Lemma 4.9 (`Sapprox_bound`)

Proved

IntMul.HvdH.SPf.Sapprox_bound

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

complexity-theoryinteger-multiplicationkappa-improvement

Lemma 4.9: the truncated fixed-point sum approximates 2^p (S' u)_k within 6m + 4.

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

theorem IntMul.HvdH.SPf.Sapprox_bound (p s t α k : ℕ) (hs : 0 < s) (ht : 0 < t) (hα : 2 ≤ α) (hαp : α ^ 2 ≤ p)
    (U : ℕ → ℤ × ℤ) (hU : ∀ i, ‖toC (U i)‖ ≤ 2 ^ p) :
    ‖toC (Sapprox p s t α U k) - 2 ^ p * Sexact s t α (fun i => toC (U i) / 2 ^ p) k‖
      ≤ 6 * sWidth p α + 4

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_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_Kappa_R6_Defs16

set_option autoImplicit false
set_option relaxedAutoImplicit false
set_option maxHeartbeats 1000000

open IntMul IntMul.HvdH IntMul.HvdH.SPf
open Complex
open IntMul.HvdH

Formal statement

theorem IntMul.HvdH.SPf.Sapprox_bound (p s t α k : ℕ) (hs : 0 < s) (ht : 0 < t) (hα : 2 ≤ α) (hαp : α ^ 2 ≤ p)
    (U : ℕ → ℤ × ℤ) (hU : ∀ i, ‖toC (U i)‖ ≤ 2 ^ p) :
    ‖toC (Sapprox p s t α U k) - 2 ^ p * Sexact s t α (fun i => toC (U i) / 2 ^ p) k‖
      ≤ 6 * sWidth p α + 4 := 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 `IntMul.HvdH.SPf.Sapprox_bound`, Solutions/KFMBaseSap.lean (lines 271–406) of the round-six development.

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