openAI-math-dataset / lean /OAI /Analysis /CoulombIonization /FieldAnalysis /ConditionalFieldIdentity.lean
| import Mathlib | |
| import OAI.Analysis.CoulombIonization.Variational.CutCoreNumber | |
| import OAI.Analysis.CoulombIonization.Variational.CorePriceExcess | |
| import OAI.Analysis.CoulombIonization.FormDomain.BlockEnergetics | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter | |
| open scoped Topology BigOperators ContDiff | |
| open MeasureTheory Filter Set Metric | |
| open scoped BigOperators ContDiff | |
| namespace CoulombAtom | |
| lemma coreCoulombAt_mass_zero {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (hm : formMass ψ = 0) (y : Space) : coreCoulombAt ψ y = 0 := by | |
| have hz (s : Spins N) : (∫ x, ‖ψ.value s x‖^2) = 0 := | |
| (Finset.sum_eq_zero_iff_of_nonneg (fun s _ => integral_nonneg fun x => sq_nonneg _)).mp hm s | |
| (Finset.mem_univ _) | |
| unfold coreCoulombAt | |
| apply Finset.sum_eq_zero | |
| intro s _ | |
| apply Finset.sum_eq_zero | |
| intro i _ | |
| have he := (integral_eq_zero_iff_of_nonneg (fun x => sq_nonneg _) | |
| (hψ.1 s).norm.integrable_sq).mp (hz s) | |
| calc | |
| _ = ∫ _x : Configuration N, (0:ℝ) := integral_congr_ae (he.mono fun x hx => by simp only [hx,Pi.zero_apply,zero_div]) | |
| _ = 0 := integral_zero _ _ | |
| lemma normalizedCoreField_mul_mass {N : ℕ} {ψ : FormVector N} (hψ : SobolevVector ψ) | |
| (Z lam : ℝ) (y : Space) : | |
| normalizedCoreField Z lam ψ y*formMass ψ = | |
| Z*formMass ψ/‖y‖-coreCoulombAt ψ y-lam*formMass ψ := by | |
| by_cases hm : formMass ψ = 0 | |
| · simp only [hm,mul_zero,zero_div,coreCoulombAt_mass_zero hψ hm,sub_zero] | |
| · unfold normalizedCoreField | |
| rw [sub_mul,div_mul_cancel₀ _ hm] | |
| lemma coreSlice_mass_div {N M : ℕ} (ψ : FormVector (N+M)) (t : Spins M) | |
| (y : Configuration M) (j : Fin M) : | |
| formMass (coreSlice ψ t y)/‖y j‖ = | |
| ∑ s : Spins N, ∫ x : Configuration N, ‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖y j‖ := by | |
| simp only [formMass,Finset.sum_div,integral_div,coreSlice] | |
| lemma coreSlice_mass_div_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (t : Spins M) (j : Fin M) : | |
| Integrable (fun y : Configuration M => formMass (coreSlice ψ t y)/‖y j‖) := by | |
| simp_rw [coreSlice_mass_div] | |
| apply integrable_finsetSum | |
| intro s _ | |
| have hh := (integrable_join (N := N) (M := M) (nuclear_integrable | |
| (finSumFinEquiv (Sum.inr j)) (hψ.1 (joinLists s t)) | |
| (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right | |
| simpa only [joinLists_right] using hh | |
| lemma coreSlice_cross_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (t : Spins M) (j : Fin M) : | |
| Integrable (fun y : Configuration M => coreCoulombAt (coreSlice ψ t y) (y j)) := by | |
| unfold coreCoulombAt | |
| apply integrable_finsetSum | |
| intro s _ | |
| apply integrable_finsetSum | |
| intro i _ | |
| have hne : (finSumFinEquiv (Sum.inl i) : Fin (N+M)) ≠ finSumFinEquiv (Sum.inr j) := by | |
| intro h | |
