openAI-math-dataset / lean /OAI /Probability /FreeIsing /Posterior /LipschitzObservationDerivative.lean
| import OAI.Probability.FreeIsing.Posterior.GaussianTimeDerivative | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology BigOperators NNReal | |
| namespace Problem367 | |
| theorem hasDerivAt_integral_sqrt_observation_of_integrable | |
| {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} [IsFiniteMeasure P] | |
| {S G : Ω → ℝ} (hS : Integrable S P) (hG : Integrable G P) | |
| {f f' : ℝ → ℝ} (hf : ∀ x, HasDerivAt f (f' x) x) (hf'm : Measurable f') | |
| {M : ℝ} (hM : 0 ≤ M) | |
| (hf'bound : ∀ x, |f' x| ≤ M) {t : ℝ} (ht : 0 < t) | |
| (hFint : Integrable (fun ω => f (t * S ω + Real.sqrt t * G ω)) P) : | |
| Integrable (fun ω => f' (t * S ω + Real.sqrt t * G ω) * | |
| (S ω + G ω / (2 * Real.sqrt t))) P ∧ | |
| HasDerivAt (fun r : ℝ => ∫ ω, f (r * S ω + Real.sqrt r * G ω) ∂P) | |
| (∫ ω, f' (t * S ω + Real.sqrt t * G ω) * | |
| (S ω + G ω / (2 * Real.sqrt t)) ∂P) t := by | |
| have hfcont : Continuous f := continuous_iff_continuousAt.mpr | |
| (fun x => (hf x).continuousAt) | |
| let F : ℝ → Ω → ℝ := fun r ω => f (r * S ω + Real.sqrt r * G ω) | |
| let F' : ℝ → Ω → ℝ := fun r ω => f' (r * S ω + Real.sqrt r * G ω) * | |
| (S ω + G ω / (2 * Real.sqrt r)) | |
| let a : ℝ := 1 / (2 * Real.sqrt (t / 2)) | |
| have ha : 0 ≤ a := by dsimp [a]; positivity | |
| have hFm (r : ℝ) : AEStronglyMeasurable (F r) P := | |
| hfcont.comp_aestronglyMeasurable | |
| ((hS.aestronglyMeasurable.const_mul r).add | |
| (hG.aestronglyMeasurable.const_mul (Real.sqrt r))) | |
| have hF'm (r : ℝ) : AEStronglyMeasurable (F' r) P := by | |
| apply AEStronglyMeasurable.mul | |
| · exact (hf'm.comp_aemeasurable | |
| ((hS.aemeasurable.const_mul r).add | |
| (hG.aemeasurable.const_mul (Real.sqrt r)))).aestronglyMeasurable | |
| · simpa only [div_eq_mul_inv, Pi.add_def] using hS.aestronglyMeasurable.add | |
| (hG.aestronglyMeasurable.mul_const (2 * Real.sqrt r)⁻¹) | |
| have hdom : Integrable (fun ω => M * (|S ω| + a * |G ω|)) P := | |
| (hS.abs.add (hG.abs.const_mul a)).const_mul M | |
| have hbound : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2), | |
| ‖F' r ω‖ ≤ M * (|S ω| + a * |G ω|) := by | |
| filter_upwards [] with ω | |
| intro r hr | |
| have hrpos : 0 < r := lt_trans (by linarith) hr | |
| have hden : 0 < 2 * Real.sqrt (t / 2) := by positivity | |
| have hrecip : 1 / (2 * Real.sqrt r) ≤ a := by | |
| apply one_div_le_one_div_of_le hden | |
| exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hr.le) (by norm_num) | |
| have hinner : |S ω + G ω / (2 * Real.sqrt r)| ≤ |S ω| + a * |G ω| := by | |
| calc | |
| _ ≤ |S ω| + |G ω / (2 * Real.sqrt r)| := abs_add_le _ _ | |
| _ = |S ω| + (1 / (2 * Real.sqrt r)) * |G ω| := by | |
| rw [abs_div, abs_of_nonneg (by positivity : 0 ≤ 2 * Real.sqrt r)] | |
| ring | |
| _ ≤ |S ω| + a * |G ω| := by gcongr | |
| dsimp [F'] | |
| rw [abs_mul] | |
| exact mul_le_mul (hf'bound _) hinner (abs_nonneg _) hM | |
| have hdiff : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2), HasDerivAt (fun q => F q ω) (F' r ω) r := by | |
| filter_upwards [] with ω | |
| intro r hr | |
| have hrpos : 0 < r := lt_trans (by linarith) hr | |
