openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /ResolventPowerDerivative.lean
| import OAI.Probability.InvariantIsing.Gaussian.MPDensityTransform | |
| import Mathlib.Analysis.Calculus.ParametricIntegral | |
| namespace OAI | |
| /-! Differentiation of bounded positive-resolvent moments. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set Filter | |
| open scoped Topology | |
| namespace InvariantIsing | |
| lemma positiveResolventTest_hasDerivAt {t : ℝ} (ht : 0 < t) (x : ℝ) : | |
| HasDerivAt (fun s => positiveResolventTest s x) (-(positiveResolventTest t x)^2) t := by | |
| have hn : t+max x 0 ≠ 0 := (add_pos_of_pos_of_nonneg ht (le_max_right x 0)).ne' | |
| convert ((hasDerivAt_id t).add_const (max x 0)).inv hn using 1 | |
| · funext s | |
| simp only [positiveResolventTest,one_div,Pi.inv_apply,id_eq] | |
| · simp only [positiveResolventTest,div_eq_mul_inv,inv_pow,one_mul,id_eq,neg_mul] | |
| lemma positiveResolventPower_hasDerivAt {t : ℝ} (ht : 0 < t) (x : ℝ) (n : ℕ) : | |
| HasDerivAt (fun s => (positiveResolventTest s x)^n) | |
| (-(n : ℝ)*(positiveResolventTest t x)^(n+1)) t := by | |
| induction n with | |
| | zero => simpa only [pow_zero,Nat.cast_zero,neg_zero,zero_mul] using hasDerivAt_const t (1 : ℝ) | |
| | succ n ih => | |
| convert ih.mul (positiveResolventTest_hasDerivAt ht x) using 1 | |
| · funext s | |
| exact pow_succ _ _ | |
| · simp only [Nat.cast_succ,pow_succ] | |
| ring | |
| lemma positiveResolventPower_integrable {t : ℝ} (ht : 0 < t) (n : ℕ) | |
| (μ : Measure ℝ) [IsFiniteMeasure μ] : | |
| Integrable (fun x => (positiveResolventTest t x)^n) μ := by | |
| apply Integrable.of_bound ((continuous_positiveResolventTest ht).pow n).aestronglyMeasurable ((1/t)^n) | |
| exact ae_of_all _ fun x => by | |
| change ‖(positiveResolventTest t x)^n‖ ≤ _ | |
| rw [Real.norm_eq_abs,abs_of_nonneg (pow_nonneg (positiveResolventTest_bound ht x).1 _)] | |
| exact pow_le_pow_left₀ (positiveResolventTest_bound ht x).1 (positiveResolventTest_bound ht x).2 n | |
| theorem positiveResolventPower_integral_hasDerivAt (μ : Measure ℝ) [IsFiniteMeasure μ] | |
| {t : ℝ} (ht : 0 < t) (n : ℕ) : | |
| HasDerivAt (fun s => ∫ x, (positiveResolventTest s x)^n ∂μ) | |
| (-(n : ℝ)*(∫ x, (positiveResolventTest t x)^(n+1) ∂μ)) t := by | |
| have hs : Ioi (t/2) ∈ 𝓝 t := Ioi_mem_nhds (by linarith) | |
| have hm : ∀ᶠ s in 𝓝 t, AEStronglyMeasurable (fun x => (positiveResolventTest s x)^n) μ := by | |
| filter_upwards [hs] with s hs' | |
| exact ((continuous_positiveResolventTest (lt_trans (half_pos ht) hs')).pow n).aestronglyMeasurable | |
| have hd : ∀ᵐ x ∂μ, ∀ s ∈ Ioi (t/2), | |
| HasDerivAt (fun u => (positiveResolventTest u x)^n) | |
| (-(n : ℝ)*(positiveResolventTest s x)^(n+1)) s := | |
| ae_of_all _ fun x s hs' => positiveResolventPower_hasDerivAt (lt_trans (half_pos ht) hs') x n | |
| have hb : ∀ᵐ x ∂μ, ∀ s ∈ Ioi (t/2), | |
| ‖-(n : ℝ)*(positiveResolventTest s x)^(n+1)‖ ≤ (n : ℝ)*(2/t)^(n+1) := by | |
| refine ae_of_all _ fun x s hs' => ?_ | |
| have hsp : 0 < s := lt_trans (half_pos ht) hs' | |
| have hr : positiveResolventTest s x ≤ 2/t := by | |
| calc | |
| _ ≤ 1/s := (positiveResolventTest_bound hsp x).2 | |
| _ ≤ 1/(t/2) := one_div_le_one_div_of_le (half_pos ht) hs'.le | |
| _ = 2/t := by ring | |
| rw [norm_mul,norm_neg,Real.norm_of_nonneg (Nat.cast_nonneg n), | |
| Real.norm_of_nonneg (pow_nonneg (positiveResolventTest_bound hsp x).1 _)] | |
| exact mul_le_mul_of_nonneg_left | |
| (pow_le_pow_left₀ (positiveResolventTest_bound hsp x).1 hr _) (Nat.cast_nonneg n) | |
| have hh := (hasDerivAt_integral_of_dominated_loc_of_deriv_le hs hm | |
| (positiveResolventPower_integrable ht n μ) | |
| (((continuous_positiveResolventTest ht).pow (n+1)).const_mul (-(n : ℝ))).aestronglyMeasurable | |
| hb (integrable_const _) hd).2 | |
| simpa only [integral_const_mul] using hh | |
| end InvariantIsing | |
| end | |
| end OAI | |