| have hh := finSumFinEquiv.injective h | |
| cases hh | |
| have hh := (integrable_join (N := N) (M := M) (pair_integrable _ _ hne | |
| (hψ.1 (joinLists s t)) (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right | |
| simpa only [coreSlice,joinLists_left,joinLists_right] using hh | |
| lemma integral_coreSlice_mass_div {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) : | |
| (∑ t : Spins M, ∫ y : Configuration M, ∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖) = | |
| outNuclear ψ := by | |
| have hi (s : Spins N) (t : Spins M) (j : Fin M) : | |
| Integrable (fun y : Configuration M => ∫ x : Configuration N, | |
| ‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖y j‖) := by | |
| simpa only [joinLists_right] using (integrable_join (N := N) (M := M) | |
| (nuclear_integrable (finSumFinEquiv (Sum.inr j)) (hψ.1 (joinLists s t)) | |
| (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right | |
| simp_rw [integral_finsetSum _ (fun j _ => coreSlice_mass_div_integrable hψ _ j), | |
| coreSlice_mass_div,integral_finsetSum _ (fun s _ => hi s _ _)] | |
| have he (s : Spins N) (t : Spins M) (j : Fin M) : | |
| (∫ y : Configuration M, ∫ x : Configuration N, | |
| ‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖y j‖) = | |
| ∫ z, ‖ψ.value (joinLists s t) z‖^2/‖z (finSumFinEquiv (Sum.inr j))‖ := by | |
| simpa only [joinLists_right] using integral_join (N := N) (M := M) | |
| (nuclear_integrable (finSumFinEquiv (Sum.inr j)) (hψ.1 (joinLists s t)) | |
| (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _)) | |
| simp_rw [he] | |
| conv_lhs => arg 2; ext t; rw [Finset.sum_comm] | |
| exact sum_spin_join (fun s => ∑ j : Fin M, ∫ z, | |
| ‖ψ.value s z‖^2/‖z (finSumFinEquiv (Sum.inr j))‖) | |
| lemma integral_coreSlice_cross {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) : | |
| (∑ t : Spins M, ∫ y : Configuration M, ∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j)) = | |
| crossRepulsion ψ := by | |
| have hi (s : Spins N) (t : Spins M) (i : Fin N) (j : Fin M) : | |
| Integrable (fun y : Configuration M => ∫ x : Configuration N, | |
| ‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖x i-y j‖) := by | |
| have hne : (finSumFinEquiv (Sum.inl i) : Fin (N+M)) ≠ finSumFinEquiv (Sum.inr j) := by | |
| intro h | |
| have hh := finSumFinEquiv.injective h | |
| cases hh | |
| simpa only [joinLists_right,joinLists_left] using (integrable_join (N := N) (M := M) | |
| (pair_integrable _ _ hne (hψ.1 (joinLists s t)) | |
| (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _))).integral_prod_right | |
| simp_rw [integral_finsetSum _ (fun j _ => coreSlice_cross_integrable hψ _ j),coreCoulombAt,coreSlice] | |
| simp_rw [integral_finsetSum _ (fun s _ => integrable_finsetSum _ (fun i _ => hi s _ i _)), | |
| integral_finsetSum _ (fun i _ => hi _ _ i _)] | |
| have he (s : Spins N) (t : Spins M) (i : Fin N) (j : Fin M) : | |
| (∫ y : Configuration M, ∫ x : Configuration N, | |
| ‖ψ.value (joinLists s t) (joinLists x y)‖^2/‖x i-y j‖) = | |
