openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /NormalizedAngularWord.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SmoothFinsetProduct | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open Set | |
| open scoped ContDiff | |
| variable {V α : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] | |
| noncomputable def normalizedAngularWordND (D : ℝ) (L : List (V →L[ℝ] V)) | |
| (f : SmoothFunctionND V) : SmoothFunctionND V := | |
| ((D:ℂ)^L.length)⁻¹ • angularWordOperatorND L f | |
| lemma normalizedAngularWordND_extract {D : ℝ} (hD : D ≠ 0) (L : List (V →L[ℝ] V)) | |
| (f : SmoothFunctionND V) : | |
| (D:ℂ)^L.length • normalizedAngularWordND D L f = angularWordOperatorND L f := by | |
| rw [normalizedAngularWordND,smul_smul,mul_inv_cancel₀ (pow_ne_zero _ (Complex.ofReal_ne_zero.mpr hD)),one_smul] | |
| lemma normalizedAngularWordND_support (D : ℝ) (L : List (V →L[ℝ] V)) | |
| (f : SmoothFunctionND V) : tsupport (normalizedAngularWordND D L f).1 ⊆ tsupport f.1 := by | |
| apply subset_trans _ (angularWordOperatorND_support L f) | |
| change tsupport (fun x => ((D:ℂ)^L.length)⁻¹ * (angularWordOperatorND L f).1 x) ⊆ _ | |
| apply closure_mono | |
| intro x hx hz | |
| exact hx (by simp [hz]) | |
| lemma smoothFinsetProductND_smul (S : Finset α) (c : α → ℂ) (f : α → SmoothFunctionND V) : | |
| smoothFinsetProductND S (fun i => c i • f i) = (∏ i ∈ S, c i) • smoothFinsetProductND S f := by | |
| apply Subtype.ext | |
| funext x | |
| change (∏ i ∈ S, c i*(f i).1 x)=(∏ i ∈ S,c i)*(∏ i ∈ S,(f i).1 x) | |
| exact Finset.prod_mul_distrib | |
| lemma assignedAngularProductND_normalized (S : Finset α) (D : α → ℝ) | |
| (k : α → List (V →L[ℝ] V)) (f : α → SmoothFunctionND V) (hD : ∀ i ∈ S, D i ≠ 0) : | |
| smoothFinsetProductND S (fun i => angularWordOperatorND (k i) (f i)) = | |
| ((∏ i ∈ S, D i ^ (k i).length : ℝ):ℂ) • | |
| smoothFinsetProductND S (fun i => normalizedAngularWordND (D i) (k i) (f i)) := by | |
| simp only [Complex.ofReal_prod,Complex.ofReal_pow] | |
| rw [← smoothFinsetProductND_smul] | |
| apply Subtype.ext | |
| funext x | |
| apply Finset.prod_congr rfl | |
| intro i hi | |
| exact congrArg (fun g : SmoothFunctionND V => g.1 x) (normalizedAngularWordND_extract (hD i hi) (k i) (f i)).symm | |
| lemma assignedAngularScaleND_le (S : Finset α) (D : α → ℝ) | |
| (k : α → List (V →L[ℝ] V)) {M : ℝ} {b : ℕ} (hM : 1 ≤ M) | |
| (hD₀ : ∀ i ∈ S, 0 ≤ D i) (hD : ∀ i ∈ S, D i ≤ M) | |
| (hk : ∑ i ∈ S, (k i).length ≤ b) : | |
| ∏ i ∈ S, D i ^ (k i).length ≤ M^b := by | |
| calc | |
| _ ≤ ∏ i ∈ S, M^(k i).length := Finset.prod_le_prod₀ | |
| (fun i hi => pow_nonneg (hD₀ i hi) _) (fun i hi => pow_le_pow_left₀ (hD₀ i hi) (hD i hi) _) | |
| _ = M^(∑ i ∈ S,(k i).length) := by rw [Finset.prod_pow_eq_pow_sum] | |
| _ ≤ M^b := pow_le_pow_right₀ hM hk | |
| lemma assignedAngularScaleND_paid (S : Finset α) (D : α → ℝ) | |
| (k : α → List (V →L[ℝ] V)) {M r : ℝ} {j : ℕ} (hM : 1 ≤ M) | |
| (hr : 0 ≤ r) (hpaid : r*M^2 ≤ 1) | |
| (hD₀ : ∀ i ∈ S, 0 ≤ D i) (hD : ∀ i ∈ S, D i ≤ M) | |
| (hk : ∑ i ∈ S, (k i).length ≤ 2*j) : | |
| r^j * (∏ i ∈ S, D i ^ (k i).length) ≤ 1 := by | |
| calc | |
| _ ≤ r^j * M^(2*j) := mul_le_mul_of_nonneg_left | |
| (assignedAngularScaleND_le S D k hM hD₀ hD hk) (pow_nonneg hr _) | |
| _ = (r*M^2)^j := by rw [mul_pow,pow_mul] | |
| _ ≤ 1 := by simpa using pow_le_pow_left₀ (by positivity) hpaid j | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |