openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /ObservedExpandedAnnular.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Localization.MasterExpandedAnnularQuery | |
| namespace OAI | |
| noncomputable section | |
| namespace CoulombBarrier | |
| open MeasureTheory ProbabilityTheory Filter Set Metric | |
| open scoped BigOperators Topology ContDiff | |
| open CoulombAtom CoulombObservation | |
| lemma original_posterior_expanded_count_band_bound {N K : ℕ} | |
| (μ : Measure (Configuration N)) [IsFiniteMeasure μ] | |
| (ell : Fin K → ℝ) (hell : ∀ k, 0 ≤ ell k) (j : ℕ) (k : Fin K) | |
| (hk : j ≤ k.val) {B r : ℝ} (hB : 1 ≤ B) (hr : 0 < r) | |
| (hnoise : Real.sqrt 3*ell k ≤ r/4) : | |
| ∀ᵐ z ∂physicalObservationLaw μ K, | |
| (∫ x, rawAnnularCount (r/2) (2*B*r) x ∂originalRawKernel μ ell j (originalDatum ell j z)) ≤ | |
| observedAnnularCount ell k (r/4) (3*B*r) z := by | |
| filter_upwards [original_posterior_annular_count_le_observed μ ell hell j k hk (r/2) (2*B*r)] with z hz | |
| apply hz.trans | |
| exact rawAnnularCount_mono (by linarith) (by nlinarith) _ | |
| theorem exists_observed_posterior_expanded_density_constant {g : Space → ℝ} (hg : Continuous g) | |
| (hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B) (hc : 0 < c₁) | |
| (hcL : c₁ < (10*(100000:ℝ))⁻¹) : | |
| ∃ D : ℝ, 0 ≤ D ∧ ∀ {N K : ℕ} (μ : Measure (Configuration N)), | |
| ∀ [IsFiniteMeasure μ] (ell : Fin K → ℝ), (∀ k, 0 ≤ ell k) → | |
| ∀ (j : ℕ) (k : Fin K), j ≤ k.val → ∀ {r₀ s r T : ℝ}, | |
| 0 < r₀ → 0 < s → s ≤ 1 → r₀ ≤ s → r₀ ≤ 2*r → r ≤ s → | |
| Real.sqrt 3*ell k ≤ r/4 → | |
| ∀ᵐ z ∂physicalObservationLaw μ K, | |
| observedAnnularCount ell k (r/4) (3*B*r) z ≤ T/r^3 → | |
| ∀ y : Space, r ≤ ‖y‖ → ‖y‖ ≤ B*r → | |
| jointMasterPosterior μ ell j c₁ r₀ s g (originalDatum ell j z) y ≤ | |
| D*T*r^(-6-3*masterExponent) := by | |
| obtain ⟨D,hD,hbound⟩ := exists_posterior_expanded_annular_density_constant hg hgs hB hc hcL | |
| refine ⟨D,hD,?_⟩ | |
| intro N K μ hμ ell hell j k hk r₀ s r T hr₀ hs hs1 hr₀s hrr hrs hnoise | |
| have hr : 0 < r := by linarith | |
| filter_upwards [original_posterior_expanded_count_band_bound μ ell hell j k hk hB hr hnoise] with z hz | |
| intro hcount y hy hy2 | |
| exact (hbound μ ell j hr₀ hs hs1 hrr hrs _ y hy hy2).trans | |
| ((mul_le_mul_of_nonneg_left (hz.trans hcount) (by positivity)).trans_eq | |
| (master_density_count_scale hr D T)) | |
| theorem exists_observed_posterior_expanded_query_constant {g : Space → ℝ} | |
| (hg : ContDiff ℝ ∞ g) (hgs : tsupport g ⊆ ball 0 1) {B c₁ : ℝ} (hB : 1 ≤ B) | |
| (hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) : | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ {N K : ℕ} (μ : Measure (Configuration N)), | |
| ∀ [IsFiniteMeasure μ] (ell : Fin K → ℝ), (∀ k, 0 ≤ ell k) → | |
| ∀ (j : ℕ) (k : Fin K), j ≤ k.val → ∀ {r₀ s r T : ℝ}, | |
