openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /PositiveResolventBounds.lean
| import OAI.Probability.InvariantIsing.Spectral.PositiveResolventSpectrum | |
| namespace OAI | |
| /-! Uniform trace and quadratic bounds for positive resolvents. -/ | |
| noncomputable section | |
| open Matrix | |
| open scoped BigOperators RealInnerProductSpace | |
| namespace InvariantIsing | |
| lemma positiveResolvent_trace_le {N : ℕ} {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) : | |
| (positiveResolvent t B).trace ≤ (N : ℝ)/t := by | |
| have he := (positiveResolvent_posDef ht hB).isHermitian.trace_eq_sum_eigenvalues | |
| simp only [RCLike.ofReal_real_eq_id,id_eq] at he | |
| rw [he] | |
| calc | |
| _ ≤ ∑ _ : Fin N, 1/t := Finset.sum_le_sum (fun i _ => (positiveResolvent_eigenvalues ht hB i).2) | |
| _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; ring | |
| lemma positiveResolvent_quadratic_le {N : ℕ} {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : EuclideanSpace ℝ (Fin N)) : | |
| gaussianQuadraticForm (positiveResolvent t B) x ≤ ‖x‖^2/t := by | |
| let hR := (positiveResolvent_posDef ht hB).isHermitian | |
| rw [gaussianQuadraticForm,hermitian_quadratic_coordinates _ hR] | |
| calc | |
| _ ≤ ∑ i, (1/t)*(hR.eigenvectorBasis.repr x i)^2 := | |
| Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_right | |
| (positiveResolvent_eigenvalues ht hB i).2 (sq_nonneg _)) | |
| _ = (1/t)*‖hR.eigenvectorBasis.repr x‖^2 := by | |
| rw [EuclideanSpace.real_norm_sq_eq,Finset.mul_sum] | |
| _ = _ := by rw [LinearIsometryEquiv.norm_map]; ring | |
| lemma positiveResolvent_quadratic_nonneg {N : ℕ} {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : EuclideanSpace ℝ (Fin N)) : | |
| 0 ≤ gaussianQuadraticForm (positiveResolvent t B) x := by | |
| rw [gaussianQuadraticForm,EuclideanSpace.inner_eq_star_dotProduct] | |
| simp only [star_trivial] | |
| rw [dotProduct_comm] | |
| exact (positiveResolvent_posDef ht hB).posSemidef.dotProduct_mulVec_nonneg x.ofLp | |
| lemma positiveResolvent_normalized_abs_bound {N : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t) | |
| {B : Matrix (Fin N) (Fin N) ℝ} (hB : B.PosSemidef) (x : EuclideanSpace ℝ (Fin N)) : | |
| |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x- | |
| (positiveResolvent t B).trace/N| ≤ (1/((N : ℝ)*t))*‖x‖^2+1/t := by | |
| have hn : (0 : ℝ) < N := by exact_mod_cast hN | |
| have hq := positiveResolvent_quadratic_nonneg ht hB x | |
| have htr := (positiveResolvent_posDef ht hB).posSemidef.trace_nonneg | |
| calc | |
| _ ≤ |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x|+ | |
| |(positiveResolvent t B).trace/N| := abs_sub _ _ | |
| _ = (1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x+ | |
| (positiveResolvent t B).trace/N := by rw [abs_of_nonneg (by positivity),abs_of_nonneg (by positivity)] | |
| _ ≤ (1/(N : ℝ))*(‖x‖^2/t)+((N : ℝ)/t)/N := add_le_add | |
| (mul_le_mul_of_nonneg_left (positiveResolvent_quadratic_le ht hB x) (by positivity)) | |
| (div_le_div_of_nonneg_right (positiveResolvent_trace_le ht hB) hn.le) | |
| _ = _ := by field_simp | |
| end InvariantIsing | |
| end | |
| end OAI | |