Per-term error (`sTerm_err`)
ProvedIntMul.HvdH.SPf.sTerm_errcomplexity-theoryinteger-multiplicationkappa-improvement
Per-term error: ‖sTerm j - 2^p β e_j u_j‖ ≤ 11/4 (in units of 2^{-p}).
The following holds (Lean statement, in the scope of module Solutions.KFMBaseSap):
theorem IntMul.HvdH.SPf.sTerm_err (p s t α k : ℕ) (ht : 0 < t) (hα : 2 ≤ α) (U : ℕ → ℤ × ℤ)
(hU : ∀ i, ‖toC (U i)‖ ≤ 2 ^ p) (j : ℤ) :
‖toC (sTerm p s t α k U j) -
2 ^ p * ((((2 * α : ℝ)⁻¹ * ew s t α k j : ℝ) : ℂ) * (toC (U (j % s).toNat) / 2 ^ p))‖
≤ 11 / 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_R6_Defs60 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.sTerm_err (p s t α k : ℕ) (ht : 0 < t) (hα : 2 ≤ α) (U : ℕ → ℤ × ℤ)
(hU : ∀ i, ‖toC (U i)‖ ≤ 2 ^ p) (j : ℤ) :
‖toC (sTerm p s t α k U j) -
2 ^ p * ((((2 * α : ℝ)⁻¹ * ew s t α k j : ℝ) : ℂ) * (toC (U (j % s).toNat) / 2 ^ p))‖
≤ 11 / 4 := by sorrySource
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.sTerm_err`, Solutions/KFMBaseSap.lean (lines 55–138) of the round-six development.