| have harg := ((hasDerivAt_id r).mul_const (S ω)).add | |
| ((Real.hasDerivAt_sqrt hrpos.ne').mul_const (G ω)) | |
| have harg' : HasDerivAt (fun q : ℝ => q * S ω + Real.sqrt q * G ω) | |
| (S ω + G ω / (2 * Real.sqrt r)) r := by | |
| simpa only [Pi.add_def, id_eq, one_mul, mul_one, one_div, div_eq_mul_inv, mul_comm] using harg | |
| exact (hf (r * S ω + Real.sqrt r * G ω)).comp r harg' | |
| exact hasDerivAt_integral_of_dominated_loc_of_deriv_le | |
| (Ioi_mem_nhds (by linarith : t / 2 < t)) | |
| (Eventually.of_forall hFm) hFint (hF'm t) hbound hdom hdiff | |
| theorem hasDerivAt_integral_sqrt_observation_vector_of_integrable | |
| {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {P : Measure Ω} [IsFiniteMeasure P] | |
| {S G : Ω → E} (hS : Integrable S P) (hG : Integrable G P) | |
| {f : E → ℝ} {f' : E → E →L[ℝ] ℝ} | |
| (hf : ∀ x, HasFDerivAt f (f' x) x) (hf'cont : Continuous f') | |
| {M : ℝ} (hM : 0 ≤ M) | |
| (hf'bound : ∀ x, ‖f' x‖ ≤ M) {t : ℝ} (ht : 0 < t) | |
| (hFint : Integrable (fun ω => f (t • S ω + Real.sqrt t • G ω)) P) : | |
| Integrable (fun ω => f' (t • S ω + Real.sqrt t • G ω) | |
| (S ω + (1 / (2 * Real.sqrt t)) • G ω)) P ∧ | |
| HasDerivAt (fun r : ℝ => ∫ ω, f (r • S ω + Real.sqrt r • G ω) ∂P) | |
| (∫ ω, f' (t • S ω + Real.sqrt t • G ω) | |
| (S ω + (1 / (2 * Real.sqrt t)) • G ω) ∂P) t := by | |
| have hfcont : Continuous f := continuous_iff_continuousAt.mpr | |
| (fun x => (hf x).continuousAt) | |
| let F : ℝ → Ω → ℝ := fun r ω => f (r • S ω + Real.sqrt r • G ω) | |
| let F' : ℝ → Ω → ℝ := fun r ω => f' (r • S ω + Real.sqrt r • G ω) | |
| (S ω + (1 / (2 * Real.sqrt r)) • G ω) | |
| let a : ℝ := 1 / (2 * Real.sqrt (t / 2)) | |
| have ha : 0 ≤ a := by dsimp [a]; positivity | |
| have hargm (r : ℝ) : AEStronglyMeasurable | |
| (fun ω => r • S ω + Real.sqrt r • G ω) P := | |
| (hS.aestronglyMeasurable.const_smul r).add | |
| (hG.aestronglyMeasurable.const_smul (Real.sqrt r)) | |
| have hFm (r : ℝ) : AEStronglyMeasurable (F r) P := | |
| hfcont.comp_aestronglyMeasurable (hargm r) | |
| have hF'm (r : ℝ) : AEStronglyMeasurable (F' r) P := | |
| (isBoundedBilinearMap_apply (𝕜 := ℝ) (E := E) (F := ℝ)).continuous | |
| |>.comp_aestronglyMeasurable | |
| ((hf'cont.comp_aestronglyMeasurable (hargm r)).prodMk | |
| (hS.aestronglyMeasurable.add | |
| (hG.aestronglyMeasurable.const_smul (1 / (2 * Real.sqrt r))))) | |
| have hdom : Integrable (fun ω => M * (‖S ω‖ + a * ‖G ω‖)) P := | |
| (hS.norm.add (hG.norm.const_mul a)).const_mul M | |
| have hbound : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2), | |
| ‖F' r ω‖ ≤ M * (‖S ω‖ + a * ‖G ω‖) := by | |
| filter_upwards [] with ω | |
| intro r hr | |
| have hrpos : 0 < r := lt_trans (by linarith) hr | |
| have hden : 0 < 2 * Real.sqrt (t / 2) := by positivity | |
| have hrecip : 1 / (2 * Real.sqrt r) ≤ a := by | |
| apply one_div_le_one_div_of_le hden | |
| exact mul_le_mul_of_nonneg_left (Real.sqrt_le_sqrt hr.le) (by norm_num) | |
| have hinner : ‖S ω + (1 / (2 * Real.sqrt r)) • G ω‖ ≤ | |
| ‖S ω‖ + a * ‖G ω‖ := by | |
| calc | |
| _ ≤ ‖S ω‖ + ‖(1 / (2 * Real.sqrt r)) • G ω‖ := norm_add_le _ _ | |
| _ = ‖S ω‖ + (1 / (2 * Real.sqrt r)) * ‖G ω‖ := by | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (by positivity)] | |
