`BookProof.TruncationGapLift.exists_galerkin_tail_decomp` (b : HilbertBasis ℕ ℂ F) (m : ℕ) {v : F} (hv : v ∈ finiteModeDomain b) : ∃ x ∈ galerkinSpan b m, ∃ w ∈ tailSpan b m, v = x

Proved

BookProof.TruncationGapLift.exists_galerkin_tail_decomp

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterTruncationGapLift.

BookProof.TruncationGapLift.exists_galerkin_tail_decomp (b : HilbertBasis ℕ ℂ F) (m : ℕ) {v : F} (hv : v ∈ finiteModeDomain b) : ∃ x ∈ galerkinSpan b m, ∃ w ∈ tailSpan b m, v = x + w

Formalization note: Lean 4 identifier BookProof.TruncationGapLift.exists_galerkin_tail_decomp.

Preamble

-- Generated from ChapterTruncationGapLift.lean — theorem BookProof.TruncationGapLift.exists_galerkin_tail_decomp
import Definitions.Def_ChapterFarisLavine
import Definitions.Def_ChapterBandEnclosure
import Definitions.Def_ChapterFockSecondQuantization
import Definitions.Def_ChapterFockOneParticleGap
import Definitions.Def_ChapterFockNumberPreservingGap
import Definitions.Def_ChapterFockInteractionStability
import Definitions.Def_ChapterYangMillsHermite
import Definitions.Def_ChapterHermiteProductCore
import Definitions.Def_ChapterYangMillsFriedrichs
import Definitions.Def_ChapterYangMillsFockGapChain
import Mathlib
import Definitions.Def_ChapterTruncationGapLift
import Definitions.Def_ChapterHermiteGalerkinFriedrichs
open BookProof.HermiteGalerkin
open BookProof.TruncationGapLift


noncomputable section


open BookProof.FarisLavine BookProof.HermiteGalerkin BookProof.BandEnclosure
open BookProof.FockSecondQuantization BookProof.FockOneParticleGap
open BookProof.FockNumberPreservingGap BookProof.FockInteractionStability
open BookProof.YangMillsHermite BookProof.HermiteProductCore
open BookProof.YangMillsFriedrichs BookProof.YangMillsFockGapChain

variable {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℂ F]

Formal statement

theorem BookProof.TruncationGapLift.exists_galerkin_tail_decomp (b : HilbertBasis ℕ ℂ F) (m : ℕ)
    {v : F} (hv : v ∈ finiteModeDomain b) :
    ∃ x ∈ galerkinSpan b m, ∃ w ∈ tailSpan b m, v = x + w := by sorry

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