11 : G rgK = [(K : ℤ)])
    (er : ∀ j, j ≠ 0 → j ≠ pW → j ≠ tD → j ≠ qB → j ≠ NLogN.KFZ.qA → j ≠ NLogN.KFZ.qT → j ≠ qI → j ≠ rgP →
      j ≠ rgH → j ≠ rgD → j ≠ rgS → j ≠ rgK → G j = F0 j) :
    NLogN.KFZ.St F0 p h d σ K td z w tb ta tt ti = G := 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.St_eq`, Solutions/KFZFull.lean (lines 29–75) of the round-six development.

View graph