| _ ≤ ‖S ω‖ + a * ‖G ω‖ := by gcongr | |
| exact ((f' (r • S ω + Real.sqrt r • G ω)).le_opNorm _).trans | |
| (mul_le_mul (hf'bound _) hinner (norm_nonneg _) hM) | |
| have hdiff : ∀ᵐ ω ∂P, ∀ r ∈ Ioi (t / 2), | |
| HasDerivAt (fun q => F q ω) (F' r ω) r := by | |
| filter_upwards [] with ω | |
| intro r hr | |
| have hrpos : 0 < r := lt_trans (by linarith) hr | |
| have harg := ((hasDerivAt_id r).smul_const (S ω)).add | |
| ((Real.hasDerivAt_sqrt hrpos.ne').smul_const (G ω)) | |
| have harg' : HasDerivAt (fun q : ℝ => q • S ω + Real.sqrt q • G ω) | |
| (S ω + (1 / (2 * Real.sqrt r)) • G ω) r := by | |
| simpa only [Pi.add_def, id_eq, one_smul, one_div] using harg | |
| exact (hf (r • S ω + Real.sqrt r • G ω)).comp_hasDerivAt r harg' | |
| exact hasDerivAt_integral_of_dominated_loc_of_deriv_le | |
| (Ioi_mem_nhds (by linarith : t / 2 < t)) | |
| (Eventually.of_forall hFm) hFint (hF'm t) hbound hdom hdiff | |
| theorem continuous_integral_sqrt_observation_lipschitz | |
| {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {P : Measure Ω} [IsFiniteMeasure P] {S G : Ω → E} | |
| (hS : Integrable S P) (hG : Integrable G P) | |
| {f : E → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) : | |
| Continuous (fun t : ℝ => ∫ ω, f (t • S ω + Real.sqrt t • G ω) ∂P) := by | |
| apply continuous_iff_continuousAt.mpr | |
| intro t | |
| let a : ℝ := |t| + 1 | |
| have ha : 0 ≤ a := by dsimp [a]; positivity | |
| have hloc : ∀ᶠ r in 𝓝 t, |r| < a := | |
| continuous_abs.continuousAt.eventually | |
| (gt_mem_nhds (by dsimp [a]; linarith : |t| < a)) | |
| apply continuousAt_of_dominated | |
| (bound := fun ω => ‖f 0‖ + (K : ℝ) * | |
| (a * ‖S ω‖ + Real.sqrt a * ‖G ω‖)) | |
| · exact Eventually.of_forall fun r => hf.continuous.comp_aestronglyMeasurable | |
| ((hS.aestronglyMeasurable.const_smul r).add | |
| (hG.aestronglyMeasurable.const_smul (Real.sqrt r))) | |
| · filter_upwards [hloc] with r hr | |
| exact Eventually.of_forall fun ω => by | |
| have harg : ‖r • S ω + Real.sqrt r • G ω‖ ≤ | |
| a * ‖S ω‖ + Real.sqrt a * ‖G ω‖ := by | |
| calc | |
| _ ≤ ‖r • S ω‖ + ‖Real.sqrt r • G ω‖ := norm_add_le _ _ | |
| _ = |r| * ‖S ω‖ + Real.sqrt r * ‖G ω‖ := by | |
| rw [norm_smul, norm_smul, Real.norm_eq_abs, Real.norm_eq_abs, | |
| abs_of_nonneg (Real.sqrt_nonneg r)] | |
| _ ≤ _ := add_le_add | |
| (mul_le_mul_of_nonneg_right hr.le (norm_nonneg _)) | |
| (mul_le_mul_of_nonneg_right | |
| (Real.sqrt_le_sqrt ((le_abs_self r).trans hr.le)) (norm_nonneg _)) | |
| calc | |
| _ ≤ ‖f 0‖ + ‖f (r • S ω + Real.sqrt r • G ω) - f 0‖ := | |
| norm_le_norm_add_norm_sub' _ _ | |
| _ ≤ ‖f 0‖ + (K : ℝ) * ‖r • S ω + Real.sqrt r • G ω‖ := by | |
| gcongr | |
| simpa only [sub_zero] using hf.norm_sub_le | |
| (r • S ω + Real.sqrt r • G ω) 0 | |
| _ ≤ _ := add_le_add le_rfl (mul_le_mul_of_nonneg_left harg K.coe_nonneg) | |
| · exact (integrable_const ‖f 0‖).add | |
| (((hS.norm.const_mul a).add (hG.norm.const_mul (Real.sqrt a))).const_mul (K : ℝ)) | |
| · exact Eventually.of_forall fun ω => (hf.continuous.comp | |
| ((continuous_id.smul continuous_const).add | |
| (Real.continuous_sqrt.smul continuous_const))).continuousAt | |