| 0 < r₀ → 0 < s → s ≤ 1 → r₀ ≤ s → r₀ ≤ 2*r → r ≤ s → | |
| Real.sqrt 3*ell k ≤ r/4 → | |
| ∀ᵐ q ∂physicalObservationLaw μ K, | |
| observedAnnularCount ell k (r/4) (3*B*r) q ≤ T/r^3 → | |
| ∀ y z : Space, r ≤ ‖y‖ → ‖y‖ ≤ B*r → r ≤ ‖z‖ → ‖z‖ ≤ B*r → | |
| ‖jointMasterPosterior μ ell j c₁ r₀ s g (originalDatum ell j q) y- | |
| jointMasterPosterior μ ell j c₁ r₀ s g (originalDatum ell j q) z‖ ≤ | |
| C*T*r^(-7-4*masterExponent)*‖y-z‖ := by | |
| obtain ⟨C,hC,hbound⟩ := exists_posterior_expanded_annular_query_constant hg hgs hB hc hcL | |
| refine ⟨C,hC,?_⟩ | |
| intro N K μ hμ ell hell j k hk r₀ s r T hr₀ hs hs1 hr₀s hrr hrs hnoise | |
| have hr : 0 < r := by linarith | |
| filter_upwards [original_posterior_expanded_count_band_bound μ ell hell j k hk hB hr hnoise] with q hq | |
| intro hcount y z hy hy2 hz hz2 | |
| exact (hbound μ ell j hr₀ hs hs1 hr₀s hrr hrs _ y z hy hy2 hz hz2).trans | |
| ((mul_le_mul_of_nonneg_left (hq.trans hcount) (by positivity)).trans_eq | |
| (master_query_count_scale hr C T ‖y-z‖)) | |
| end CoulombBarrier | |
| open MeasureTheory Filter Set | |
| open scoped Topology ContDiff | |
| namespace CoulombAtom | |
| open CoulombAnalysis CoulombObservation CoulombBarrier | |
| attribute [local irreducible] graphComponent graphFormVector fermionGraph weakGraph fermionGraphValue | |
| lemma rpow_half_scale {u : ℝ} (hu : 0 ≤ u) (e : ℝ) : | |
| (u/2)^e = (1/2:ℝ)^e*u^e := by | |
| rw [show u/2 = (1/2:ℝ)*u by ring, Real.mul_rpow (by norm_num) hu] | |
| theorem exists_actual_expanded_conditional_regularity_constants {B c₁ : ℝ} | |
| (hB : 1 ≤ B) (hc : 0 < c₁) (hcL : c₁ < (10*(100000:ℝ))⁻¹) : | |
| ∃ T D L : ℝ, 0 < T ∧ 0 ≤ D ∧ 0 ≤ L ∧ ∀ {ι : Type*} {l : Filter ι} | |
| {r₀ s Z lam : ι → ℝ} {N K : ι → ℕ} {F : ∀ i, fermionGraph (N i)} {δ : ℝ}, | |
| Tendsto s l (𝓝 0) → (∀ᶠ i in l, 0 < r₀ i) → | |
| (∀ᶠ i in l, 0 ≤ Z i ∧ 0 < lam i ∧ | |
| OwnProbabilityTailTiltState (Z i) (lam i) (r₀ i) (K i) | |
| (fun k => tinyProbabilityFloor (Z i) ((2:ℝ)^k.val*r₀ i)) δ (F i)) → | |
| ∀ᶠ i in l, ∀ j : Fin (K i), (2:ℝ)^j.val*r₀ i ≤ s i → | |
| let u := (2:ℝ)^j.val*r₀ i | |
| let count := observedAnnularCount | |
| (fun k : Fin (K i) => dyadicObservationWidth (r₀ i) k) j (u/8) (3*B*u) | |
| Measurable[observationInformation | |
| (fun k : Fin (K i) => dyadicObservationWidth (r₀ i) k) j] count ∧ | |
| (physicalObservationLaw (graphRawLaw (F i)) (K i)).real | |
| {z | T/u^3 < count z} < u^40 ∧ | |
| ∀ᵐ q ∂physicalObservationLaw (graphRawLaw (F i)) (K i), | |
| count q ≤ T/u^3 → | |
| (∀ y : Space, u/2 ≤ ‖y‖ → ‖y‖ ≤ B*u → | |
| originalQueryDensity (F i) (r₀ i) j c₁ (r₀ i) (s i) q y ≤ | |
| D*u^(-6-3*masterExponent)) ∧ | |
| (∀ y z : Space, u/2 ≤ ‖y‖ → ‖y‖ ≤ B*u → u/2 ≤ ‖z‖ → ‖z‖ ≤ B*u → | |
| ‖originalQueryDensity (F i) (r₀ i) j c₁ (r₀ i) (s i) q y- | |
| originalQueryDensity (F i) (r₀ i) j c₁ (r₀ i) (s i) q z‖ ≤ | |
| L*u^(-7-4*masterExponent)*‖y-z‖) := by | |
