DirichletKernel.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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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

13 declarations, 0 contain sorry, 0 declared as axioms

Imports (1)

Carleson.Classical.Basic

Imported by 2 files

def dirichletKernel line 13 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex
Proof uses: Finset.sum, Int, Finset.Icc, Neg.neg, Int.ofNat, ContinuousMap, AddCircle, HMul.hMul, OfNat.ofNat, Real.pi, AddSubgroup.zmultiples, fourier
30 instances, coercions and auxiliary constants left out.
def dirichletKernel' line 16 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex
Proof uses: HAdd.hAdd, HDiv.hDiv, Complex.exp, HMul.hMul, Complex.I, HSub.hSub, OfNat.ofNat, Neg.neg
15 instances, coercions and auxiliary constants left out.
lemma continuous_dirichletKernel line 23 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Continuous, Real, Complex, dirichletKernel
Proof uses: id, continuous_finsetSum, Int, ContinuousMap, AddCircle, HMul.hMul, OfNat.ofNat, Real.pi, AddSubgroup.zmultiples, fourier, Finset.Icc, Neg.neg, Int.ofNat, Membership.mem, Finset, Continuous.eval, continuous_const, QuotientAddGroup.continuous_mk
74 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma dirichletKernel_periodic line 28 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Function.Periodic, Real, Complex, dirichletKernel, HMul.hMul, OfNat.ofNat, Real.pi
Proof uses: of_eq_true, Eq, HAdd.hAdd, Eq.trans, Finset.sum, Int, Finset.Icc, Neg.neg, Complex.exp, HDiv.hDiv, Complex.I, True, congr, congrArg, Finset.sum_congr, Int.ofNat, ContinuousMap, AddCircle, AddSubgroup.zmultiples, fourier and 14 more
73 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma dirichletKernel'_periodic line 33 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Function.Periodic, Real, Complex, dirichletKernel', HMul.hMul, OfNat.ofNat, Real.pi
Proof uses: id, Eq, HAdd.hAdd, Eq.mpr, HDiv.hDiv, Complex.exp, Complex.I, HSub.hSub, Neg.neg, congrFun', congrArg, congr, Eq.trans, Complex.ofReal_add, Complex.ofReal_mul, HAdd, Eq.rec, Eq.refl, HDiv, mul_add and 49 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 33 steps
Open to load the checked before-and-after goal states.
lemma dirichletKernel'_measurable line 65 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Measurable, Real, Complex, dirichletKernel'
Proof uses: Measurable.add, HDiv.hDiv, Complex.exp, HMul.hMul, Complex.I, HSub.hSub, OfNat.ofNat, Neg.neg, Measurable.fun_div, Measurable.cexp, Measurable.const_mul, Measurable.complex_ofReal, measurable_id', Measurable.const_sub
70 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma dirichletKernel_eq Second part of Lemma 11.1.8 (Dirichlet kernel) from the paper line 69 source
Source documentation
Second part of Lemma 11.1.8 (Dirichlet kernel) from the paper.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Ne, Complex, Complex.exp, HMul.hMul, Complex.I, OfNat.ofNat, Eq, dirichletKernel, dirichletKernel'
Proof uses: HSub.hSub, HDiv.hDiv, Neg.neg, HAdd.hAdd, Trans.trans, Finset.sum, Int, Finset.Icc, rfl, Eq.mpr, Int.ofNat, ContinuousMap, AddCircle, Real.pi, AddSubgroup.zmultiples, fourier, id, congrArg, dirichletKernel.eq_1, Finset.mul_sum and 167 more
247 instances, coercions and auxiliary constants left out.

