openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CompactMomentPerturbation.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.CompactFiberPrimitive | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Matrix Filter Topology | |
| variable {ι P : Type*} | |
| [TopologicalSpace P] [FirstCountableTopology P] [LocallyCompactSpace P] | |
| def compactCrossMoment (χ : ℝ → ℝ) (w v : ℝ → ι → ℝ) : Matrix ι ι ℝ := | |
| fun i j => ∫ s, χ s*w s j*v s i | |
| lemma compactCrossMoment_restrict (χ : ℝ → ℝ) (w v : ℝ → ι → ℝ) (i j : ι) : | |
| compactCrossMoment χ w v i j=∫ s in tsupport χ, χ s*w s j*v s i := by | |
| dsimp only [compactCrossMoment] | |
| nth_rw 1 [←setIntegral_univ] | |
| apply setIntegral_eq_of_subset_of_forall_sdiff_eq_zero MeasurableSet.univ (subset_univ _) | |
| intro s hs | |
| simp only [image_eq_zero_of_notMem_tsupport hs.2,zero_mul] | |
| lemma compactCrossMoment_continuous {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} | |
| {v : P → ℝ → ι → ℝ} (hc : Continuous χ) (hs : HasCompactSupport χ) | |
| (hw : ∀ j,Continuous (fun s => w s j)) | |
| (hv : ∀ i,Continuous (fun ps : P × ℝ => v ps.1 ps.2 i)) : | |
| Continuous (fun p => compactCrossMoment χ w (v p)) := by | |
| apply continuous_pi; intro i | |
| apply continuous_pi; intro j | |
| simp only [compactCrossMoment_restrict] | |
| apply continuous_parametric_integral_of_continuous (μ := volume) _ hs | |
| exact ((hc.comp continuous_snd).mul ((hw j).comp continuous_snd)).mul (hv i) | |
| lemma compactCrossMoment_self (χ : ℝ → ℝ) (w : ℝ → ι → ℝ) : | |
| compactCrossMoment χ w w=compactMomentGram χ w := by | |
| ext i j | |
| apply integral_congr_ae | |
| exact Filter.Eventually.of_forall (fun s => by dsimp; ring) | |
| variable [Fintype ι] [DecidableEq ι] | |
| lemma compactCrossMoment_solve {χ : ℝ → ℝ} {w v : ℝ → ι → ℝ} | |
| (hc : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (hs : HasCompactSupport χ) | |
| (hw : ∀ j,ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => w s j)) | |
| (hv : ∀ i,Continuous (fun s => v s i)) | |
| (hM : IsUnit (compactCrossMoment χ w v)) (b : ι → ℝ) : | |
| ∃ a : ι → ℝ, compactCrossMoment χ w v*ᵥa=b ∧ | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => χ s*(a ⬝ᵥ w s)) ∧ | |
| HasCompactSupport (fun s => χ s*(a ⬝ᵥ w s)) ∧ | |
| tsupport (fun s => χ s*(a ⬝ᵥ w s))⊆tsupport χ ∧ | |
| (∀ i,(∫ s,χ s*(a ⬝ᵥ w s)*v s i)=b i) := by | |
| obtain ⟨a,ha⟩ := Matrix.mulVec_surjective_iff_isUnit.mpr hM b | |
| refine ⟨a,ha,hc.mul (ContDiff.sum fun j _ => contDiff_const.mul (hw j)), | |
| hs.mul_right,tsupport_mul_subset_left,?_⟩ | |
| intro i | |
| rw [←congrFun ha i] | |
| change (∫ s,χ s*(a ⬝ᵥ w s)*v s i)=∑ j,compactCrossMoment χ w v i j*a j | |
| calc | |
| _ = ∫ s,∑ j,(χ s*w s j*v s i)*a j := by | |
| apply integral_congr_ae | |
| exact Filter.Eventually.of_forall fun s => by | |
| simp only [dotProduct,Finset.mul_sum,Finset.sum_mul] | |
| apply Finset.sum_congr rfl; intro j _; ring | |
| _ = _ := by | |
| have hi (j : ι) : Integrable (fun s => (χ s*w s j*v s i)*a j) := by | |
| exact (((hc.continuous.mul (hw j).continuous).mul (hv i)).integrable_of_hasCompactSupport | |
| (hs.mul_right.mul_right)).mul_const _ | |
| rw [integral_finsetSum _ (fun j _ => hi j)] | |
| simp only [integral_mul_const,compactCrossMoment] | |
| theorem compactMoment_perturbed_surjective {χ : ℝ → ℝ} {w : ℝ → ι → ℝ} | |
| {v : P → ℝ → ι → ℝ} {p₀ : P} | |
| (hc : ContDiff ℝ (↑(⊤ : ℕ∞)) χ) (hs : HasCompactSupport χ) | |
| (hw : ∀ j,ContDiff ℝ (↑(⊤ : ℕ∞)) (fun s => w s j)) | |
| (hv : ∀ i,Continuous (fun ps : P × ℝ => v ps.1 ps.2 i)) | |
| (hlim : v p₀=w) (hG : (compactMomentGram χ w).PosDef) : | |
| ∀ᶠ p in 𝓝 p₀,∀ b : ι → ℝ, | |
| ∃ f : ℝ → ℝ,ContDiff ℝ (↑(⊤ : ℕ∞)) f ∧ HasCompactSupport f ∧ | |
| tsupport f⊆tsupport χ ∧ (∀ i,(∫ s,f s*v p s i)=b i) := by | |
| have hM := compactCrossMoment_continuous hc.continuous hs (fun j => (hw j).continuous) hv | |
| have hM₀ : IsUnit (compactCrossMoment χ w (v p₀)) := by | |
| rw [hlim,compactCrossMoment_self] | |
| exact hG.isUnit | |
| have hd₀ : (compactCrossMoment χ w (v p₀)).det≠0 := | |
| isUnit_iff_ne_zero.mp ((Matrix.isUnit_iff_isUnit_det _).mp hM₀) | |
| have hed : ∀ᶠ p in 𝓝 p₀,(compactCrossMoment χ w (v p)).det≠0 := | |
| (isOpen_ne_fun hM.matrix_det continuous_const).mem_nhds hd₀ | |
| have he : ∀ᶠ p in 𝓝 p₀,IsUnit (compactCrossMoment χ w (v p)) := | |
| hed.mono (fun p hp => (Matrix.isUnit_iff_isUnit_det _).mpr (isUnit_iff_ne_zero.mpr hp)) | |
| filter_upwards [he] with p hp | |
| intro b | |
| obtain ⟨a,ha,hf,hfs,hsub,hb⟩ := compactCrossMoment_solve hc hs hw | |
| (fun i => (hv i).comp (continuous_const.prodMk continuous_id)) hp b | |
| exact ⟨_,hf,hfs,hsub,hb⟩ | |
| end ScalarConductivity | |
| end | |
| end OAI | |