| obtain ⟨T,hT,hcount⟩ := exists_actual_uniform_annular_good_constant | |
| (by norm_num : (0:ℝ) < 1/8) (by linarith : (0:ℝ) < 3*B) | |
| obtain ⟨D,hD,hdensity⟩ := exists_observed_posterior_expanded_density_constant | |
| canonicalRealPacket_smooth.continuous canonicalRealPacket_support (by linarith : 1 ≤ 2*B) hc hcL | |
| obtain ⟨L,hL,hquery⟩ := exists_observed_posterior_expanded_query_constant | |
| canonicalRealPacket_smooth canonicalRealPacket_support (by linarith : 1 ≤ 2*B) hc hcL | |
| refine ⟨T,D*T*(1/2:ℝ)^(-6-3*masterExponent),L*T*(1/2:ℝ)^(-7-4*masterExponent), | |
| hT,by positivity,by positivity,?_⟩ | |
| intro ι l r₀ s Z lam N K F δ hs0 hr₀ hstate | |
| have hn := uniform_observation_noise_eventually (by norm_num : (0:ℝ) < 1/8) hs0 | |
| filter_upwards [hcount hs0 hr₀ hstate,hr₀,hn, | |
| hs0.eventually (gt_mem_nhds (by norm_num : (0:ℝ) < 1))] with i hci hri hni hsi | |
| intro j hjs | |
| let u := (2:ℝ)^j.val*r₀ i | |
| have hru : r₀ i ≤ u := le_mul_of_one_le_left hri.le (one_le_pow₀ (by norm_num)) | |
| have hu : 0 < u := hri.trans_le hru | |
| have hs : 0 < s i := hu.trans_le hjs | |
| have hnoise : Real.sqrt 3*dyadicObservationWidth (r₀ i) j ≤ (u/2)/4 := by | |
| have hh := hni u hu hjs | |
| simpa only [dyadicObservationWidth,u,show (1.01:ℝ) = 101/100 by norm_num, | |
| one_div_mul_eq_div,div_div,show (2:ℝ)*4 = 8 by norm_num] using hh | |
| have hr₀s := hru.trans hjs | |
| have hrr : r₀ i ≤ 2*(u/2) := by linarith | |
| have hrs : u/2 ≤ s i := by linarith | |
| have henn : 0 ≤ u/2 := by positivity | |
| have hshape : (u/2)/4 = u/8 := by ring | |
| have hshape' : 3*(2*B)*(u/2) = 3*B*u := by ring | |
| have hshape'' : (2*B)*(u/2) = B*u := by ring | |
| refine ⟨observedAnnularCount_information_measurable _ j le_rfl _ _,?_,?_⟩ | |
| · simpa only [one_div_mul_eq_div] using hci j hjs | |
| · filter_upwards [hdensity (graphRawLaw (F i)) _ (fun k => (dyadicObservationWidth_pos hri k).le) | |
| j j le_rfl hri hs hsi.le hr₀s hrr hrs hnoise, | |
| hquery (graphRawLaw (F i)) _ (fun k => (dyadicObservationWidth_pos hri k).le) | |
| j j le_rfl hri hs hsi.le hr₀s hrr hrs hnoise] with q hd hq | |
| intro hcut | |
| have hcut' : observedAnnularCount (fun k : Fin (K i) => dyadicObservationWidth (r₀ i) k) | |
| j ((u/2)/4) (3*(2*B)*(u/2)) q ≤ T/(u/2)^3 := by | |
| rw [hshape,hshape'] | |
| exact hcut.trans (div_le_div_of_nonneg_left hT.le (by positivity) | |
| (pow_le_pow_left₀ henn (by linarith : u/2 ≤ u) 3)) | |
| have hd' := hd hcut' | |
| have hq' := hq hcut' | |
| rw [hshape''] at hd' hq' | |
| constructor | |
| · intro y hyl hyh | |
| simpa only [rpow_half_scale hu.le,mul_assoc,u,originalQueryDensity] using hd' y hyl hyh | |
| · intro y z hyl hyh hzl hzh | |
| simpa only [rpow_half_scale hu.le,mul_assoc,u,originalQueryDensity] using hq' y z hyl hyh hzl hzh | |
| end CoulombAtom | |
| end | |
| end OAI | |