| theorem tendsto_integral_sqrt_observation_zero_lipschitz | |
| {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {P : Measure Ω} [IsFiniteMeasure P] {S G : Ω → E} | |
| (hS : Integrable S P) (hG : Integrable G P) | |
| {f : E → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) : | |
| Tendsto (fun t : ℝ => ∫ ω, f (t • S ω + Real.sqrt t • G ω) ∂P) | |
| (𝓝 0) (𝓝 (P.real Set.univ * f 0)) := by | |
| simpa only [zero_smul, Real.sqrt_zero, zero_add, integral_const, smul_eq_mul] using | |
| (continuous_integral_sqrt_observation_lipschitz hS hG hf).tendsto 0 | |
| namespace FinitePosterior | |
| variable {V Ω : Type*} [Fintype V] [MeasurableSpace Ω] | |
| theorem hasDerivAt_integral_gaussianObservationCurve_of_integrable | |
| (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (f : (V → ℝ) → ℝ) (g : (V → ℝ) → V → ℝ) | |
| (hdf : ∀ x, HasFDerivAt f (∑ v, g x v • | |
| (ContinuousLinearMap.proj v : (V → ℝ) →L[ℝ] ℝ)) x) | |
| (hg : ∀ v, Continuous (fun x => g x v)) | |
| (C : ℝ) (hC : 0 ≤ C) (hgbound : ∀ x v, |g x v| ≤ C) | |
| (signal noise : Ω → V → ℝ) | |
| (hsignal : Measurable signal) (hnoise : Measurable noise) | |
| (hspin : ∀ ω v, |signal ω v| ≤ 1) | |
| (hnoiseInt : ∀ v, Integrable (fun ω => noise ω v) μ) | |
| {t : ℝ} (ht : 0 < t) | |
| (hFint : Integrable (fun ω => | |
| f (gaussianObservationCurve (signal ω) (noise ω) t)) μ) : | |
| HasDerivAt (fun r => ∫ ω, f (gaussianObservationCurve (signal ω) (noise ω) r) ∂μ) | |
| (∫ ω, ∑ v, g (gaussianObservationCurve (signal ω) (noise ω) t) v * | |
| gaussianObservationVelocity (signal ω) (noise ω) t v ∂μ) t := by | |
| let D : (V → ℝ) → (V → ℝ) →L[ℝ] ℝ := fun x => ∑ v, g x v • | |
| (ContinuousLinearMap.proj v : (V → ℝ) →L[ℝ] ℝ) | |
| have hDcont : Continuous D := by | |
| apply continuous_finsetSum | |
| intro v _ | |
| exact (hg v).smul continuous_const | |
| have hDbound (x : V → ℝ) : ‖D x‖ ≤ (Fintype.card V : ℝ) * C := by | |
| apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (by positivity) hC) | |
| intro y | |
| simp only [D, sum_apply, smul_apply, ContinuousLinearMap.proj_apply, smul_eq_mul] | |
| calc | |
| ‖∑ v, g x v * y v‖ ≤ ∑ v, ‖g x v * y v‖ := norm_sum_le _ _ | |
| _ ≤ ∑ v : V, C * ‖y‖ := by | |
| apply Finset.sum_le_sum | |
| intro v _ | |
| rw [norm_mul, Real.norm_eq_abs (g x v)] | |
| exact mul_le_mul (hgbound x v) (norm_le_pi_norm y v) (norm_nonneg _) hC | |
| _ = _ := by simp [Finset.sum_const, nsmul_eq_mul, mul_assoc] | |
| have hsInt : Integrable signal μ := Integrable.of_eval fun v => | |
| Integrable.of_bound ((measurable_pi_apply v).comp hsignal).aestronglyMeasurable 1 | |
| (Filter.Eventually.of_forall fun ω => by simpa only [Real.norm_eq_abs] using hspin ω v) | |
| have hnInt : Integrable noise μ := | |
| ⟨hnoise.aestronglyMeasurable, (Integrable.of_eval hnoiseInt).hasFiniteIntegral⟩ | |
| have hd := (hasDerivAt_integral_sqrt_observation_vector_of_integrable hsInt hnInt hdf hDcont | |
| (mul_nonneg (by positivity) hC) hDbound ht hFint).2 | |
| simpa only [gaussianObservationCurve, gaussianObservationVelocity, D, sum_apply, | |
| smul_apply, ContinuousLinearMap.proj_apply, smul_eq_mul] using hd | |
| end FinitePosterior | |
| end Problem367 | |
| end | |
| end OAI | |