openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /Localization /InverseNetEventsBarrier.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.RadialBounds.ActualBandInverseBarrier | |
| import OAI.Analysis.CoulombIonization.RadialBounds.OwnProbabilityRareCapBarrier | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter Set Metric | |
| open scoped Topology BigOperators | |
| namespace CoulombAtom | |
| open CoulombAnalysis CoulombObservation CoulombBarrier | |
| attribute [local irreducible] graphComponent graphFormVector fermionGraph weakGraph fermionGraphValue | |
| attribute [local irreducible] physicalObservationLaw jointMasterPosterior | |
| lemma originalQueryDensity_info_measurable {N K : ℕ} (F : fermionGraph N) | |
| (r : ℝ) (j : ℕ) {c₁ r₀ s : ℝ} (hc : 0 < c₁) (hr₀ : 0 < r₀) | |
| (hs : 0 < s) (y : Space) : | |
| Measurable[observationInformation (fun k : Fin K => dyadicObservationWidth r k) j] | |
| (fun z => originalQueryDensity F r j c₁ r₀ s z y) := by | |
| exact (jointMasterPosterior_measurable (graphRawLaw F) | |
| (fun k : Fin K => dyadicObservationWidth r k) j hc hr₀ hs | |
| canonicalRealPacket_smooth.continuous).comp (measurable_id.prodMk measurable_const) | |
| lemma original_inverse_failure_info_measurable {N K : ℕ} (F : fermionGraph N) | |
| (Z lam r₀ s c₁ h xi : ℝ) (j : Fin K) (y : Space) | |
| (hc : 0 < c₁) (hr₀ : 0 < r₀) (hs : 0 < s) : | |
| MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r₀ k) j] | |
| (originalLowFailure F Z lam r₀ s c₁ h xi j y ∪ | |
| originalHighFailure F Z lam r₀ s c₁ h xi j y) := by | |
| have hf := originalQueryField_info_measurable (K := K) F Z lam r₀ j hc hr₀ hs y | |
| have hd := originalQueryDensity_info_measurable (K := K) F r₀ j hc hr₀ hs y | |
| exact ((measurableSet_le measurable_const hf).inter (measurableSet_le hd measurable_const)).union | |
| ((measurableSet_le hf measurable_const).inter (measurableSet_le measurable_const hd)) | |
| lemma original_inverse_height_net_union_bound {N K : ℕ} {F : fermionGraph N} | |
| {Z lam r₀ s c₁ xi : ℝ} {j : Fin K} (Q : Finset Space) (H : Finset ℝ) | |
| (hQ : ∀ h ∈ H, ∀ y ∈ Q, OriginalPointInverseBounds F Z lam r₀ s c₁ h xi j y) : | |
| (physicalObservationLaw (graphRawLaw F) K).real | |
| (⋃ h ∈ H, ⋃ y ∈ Q, originalLowFailure F Z lam r₀ s c₁ h xi j y ∪ | |
| originalHighFailure F Z lam r₀ s c₁ h xi j y) ≤ | |
| 4*(H.card:ℝ)*(Q.card:ℝ)*((2:ℝ)^j.val*r₀)^40 := by | |
| apply (measureReal_biUnion_finset_le H _).trans | |
| calc | |
| _ ≤ ∑ _h ∈ H, 4*(Q.card:ℝ)*((2:ℝ)^j.val*r₀)^40 := by | |
| exact Finset.sum_le_sum (fun h hh => original_inverse_net_union_bound Q (hQ h hh)) | |
| _ = _ := by simp only [Finset.sum_const, nsmul_eq_mul]; ring | |
| lemma exists_inverse_band_spatial_net {u : ℝ} (hu : 0 < u) (hu1 : u ≤ 1) : | |
| ∃ Q : Finset Space, | |
| (∀ x ∈ Q, 11*u/8 ≤ ‖x‖ ∧ ‖x‖ ≤ 13*u/8) ∧ | |
| (∀ y : Space, 11*u/8 ≤ ‖y‖ → ‖y‖ ≤ 13*u/8 → | |
| ∃ x ∈ Q, ‖x-y‖ ≤ u^2) ∧ | |
| (Q.card:ℝ) ≤ 4913/u^3 := by | |
| obtain ⟨Q,hQ,hcover,hcard⟩ := exists_spatial_grid_net | |
| (A := {x : Space | 11*u/8 ≤ ‖x‖ ∧ ‖x‖ ≤ 13*u/8}) | |
| (R := 2*u) (h := u^2/3) (by positivity) (by positivity) | |
| (fun _ hx => by linarith [hx.2]) | |
| refine ⟨Q,hQ,?_,hcard.trans ?_⟩ | |
| · intro y hyl hyh | |
| obtain ⟨x,hx,hxy⟩ := hcover y ⟨hyl,hyh⟩ | |
| refine ⟨x,hx,?_⟩ | |
| rw [norm_sub_rev] at hxy | |
| nlinarith | |
| · have hlin : 2*(2*u/(u^2/3))+5 ≤ 17/u := by | |
| apply (le_div_iff₀ hu).mpr | |
| field_simp | |
| nlinarith | |
| calc _ ≤ (17/u)^3 := pow_le_pow_left₀ (by positivity) hlin 3 | |
| _ = _ := by ring | |
| lemma divided_net_field_variation {a b alpha beta t : ℝ} | |
| (ht : 1 ≤ t) (ha : 0 ≤ alpha) (hv : |b-a| ≤ alpha+beta*max (-a) 0) : | |
| |b/t-a/t| ≤ alpha+beta*max (-(a/t)) 0 := by | |
| have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht | |
| have hm : max (-(a/t)) 0 = max (-a) 0/t := by | |
| simpa only [neg_div,zero_div] using max_div_div_right ht0.le (-a) 0 | |
| rw [←sub_div,abs_div,abs_of_pos ht0,hm] | |
| calc | |
| _ ≤ (alpha+beta*max (-a) 0)/t := div_le_div_of_nonneg_right hv ht0.le | |
| _ = alpha/t+beta*(max (-a) 0/t) := by ring | |
| _ ≤ _ := add_le_add (div_le_self ha ht) le_rfl | |
| lemma localCellRadius_normalized {y : Space} (p : ℕ) (a : ℝ) : | |
| (localCellRadius y)^p*a = (‖y‖^p*a)/(100000:ℝ)^p := by | |
| simp only [localCellRadius,div_pow] | |
| ring | |
| end CoulombAtom | |
| end | |
| end OAI | |