| ∫ z, ‖ψ.value (joinLists s t) z‖^2/ | |
| ‖z (finSumFinEquiv (Sum.inl i))-z (finSumFinEquiv (Sum.inr j))‖ := by | |
| have hne : (finSumFinEquiv (Sum.inl i) : Fin (N+M)) ≠ finSumFinEquiv (Sum.inr j) := by | |
| intro h | |
| have hh := finSumFinEquiv.injective h | |
| cases hh | |
| simpa only [joinLists_right,joinLists_left] using integral_join (N := N) (M := M) | |
| (pair_integrable _ _ hne (hψ.1 (joinLists s t)) | |
| (hψ.2.1 (joinLists s t) _) (hψ.2.2 (joinLists s t) _)) | |
| simp_rw [he] | |
| conv_lhs => arg 2; ext t; rw [Finset.sum_comm]; arg 2; ext s; rw [Finset.sum_comm] | |
| exact sum_spin_join (fun s => ∑ i : Fin N, ∑ j : Fin M, ∫ z, | |
| ‖ψ.value s z‖^2/‖z (finSumFinEquiv (Sum.inl i))-z (finSumFinEquiv (Sum.inr j))‖) | |
| def conditionalFieldSum {N M : ℕ} (Z lam : ℝ) (ψ : FormVector (N+M)) | |
| (t : Spins M) (y : Configuration M) : ℝ := | |
| formMass (coreSlice ψ t y)*∑ j : Fin M, | |
| normalizedCoreField Z lam (coreSlice ψ t y) (y j) | |
| lemma conditionalFieldSum_ae {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) (t : Spins M) : | |
| conditionalFieldSum Z lam ψ t =ᵐ[volume] | |
| fun y => Z*(∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖)- | |
| (∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j))- | |
| lam*M*formMass (coreSlice ψ t y) := by | |
| apply (hψ.ae_coreSlice t).mono | |
| intro y hy | |
| unfold conditionalFieldSum | |
| rw [Finset.mul_sum] | |
| simp_rw [mul_comm (formMass _),normalizedCoreField_mul_mass hy,mul_div_assoc] | |
| simp only [Finset.sum_sub_distrib,←Finset.mul_sum,Finset.sum_const, | |
| Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] | |
| ring | |
| lemma conditionalFieldSum_integrable {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) (t : Spins M) : | |
| Integrable (conditionalFieldSum Z lam ψ t) := by | |
| have hi := ((integrable_finsetSum Finset.univ (fun j _ => coreSlice_mass_div_integrable hψ t j)).const_mul Z).sub | |
| (integrable_finsetSum Finset.univ (fun j _ => coreSlice_cross_integrable hψ t j)) | |
| exact (hi.sub ((hψ.coreSlice_mass_integrable t).const_mul (lam*M))).congr | |
| (conditionalFieldSum_ae hψ Z lam t).symm | |
| lemma integral_conditionalFieldSum {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) : | |
| (∑ t : Spins M, ∫ y, conditionalFieldSum Z lam ψ t y) = | |
| Z*outNuclear ψ-crossRepulsion ψ-lam*M*formMass ψ := by | |
| have ht (t : Spins M) : (∫ y, conditionalFieldSum Z lam ψ t y) = | |
| Z*(∫ y, ∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖)- | |
| (∫ y, ∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j))- | |
| lam*M*(∫ y, formMass (coreSlice ψ t y)) := by | |
| rw [integral_congr_ae (conditionalFieldSum_ae hψ Z lam t)] | |
| have hi := (integrable_finsetSum Finset.univ (fun j _ => coreSlice_mass_div_integrable hψ t j)).const_mul Z | |
| have hj := integrable_finsetSum Finset.univ (fun j _ => coreSlice_cross_integrable hψ t j) | |
| rw [integral_sub (f := fun y => Z*(∑ j : Fin M, formMass (coreSlice ψ t y)/‖y j‖)- | |
