| import Mathlib |
| import OAI.Probability.SKFluctuations.ClockSampledRate |
|
|
| namespace OAI |
|
|
| /-! Clock Path Pressure Sup. -/ |
|
|
| noncomputable section |
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|
|
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
|
|
| omit [Nonempty ι] in |
| theorem clockL1Distance_le (D C : ι → ℝ) (q r : ℝ → ι → ℝ≥0) |
| (hq : ∀ i, MonotoneOn (fun u => q u i) (Icc (0:ℝ) 1)) |
| (hr : ∀ i, MonotoneOn (fun u => r u i) (Icc (0:ℝ) 1)) |
| (hC : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |(q u i:ℝ)-(r u i:ℝ)|≤C i) : |
| clockL1Distance D q r≤(1/2:ℝ)*∑ i, C i*(D i)^2 := by |
| unfold clockL1Distance |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| apply Finset.sum_le_sum |
| intro i _ |
| have hi : (∫ u in (0:ℝ)..1, |(q u i:ℝ)-(r u i:ℝ)|)≤C i := by |
| calc |
| _ ≤ ∫ u in (0:ℝ)..1, C i := |
| intervalIntegral.integral_mono_on_of_le_Ioo (by norm_num) |
| (clockDirection_intervalIntegrable r q hr hq i).abs intervalIntegrable_const |
| (fun u hu => hC u ⟨hu.1.le,hu.2.le⟩ i) |
| _ = _ := by simp |
| simpa only [mul_comm] using mul_le_mul_of_nonneg_left hi (sq_nonneg (D i)) |
|
|
| theorem pathDisorderPressure_sup_comparison {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (C D M : ι → ℝ) |
| (hself : ∀ σ i, (φ σ (Pi.single i 1))^2=C i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (q r : ℝ → ι → ℝ≥0) |
| (hq : ∀ i, MonotoneOn (fun u => q u i) (Icc (0:ℝ) 1)) |
| (hr : ∀ i, MonotoneOn (fun u => r u i) (Icc (0:ℝ) 1)) |
| (hM : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |(q u i:ℝ)-(r u i:ℝ)|≤M i) (x : ι → ℝ) : |
| |pathDisorderPressure w φ C q x-pathDisorderPressure w φ C r x| ≤ |
| (1/2:ℝ)*∑ i, M i*(D i)^2 := |
| (pathDisorderPressure_l1_comparison hw φ C D hself (fun i σ => hφ σ i) q r hq hr x).trans |
| (clockL1Distance_le D M q r hq hr hM) |
|
|
| omit [Fintype ι] [Nonempty ι] in |
| theorem affineClockPath_zero (q r : ℝ → ι → ℝ≥0) : affineClockPath q r 0=q := by |
| funext u i |
| simp [affineClockPath] |
| omit [Fintype ι] [Nonempty ι] in |
| theorem affineClockPath_one (q r : ℝ → ι → ℝ≥0) : affineClockPath q r 1=r := by |
| funext u i |
| simp [affineClockPath] |
|
|
| omit [Fintype ι] [Nonempty ι] in |
| theorem affineClockPath_difference (q r : ℝ → ι → ℝ≥0) {s t : ℝ} |
| (hs : s ∈ Icc (0:ℝ) 1) (ht : t ∈ Icc (0:ℝ) 1) (u : ℝ) (i : ι) : |
| |(affineClockPath q r s u i:ℝ)-(affineClockPath q r t u i:ℝ)|= |
| |s-t| *|(r u i:ℝ)-(q u i:ℝ)| := by |
| rw [affineClockPath_coe q r hs,affineClockPath_coe q r ht,←abs_mul] |
| congr 1 |
| ring |
|
|
| theorem pathDisorderPressure_affine_continuousOn {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (C D : ι → ℝ) |
| (hself : ∀ σ i, (φ σ (Pi.single i 1))^2=C i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (q r : ℝ → ι → ℝ≥0) |
| (hq : ∀ i, MonotoneOn (fun u => q u i) (Icc (0:ℝ) 1)) |
| (hr : ∀ i, MonotoneOn (fun u => r u i) (Icc (0:ℝ) 1)) (x : ι → ℝ) : |
| ContinuousOn (fun t => pathDisorderPressure w φ C (affineClockPath q r t) x) (Icc (0:ℝ) 1) := by |
