SpectralProjectionBound.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 981 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Pinned Carleson blueprint

6 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Carleson.Classical.Basic

Imported by 2 files

lemma L2norm_sq_eq TODO: move somewhere else? line 16 source
Source comment
TODO: move somewhere else?

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Fact, LT.lt, OfNat.ofNat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, Nat, Eq, HPow.hPow, Norm.norm, MeasureTheory.integral
Proof uses: Eq.mpr, AddMonoidHom, AddZero.mk, AddMonoid.toZero, RCLike.re, Inner.inner, id, congrArg, InnerProductSpace.norm_sq_eq_re_inner, MeasureTheory.L2.inner_def, Eq.symm, integral_re, MeasureTheory.L2.integrable_inner, of_eq_true, Eq.trans, True, congrFun', funext, eq_self
111 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma L2norm_sq_eq {T : ℝ} [hT : Fact (0 < T)] (f : Lp ℂ 2 <| @haarAddCircle T hT) :
    ‖f‖ ^ 2 = ∫ (x : AddCircle T), ‖f x‖ ^ 2 ∂haarAddCircle := by
…
```
Mentions: T
lemma fourierCoeff_eq_innerProduct line 22 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Fact, LT.lt, OfNat.ofNat, LE.le, ENNReal, Nat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, Int, Eq, fourierCoeff, Inner.inner and 1 more
Proof uses: Eq.mpr, HilbertBasis, fourierBasis, id, congrArg, Eq.symm, coe_fourierBasis, PreLp, lp, LinearIsometryEquiv, RingHom.id, HilbertBasis.repr, fourierBasis_repr, HilbertBasis.repr_apply_apply
100 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma partialFourierSumL2_norm line 27 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Fact, LT.lt, OfNat.ofNat, LE.le, ENNReal, Nat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, Eq, HPow.hPow, Norm.norm, partialFourierSumLp and 6 more
Proof uses: Trans.trans, PreLp, lp, HSMul.hSMul, LinearIsometryEquiv, RingHom.id, HilbertBasis.repr, fourierBasis, fourierLp, rfl, Eq.mpr, id, congrArg, Eq.symm, Real.rpow_natCast, Eq.refl, LinearIsometryEquiv.norm_map, partialFourierSumLp.eq_1, map_sum, RingHom and 24 more
184 instances, coercions and auxiliary constants left out.

Tactic trace: 18 steps
Open to load the checked before-and-after goal states.
lemma spectral_projection_bound_sq line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real, Fact, LT.lt, OfNat.ofNat, Nat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, LE.le, HPow.hPow, Norm.norm, partialFourierSumLp
Proof uses: Eq.mpr, Finset.sum, Int, Finset.Icc, Neg.neg, Int.ofNat, fourierCoeff, id, Eq, congrArg, partialFourierSumL2_norm, Inner.inner, fourierLp, congrFun', Finset.sum_congr, Eq.refl, Finset, fourierCoeff_eq_innerProduct, Orthonormal.sum_inner_products_le, orthonormal_fourier
87 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma spectral_projection_bound_sq_integral line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Fact, LT.lt, OfNat.ofNat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, LE.le, MeasureTheory.integral, HPow.hPow, Norm.norm, partialFourierSumLp
Proof uses: Eq.mpr, id, Eq, congrArg, Eq.symm, L2norm_sq_eq, spectral_projection_bound_sq
77 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma spectral_projection_bound_lp line 59 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Fact, LT.lt, OfNat.ofNat, Subtype, MeasureTheory.AEEqFun, AddCircle, Complex, AddSubgroup.zmultiples, AddCircle.haarAddCircle, Membership.mem, AddSubgroup, MeasureTheory.Lp, ENNReal, LE.le, Norm.norm, partialFourierSumLp
Proof uses: Eq.mpr, abs, id, Eq, congrArg, Eq.symm, abs_norm, HPow.hPow, propext, sq_le_sq, spectral_projection_bound_sq
81 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.