| (∑ j : Fin M, coreCoulombAt (coreSlice ψ t y) (y j))) | |
| (g := fun y => lam*M*formMass (coreSlice ψ t y)) | |
| (hi.sub hj) ((hψ.coreSlice_mass_integrable t).const_mul (lam*M)), | |
| integral_sub hi hj,integral_const_mul,integral_const_mul] | |
| simp_rw [ht] | |
| rw [Finset.sum_sub_distrib,Finset.sum_sub_distrib,←Finset.mul_sum,←Finset.mul_sum, | |
| integral_coreSlice_mass_div hψ,integral_coreSlice_cross hψ,hψ.integral_coreSlice_mass] | |
| theorem priced_conditional_energy_identity {N M : ℕ} {ψ : FormVector (N+M)} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) : | |
| formEnergy Z ψ+lam*(N+M)*formMass ψ = | |
| (∑ t : Spins M, ∫ y, formEnergy Z (coreSlice ψ t y)+ | |
| lam*N*formMass (coreSlice ψ t y))+ | |
| outKinetic ψ-(∑ t : Spins M, ∫ y, conditionalFieldSum Z lam ψ t y)+ | |
| outRepulsion ψ := by | |
| have ht (t : Spins M) : (∫ y, formEnergy Z (coreSlice ψ t y)+ | |
| lam*N*formMass (coreSlice ψ t y)) = | |
| (∫ y, formEnergy Z (coreSlice ψ t y))+lam*N*(∫ y, formMass (coreSlice ψ t y)) := by | |
| rw [integral_add (hψ.coreSlice_energy_integrable Z t) | |
| ((hψ.coreSlice_mass_integrable t).const_mul (lam*N)),integral_const_mul] | |
| simp_rw [ht] | |
| rw [Finset.sum_add_distrib,←Finset.mul_sum,hψ.integral_coreSlice_mass, | |
| integrated_core_parts hψ Z,integral_conditionalFieldSum hψ Z lam, | |
| formEnergy_core_out_cross] | |
| ring | |
| theorem fresh_cut_priced_localization_identity {L : ℕ} | |
| (p : Fin 2 → SmoothMultiplier spaceDirections) | |
| (hp : ∀ x, ∑ a, (p a).value x^2 = 1) {ψ : FormVector L} | |
| (hψ : SobolevVector ψ) (Z lam : ℝ) : | |
| formEnergy Z ψ+lam*L*formMass ψ+ | |
| (1/2:ℝ)*weightedParticleCount ψ (spatialErrorWeight p) = | |
| (∑ b : Fin L → Fin 2, ∑ t : Spins (cutOutNumber b), ∫ y, | |
| formEnergy Z (coreSlice (orderedCutForm p hp ψ b) t y)+ | |
| lam*cutCoreNumber b*formMass (coreSlice (orderedCutForm p hp ψ b) t y))+ | |
| (∑ b : Fin L → Fin 2, outKinetic (orderedCutForm p hp ψ b))- | |
| (∑ b : Fin L → Fin 2, ∑ t : Spins (cutOutNumber b), ∫ y, | |
| conditionalFieldSum Z lam (orderedCutForm p hp ψ b) t y)+ | |
| (∑ b : Fin L → Fin 2, outRepulsion (orderedCutForm p hp ψ b)) := by | |
| have hb (b : Fin L → Fin 2) := priced_conditional_energy_identity | |
| (orderedCutForm_sobolev p hp hψ b) Z lam | |
| have hcard (b : Fin L → Fin 2) : (cutCoreNumber b : ℝ)+cutOutNumber b = L := by | |
| exact_mod_cast cutNumbers_sum b | |
| simp_rw [hcard,orderedCutForm_energy,orderedCutForm_mass] at hb | |
| have hsum := congrArg (fun f : (Fin L → Fin 2) → ℝ => ∑ b, f b) (funext hb) | |
| simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib] at hsum | |
| rw [←Finset.mul_sum,spatial_cut_mass p hp hψ,spatial_cut_ims p hp hψ Z] at hsum | |
| have hw : (∑ s : Spins L, ∑ i : Fin L, ∫ x, | |
| spatialErrorWeight p (x i)*‖ψ.value s x‖^2) = | |
| weightedParticleCount ψ (spatialErrorWeight p) := rfl | |
| rw [hw] at hsum | |
| linarith | |
| end CoulombAtom | |
| end | |
| end OAI | |