Tactic trace: 63 steps
Open to load the checked before-and-after goal states.
```
lemma dirichletKernel_eq {x : ℝ} (h : cexp (I * x) ≠ 1) :
    dirichletKernel N x = dirichletKernel' N x := by
…
```
Mentions: dirichletKernel, dirichletKernel'
lemma dirichletKernel'_eq_zero line 134 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Eq, Complex, Complex.exp, HMul.hMul, Complex.I, OfNat.ofNat, dirichletKernel'
Proof uses: of_eq_true, Eq.trans, True, congrFun', congrArg, HAdd.hAdd, HDiv.hDiv, HSub.hSub, Neg.neg, congr, Inv.inv, neg_mul, Complex.exp_neg, inv_one, sub_self, div_zero, add_zero, eq_self
57 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma dirichletKernel_eq_ae "a.e." version of previous lemma line 138 source
Source comment
"a.e." version of previous lemma.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Filter.Eventually, Real, Eq, Complex, dirichletKernel, dirichletKernel', MeasureTheory.ae, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume
Proof uses: LE.le, Set, Set.ofPred, Not, Set.range, Int, HMul.hMul, OfNat.ofNat, Real.pi, Membership.mem, Mathlib.Tactic.Contrapose.contrapose₂, dirichletKernel_eq, Eq.mpr, Ne, Complex.exp, Complex.I, id, Eq.trans, congrFun', congrArg and 18 more
57 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma dirichletKernel_eq_ae : ∀ᵐ (x : ℝ), dirichletKernel N x = dirichletKernel' N x := by
…
```
Mentions: dirichletKernel, dirichletKernel'
lemma norm_dirichletKernel_le line 154 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, Norm.norm, Complex, dirichletKernel, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: Eq.mpr, Finset.sum, Int, Finset.Icc, Neg.neg, Int.ofNat, ContinuousMap, AddCircle, Real.pi, AddSubgroup.zmultiples, fourier, id, Eq, congrArg, dirichletKernel.eq_1, Trans.trans, rfl, norm_sum_le, Finset.sum_le_sum, Membership.mem and 46 more
136 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma norm_dirichletKernel'_le line 168 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, LE.le, Norm.norm, Complex, dirichletKernel', HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: dite, Ne, Complex.exp, Complex.I, Eq, Eq.mpr, dirichletKernel, id, congrFun', congrArg, of_eq_true, Not, Eq.trans, False, True, eq_false, not_false_eq_true, norm_dirichletKernel_le, dirichletKernel'_eq_zero, Eq.mp and 65 more
101 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma partialFourierSum_eq_conv_dirichletKernel First part of lemma 11.1.8 (Dirichlet kernel) from the blueprint line 176 source
Source documentation
First part of lemma 11.1.8 (Dirichlet kernel) from the blueprint.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, IntervalIntegrable, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, HMul.hMul, Real.pi, Eq, partialFourierSum, HDiv.hDiv, intervalIntegral, dirichletKernel, HSub.hSub
Proof uses: Trans.trans, Finset.sum, Int, Finset.Icc, Neg.neg, ContinuousMap, AddCircle, AddSubgroup.zmultiples, fourier, HSMul.hSMul, fourierCoeffOn, Real.two_pi_pos, rfl, Eq.mpr, id, congrArg, partialFourierSum.eq_1, Eq.refl, AddCommMonoid, Finset and 125 more
220 instances, coercions and auxiliary constants left out.

Tactic trace: 22 steps
Open to load the checked before-and-after goal states.
```
lemma partialFourierSum_eq_conv_dirichletKernel {f : ℝ → ℂ} {x : ℝ}
    (h : IntervalIntegrable f volume 0 (2 * π)) :
    partialFourierSum N f x =
      (1 / (2 * π)) * ∫ (y : ℝ) in (0 : ℝ)..(2 * π), f y * dirichletKernel N (x - y) := by
…
```
Mentions: partialFourierSum, dirichletKernel
lemma partialFourierSum_eq_conv_dirichletKernel' line 211 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Real, Complex, IntervalIntegrable, MeasureTheory.MeasureSpace.volume, OfNat.ofNat, HMul.hMul, Real.pi, Eq, partialFourierSum, HDiv.hDiv, intervalIntegral, dirichletKernel', HSub.hSub
Proof uses: Eq.mpr, dirichletKernel, id, congrArg, partialFourierSum_eq_conv_dirichletKernel, Trans.trans, rfl, HMul, Eq.rec, Eq.refl, Eq.symm, intervalIntegral.integral_comp_sub_left, of_eq_true, Eq.trans, True, congrFun', MeasureTheory.Measure, funext, sub_sub_cancel, eq_self and 18 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 13 steps
Open to load the checked before-and-after goal states.