openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /AffineIntegrability.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.AffineDensityDerivative | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section AffineIntegrability | |
| open Classical MeasureTheory Matrix | |
| theorem integrable_matrix_affine {ι : Type*} [Fintype ι] | |
| (A : Matrix ι ι ℝ) (a : ι → ℝ) (hA : IsUnit A.det) | |
| {f : (ι → ℝ) → ℝ} (hf : Integrable f) : | |
| Integrable (fun x => f (a+A*ᵥx)) := by | |
| have hm : Integrable f (Measure.map (matrixAffineEquiv A a hA) volume) := by | |
| rw [show (matrixAffineEquiv A a hA : (ι → ℝ) → (ι → ℝ))= | |
| (fun x => a+A*ᵥx) from funext (fun x => matrixAffineEquiv_apply A a x hA), | |
| map_matrix_affine_volume A a hA] | |
| exact hf.smul_measure ENNReal.ofReal_ne_top | |
| convert! (integrable_map_equiv (matrixAffineEquiv A a hA) f).mp hm using 1 | |
| theorem integrable_matrix_affine_inverse {ι : Type*} [Fintype ι] | |
| (A : Matrix ι ι ℝ) (a : ι → ℝ) (hA : IsUnit A.det) | |
| {f : (ι → ℝ) → ℝ} (hf : Integrable f) : | |
| Integrable (fun y => f (A⁻¹*ᵥ(y-a))) := by | |
| have h := integrable_matrix_affine A⁻¹ (-(A⁻¹*ᵥa)) | |
| (isUnit_iff_ne_zero.mpr (by | |
| rw [Matrix.det_nonsing_inv,Ring.inverse_eq_inv] | |
| exact inv_ne_zero hA.ne_zero)) hf | |
| convert! h using 1 | |
| ext y | |
| rw [Matrix.mulVec_sub] | |
| simp only [sub_eq_add_neg,add_comm] | |
| theorem affineNoiseRoot_integrable_square {ι : Type*} [Fintype ι] | |
| {f : ℝ → ℝ} (hf : Continuous f) (hc : HasCompactSupport f) | |
| (A : Matrix ι ι ℝ) (a : ι → ℝ) (hA : IsUnit A.det) : | |
| Integrable (fun y : ι → ℝ => affineNoiseRoot f A a y^2) := by | |
| have ht : Integrable (fun x : ι → ℝ => noiseTensor (fun _ : ι => f) x^2) := by | |
| convert! noiseTensor_integrable_inner (fun _ : ι => f) | |
| (fun _ : ι => f) (fun _ => ((hf.mul hf).integrable_of_hasCompactSupport | |
| (hc.mul_right (f' := f)))) using 1 | |
| ext x | |
| simp only [pow_two] | |
| simpa only [affineNoiseRoot,div_pow] using | |
| (integrable_matrix_affine_inverse A a hA ht).div_const ((Real.sqrt A.det)^2) | |
| theorem integrable_square_add {X : Type*} [MeasurableSpace X] | |
| (μ : Measure X) {f g : X → ℝ} (hf : AEStronglyMeasurable f μ) | |
| (hg : AEStronglyMeasurable g μ) | |
| (hf2 : Integrable (fun x => f x^2) μ) (hg2 : Integrable (fun x => g x^2) μ) : | |
| Integrable (fun x => (f x+g x)^2) μ := by | |
| apply ((hf2.add hg2).const_mul 2).mono' ((hf.add hg).pow 2) | |
| filter_upwards with x | |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] | |
| simp only [Pi.add_apply,Pi.pow_apply] | |
| nlinarith [sq_nonneg (f x-g x)] | |
| end AffineIntegrability | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |