openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /CellFieldNormalizationBarrier.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.RadialBounds.CellCountSupremumBarrier | |
| import OAI.Analysis.CoulombIonization.FieldAnalysis.ThinFieldConstantsBarrier | |
| import OAI.Analysis.CoulombIonization.RadialBounds.CellScaleBarrier | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| namespace CoulombAtom | |
| lemma thinFieldBase_antitone {a A b D B : ℝ} (ha : 0 < a) (haA : a ≤ A) | |
| (hb : 0 < b) (hD : 0 ≤ D) (hB : 0 ≤ B) : | |
| thinFieldBase A b D B ≤ thinFieldBase a b D B := by | |
| have hA := lt_of_lt_of_le ha haA | |
| have hI := localizationIMSConstant_nonneg | |
| unfold thinFieldBase localFieldScale | |
| gcongr | |
| lemma sqrt_count_scale {B C P m : ℝ} (_hB : 0 ≤ B) (hC : 1 ≤ C) (hP : 0 ≤ P) | |
| (hm : 0 ≤ m) (hh : B ≤ C*P*m^2) : Real.sqrt B ≤ C*Real.sqrt P*m := by | |
| have hC0 : 0 ≤ C := le_trans zero_le_one hC | |
| have hcc : C ≤ C^2 := by nlinarith | |
| have hs := Real.sq_sqrt hP | |
| have he : (C*Real.sqrt P*m)^2 = C^2*P*m^2 := by rw [mul_pow,mul_pow,hs] | |
| apply (Real.sqrt_le_iff).mpr | |
| refine ⟨by positivity,hh.trans ?_⟩ | |
| rw [he] | |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right hcc hP) (sq_nonneg m) | |
| lemma thinFieldBase_normalized {a m e D B C P : ℝ} | |
| (ha : 0 < a) (hm : 1 ≤ m) (h3 : 1/a^3 ≤ m) (he : 0 < e) | |
| (_hD : 0 ≤ D) (hDm : D*a ≤ m^2) (hB : 0 ≤ B) (hC : 1 ≤ C) | |
| (hP : 1 ≤ P) (hBC : B ≤ C*P*m^2) : | |
| thinFieldBase a (e*a) D B ≤ | |
| (6+C+localizationIMSConstant*C)*(m/a)^2*(P+1/e^4) := by | |
| let S := (m/a)^2 | |
| let V := P+1/e^4 | |
| have hm0 : 0 ≤ m := le_trans zero_le_one hm | |
| have hP0 : 0 ≤ P := le_trans zero_le_one hP | |
| have hC0 : 0 ≤ C := le_trans zero_le_one hC | |
| have hS : 0 ≤ S := sq_nonneg _ | |
| have hV : 1 ≤ V := by dsimp [V]; have := one_div_nonneg.mpr (by positivity : 0 ≤ e^4); linarith | |
| have hPV : P ≤ V := by dsimp [V]; have := one_div_nonneg.mpr (by positivity : 0 ≤ e^4); linarith | |
| have hI := localizationIMSConstant_nonneg | |
| have hrec := cell_reciprocal_bounds ha hm h3 | |
| have ham := cell_mass_product ha hm h3 | |
| have hma : 0 ≤ m/a := div_nonneg hm0 ha.le | |
| have h4 : 1/a^4 ≤ m/a := by | |
| have hh := div_le_div_of_nonneg_right h3 ha.le | |
| calc _ = (1/a^3)/a := by ring | |
| _ ≤ m/a := hh | |
| have h1 : 1/a ≤ m/a := div_le_div_of_nonneg_right hm ha.le | |
| have hfirst : (1/a^4+1/a)^2 ≤ 4*S := by | |
| have h := pow_le_pow_left₀ (by positivity : 0 ≤ 1/a^4+1/a) (add_le_add h4 h1) 2 | |
| dsimp [S] | |
| nlinarith | |
| have hsecond : 1/a^4 ≤ S := by | |
| have h := pow_le_pow_left₀ (by positivity : 0 ≤ 1/a) hrec.1 2 | |
| have hh := div_le_div_of_nonneg_right h (sq_nonneg a) | |
| calc _ = (1/a)^2/a^2 := by ring | |
| _ ≤ m^2/a^2 := hh | |
| _ = S := by dsimp [S]; ring | |
| have hthird : D/a ≤ S := by | |
| have h := div_le_div_of_nonneg_right hDm (sq_nonneg a) | |
| calc _ = (D*a)/a^2 := by field_simp | |
| _ ≤ m^2/a^2 := h | |
| _ = S := by dsimp [S]; ring | |
| have hfourth : B/a^2 ≤ C*P*S := by | |
| have h := div_le_div_of_nonneg_right hBC (sq_nonneg a) | |
| calc _ ≤ (C*P*m^2)/a^2 := h | |
| _ = C*P*S := by dsimp [S]; ring | |
| have hsqrt := sqrt_count_scale hB hC hP0 hm0 hBC | |
| have hroot : Real.sqrt P/e^2 ≤ V := by | |
| have hs := Real.sq_sqrt hP0 | |
| have hx : (1/e^2)^2 = 1/e^4 := by ring | |
| have hh := sq_nonneg (Real.sqrt P-1/e^2) | |
| calc _ = Real.sqrt P*(1/e^2) := by ring | |
| _ ≤ V := by dsimp [V]; nlinarith | |
| have hnorm : m/a^3 ≤ S := by | |
| have hh := mul_le_mul_of_nonneg_right ham (show 0 ≤ m/a^3 by positivity) | |
| calc _ ≤ (a*m)*(m/a^3) := by simpa using hh | |
| _ = S := by dsimp [S]; field_simp | |
| have hims : localizationIMSConstant*Real.sqrt B/((e*a)^2*a) ≤ | |
| localizationIMSConstant*C*V*S := by | |
| calc _ ≤ localizationIMSConstant*(C*Real.sqrt P*m)/((e*a)^2*a) := by gcongr | |
| _ = localizationIMSConstant*C*(Real.sqrt P/e^2)*(m/a^3) := by ring | |
| _ ≤ localizationIMSConstant*C*V*S := by gcongr | |
| have ht1 := mul_le_mul_of_nonneg_right hV hS | |
| have ht2 := mul_le_mul_of_nonneg_right hPV (mul_nonneg hC0 hS) | |
| unfold thinFieldBase localFieldScale | |
| change (1/a^4+1/a)^2+1/a^4+D/a+B/a^2+_ ≤ (6+C+localizationIMSConstant*C)*S*V | |
| nlinarith | |
| def cellFieldNormConstant : ℝ := 1+thinFieldUniversalConstant* | |
| (6+localCountCoverConstant+localizationIMSConstant*localCountCoverConstant) | |
| lemma cellFieldNormConstant_one_le : 1 ≤ cellFieldNormConstant := by | |
| have hC := localCountCoverConstant_one_le | |
| have hI := localizationIMSConstant_nonneg | |
| have hT := thinFieldUniversalConstant_pos | |
| unfold cellFieldNormConstant | |
| have : 0 ≤ thinFieldUniversalConstant*(6+localCountCoverConstant+localizationIMSConstant*localCountCoverConstant) := by positivity | |
| linarith | |
| lemma cellField_norm_square {a m e D B P : ℝ} | |
| (ha : 0 < a) (hm : 1 ≤ m) (h3 : 1/a^3 ≤ m) (he : 0 < e) | |
| (hD : 0 ≤ D) (hDm : D*a ≤ m^2) (hB : 0 ≤ B) | |
| (hP : 1 ≤ P) (hBC : B ≤ localCountCoverConstant*P*m^2) : | |
| thinFieldUniversalConstant*thinFieldBase (2*a) (e*a) D B ≤ | |
| (cellFieldNormConstant*(m/a)*(Real.sqrt P+1/e^2))^2 := by | |
| have hm0 : 0 ≤ m := le_trans zero_le_one hm | |
| have hP0 : 0 ≤ P := le_trans zero_le_one hP | |
| have hC := localCountCoverConstant_one_le | |
| have hI := localizationIMSConstant_nonneg | |
| have hT := thinFieldUniversalConstant_pos.le | |
| let K := thinFieldUniversalConstant*(6+localCountCoverConstant+localizationIMSConstant*localCountCoverConstant) | |
| have hK : 0 ≤ K := by dsimp [K]; positivity | |
| have hF : K ≤ cellFieldNormConstant^2 := by change K ≤ (1+K)^2; nlinarith | |
| have hthin := (thinFieldBase_antitone ha (by linarith : a ≤ 2*a) (mul_pos he ha) hD hB).trans | |
| (thinFieldBase_normalized ha hm h3 he hD hDm hB hC hP hBC) | |
| have hV : P+1/e^4 ≤ (Real.sqrt P+1/e^2)^2 := by | |
| have hs := Real.sq_sqrt hP0 | |
| have hx : (1/e^2)^2 = 1/e^4 := by ring | |
| have hh : 0 ≤ Real.sqrt P*(1/e^2) := by positivity | |
| nlinarith | |
| calc _ ≤ K*(m/a)^2*(P+1/e^4) := by | |
| have hh := mul_le_mul_of_nonneg_left hthin hT | |
| calc _ ≤ thinFieldUniversalConstant* | |
| ((6+localCountCoverConstant+localizationIMSConstant*localCountCoverConstant)*(m/a)^2*(P+1/e^4)) := hh | |
| _ = K*(m/a)^2*(P+1/e^4) := by dsimp [K]; ring | |
| _ ≤ cellFieldNormConstant^2*(m/a)^2*(Real.sqrt P+1/e^2)^2 := by gcongr | |
| _ = _ := by ring | |
| theorem cell_fresh_field_mass {N : ℕ} {ψ : FormVector N} | |
| (hψ : SobolevFermion ψ) (hm : formMass ψ = 1) {y : Space} (hy : y ≠ 0) | |
| {e r Z lam : ℝ} (he : 0 < e) (he1 : e ≤ 1/8) | |
| (hr : r ∈ Set.Icc (10*localCellRadius y) (12*localCellRadius y)) | |
| (hZ : 0 ≤ Z) (hlam : 0 < lam) : | |
| let a := localCellRadius y | |
| let D := max (corePriceExcess Z lam ψ) 0 | |
| let m := localOffsetMass D y | |
| let P := localCountSupremum ψ D | |
| ∃ hr0 : 0 ≤ r, | |
| freshOutMaximumSecondMoment (coreFirstRadialCut y hr0 (mul_pos he (localCellRadius_pos hy))) | |
| (coreFirstRadialCut_partition y hr0 (mul_pos he (localCellRadius_pos hy))) ψ Z lam ≤ | |
| (cellFieldNormConstant*(m/a)*(Real.sqrt P+1/e^2))^2 ∧ | |
| freshPatchMass (coreFirstRadialCut y hr0 (mul_pos he (localCellRadius_pos hy))) | |
| (coreFirstRadialCut_partition y hr0 (mul_pos he (localCellRadius_pos hy))) ψ Z lam y (r-4*(e*a)) ≤ | |
| 1536*cellFieldNormConstant*m*(Real.sqrt P+1/e^2) := by | |
| dsimp only | |
| let a := localCellRadius y | |
| let D := max (corePriceExcess Z lam ψ) 0 | |
| let m := localOffsetMass D y | |
| let P := localCountSupremum ψ D | |
| have ha : 0 < a := localCellRadius_pos hy | |
| have hr0 : 0 ≤ r := le_trans (by positivity : 0 ≤ 10*a) hr.1 | |
| have hb : 0 < e*a := mul_pos he ha | |
| have hba : e*a ≤ 2*a := by nlinarith | |
| have hsep : 20*(2*a) ≤ ‖y‖ := by dsimp [a,localCellRadius]; nlinarith [norm_nonneg y] | |
| have hrb : 4*(e*a) < r := by nlinarith [hr.1] | |
| let Q := cellFieldNormConstant*(m/a)*(Real.sqrt P+1/e^2) | |
| have hQ : 0 ≤ Q := by | |
| have hF := cellFieldNormConstant_one_le | |
| have hm1 := localOffsetMass_one_le D y | |
| dsimp [Q] | |
| positivity | |
| have hsq := cellField_norm_square ha (localOffsetMass_one_le D y) (localOffsetMass_cube_le D y) he | |
| (le_max_right _ _) (localOffsetMass_offset (le_max_right _ _) hy) | |
| (rawCountMoment_nonneg ψ y (32*a)) (localCountSupremum_one_le ψ D) | |
| (enlarged_cell_count_le hψ.sobolevVector (le_max_right _ _) hy) | |
| change thinFieldUniversalConstant*thinFieldBase (2*a) (e*a) D (rawCountMoment ψ y (32*a)) ≤ Q^2 at hsq | |
| have hscale : 16*(2*a) = 32*a := by ring | |
| have hh := freshOutMaximum_thin_bound hψ hm y (by positivity : 0 < 2*a) hr0 (by nlinarith [hr.2]) hb hba hsep hZ hlam | |
| have hh' := freshPatchMass_thin_bound hψ hm y (by positivity : 0 < 2*a) hr0 (by nlinarith [hr.2]) hb hrb hba hsep hZ hlam | |
| rw [hscale] at hh hh' | |
| refine ⟨hr0,hh.trans hsq,?_⟩ | |
| have hs : Real.sqrt (thinFieldUniversalConstant*thinFieldBase (2*a) (e*a) D (rawCountMoment ψ y (32*a))) ≤ Q := | |
| Real.sqrt_le_iff.mpr ⟨hQ,hsq⟩ | |
| calc _ ≤ 768*(2*a)*Real.sqrt (thinFieldUniversalConstant*thinFieldBase (2*a) (e*a) D (rawCountMoment ψ y (32*a))) := hh' | |
| _ ≤ 768*(2*a)*Q := mul_le_mul_of_nonneg_left hs (by positivity) | |
| _ = 1536*cellFieldNormConstant*m*(Real.sqrt P+1/e^2) := by dsimp [Q]; field_simp; ring | |
| end CoulombAtom | |
| end | |
| end OAI | |