`BookProof.TensorCore.exists_core_approx` (hcore : IsGraphCore D A) (n : ℕ) (x : ((domSpace Hs D₂).pow n)) {ε : ℝ} (hε : 0 < ε) : ∃ y ∈ corePow Hs D₂ D n, ‖inclPow Hs D₂ n x - incl

Proved

BookProof.TensorCore.exists_core_approx

by leonardopedro · Oct 10, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTensorGraphCore.

BookProof.TensorCore.exists_core_approx (hcore : IsGraphCore D A) (n : ℕ) (x : ((domSpace Hs D₂).pow n)) {ε : ℝ} (hε : 0 < ε) : ∃ y ∈ corePow Hs D₂ D n, ‖inclPow Hs D₂ n x - inclPow Hs D₂ n y‖ < ε ∧ ‖derPow Hs D₂ A n x - derPow Hs D₂ A n y‖ < ε

Formalization note: Lean 4 identifier BookProof.TensorCore.exists_core_approx.

Preamble

-- Generated from ChapterTensorGraphCore.lean — theorem BookProof.TensorCore.exists_core_approx
import Definitions.Def_ChapterFarisLavine
import Mathlib
import Definitions.Def_ChapterTensorGraphCore
import Definitions.Def_ChapterGraphCoreTransfer
open BookProof.GraphCore
open BookProof.TensorCore



open scoped TensorProduct
open BookProof.FarisLavine BookProof.GraphCore

noncomputable section

variable (Hs : IPSpace) (D₂ : Submodule ℂ Hs.carrier)
variable (A : D₂ →ₗ[ℂ] Hs.carrier)
variable (D : Submodule ℂ Hs.carrier)

Formal statement

theorem BookProof.TensorCore.exists_core_approx (hcore : IsGraphCore D A) (n : ℕ)
    (x : ((domSpace Hs D₂).pow n)) {ε : ℝ} (hε : 0 < ε) :
    ∃ y ∈ corePow Hs D₂ D n,
      ‖inclPow Hs D₂ n x - inclPow Hs D₂ n y‖ < ε ∧
        ‖derPow Hs D₂ A n x - derPow Hs D₂ A n y‖ < ε := by sorry

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