Round-six lemma `atanP_spec`

Proved

NLogN.atanP_spec

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.HNumAtan):

theorem NLogN.atanP_spec (x N K : ℕ) (σ : ℤ) (hx : 0 < x) (s : IState HS) (L0 L1 : List Item)
    (h0 : s.G 0 = .int N :: L0) (h1 : s.G 1 = .int K :: L1) :
    (atanP x σ).ok s ∧ (atanP x σ).run s = s.upd 2 (.int (atanSum x N σ K) :: s.G 2)

Context. atanInv / atanhInv as a register program

Inputs N on register 0, K on register 1; the result is pushed on register 2. Scratch registers 40 … 45.

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 Mathlib
import Definitions.Def_IntMul_HvdH_M_Defs8
import Definitions.Def_IntMul_MultitapeModel
import Definitions.Def_IntMul_HvdH_Numeric

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open NLogN
open IntMul NLogN.RS

Formal statement

theorem NLogN.atanP_spec (x N K : ℕ) (σ : ℤ) (hx : 0 < x) (s : IState HS) (L0 L1 : List Item)
    (h0 : s.G 0 = .int N :: L0) (h1 : s.G 1 = .int K :: L1) :
    (atanP x σ).ok s ∧ (atanP x σ).run s = s.upd 2 (.int (atanSum x N σ K) :: s.G 2) := 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.atanP_spec`, Solutions/HNumAtan.lean (lines 80–152) of the round-six development.

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