| let A : ℝ≥0 := ⟨(1/2:ℝ)*∑ i, ((r 1 i:ℝ)+(q 1 i:ℝ))*(D i)^2, by positivity⟩ |
| apply (show LipschitzOnWith A (fun t => pathDisorderPressure w φ C (affineClockPath q r t) x) |
| (Icc (0:ℝ) 1) from ?_).continuousOn |
| apply LipschitzOnWith.of_dist_le_mul |
| intro s hs t ht |
| rw [Real.dist_eq,Real.dist_eq] |
| have he := pathDisorderPressure_sup_comparison hw φ C D |
| (fun i => |s-t| *((r 1 i:ℝ)+(q 1 i:ℝ))) hself hφ |
| (affineClockPath q r s) (affineClockPath q r t) |
| (affineClockPath_monotone q r hq hr s) (affineClockPath_monotone q r hq hr t) |
| (fun u hu i => by |
| rw [affineClockPath_difference q r hs ht] |
| exact mul_le_mul_of_nonneg_left (clockDirection_bound q r hq hr hu i) (abs_nonneg _)) x |
| convert he using 1 |
| change ((1/2:ℝ)*∑ i, ((r 1 i:ℝ)+(q 1 i:ℝ))*(D i)^2) * |s-t| = _ |
| simp only [Finset.mul_sum,Finset.sum_mul] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
|
|
| end SKFluctuations.Hierarchy |
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| |
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| |
|
|
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
|
|
| theorem intervalIntegral_mul_abs_le {f g : ℝ → ℝ} |
| (hf : IntervalIntegrable f volume 0 1) (hg : IntervalIntegrable g volume 0 1) |
| {A B : ℝ} (hA : 0≤A) (_hB : 0≤B) |
| (hfB : ∀ u ∈ Ioc (0:ℝ) 1, |f u|≤A) (hgB : ∀ u ∈ Ioc (0:ℝ) 1, |g u|≤B) : |
| |∫ u in (0:ℝ)..1, f u*g u|≤A*B := by |
| have hi := intervalIntegrable_mul_bounded hf hg hgB |
| calc |
| _ ≤ ∫ u in (0:ℝ)..1, |f u*g u| := by |
| simpa only [Real.norm_eq_abs] using intervalIntegral.norm_integral_le_integral_norm |
| (f := fun u => f u*g u) (by norm_num : (0:ℝ)≤1) |
| _ ≤ ∫ _u in (0:ℝ)..1, A*B := by |
| apply intervalIntegral.integral_mono_on_of_le_Ioo (by norm_num) hi.abs intervalIntegrable_const |
| intro u hu |
| rw [abs_mul] |
| exact mul_le_mul (hfB u ⟨hu.1,hu.2.le⟩) (hgB u ⟨hu.1,hu.2.le⟩) (abs_nonneg _) hA |
| _ = _ := by simp |
|
|
| theorem pathPressureRate_sup_comparison {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D C M : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p p' q r : ℝ → ι → ℝ≥0) |
| (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| (hp' : ∀ i, MonotoneOn (fun u => p' u i) (Icc (0:ℝ) 1)) |
| (hq : ∀ i, MonotoneOn (fun u => q u i) (Icc (0:ℝ) 1)) |
| (hr : ∀ i, MonotoneOn (fun u => r u i) (Icc (0:ℝ) 1)) |
| (hC : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |(r u i:ℝ)-(q u i:ℝ)|≤C i) |
| (hM : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |(p u i:ℝ)-(p' u i:ℝ)|≤M i) : |
| |pathPressureRate w φ p (fun u i => (r u i:ℝ)-(q u i:ℝ))- |
| pathPressureRate w φ p' (fun u i => (r u i:ℝ)-(q u i:ℝ))| ≤ |
| (3/2:ℝ)*ClockTree.errorJetVariation 2 0 0*(∑ i, M i*(D i)^2)*(∑ i, C i*(D i)^2) := by |
| have hC0 (i : ι) : 0≤C i := (abs_nonneg _).trans (hC 0 (by constructor <;> norm_num) i) |
| have hM0 (i : ι) : 0≤M i := (abs_nonneg _).trans (hM 0 (by constructor <;> norm_num) i) |
| have hA : 0≤∑ i, M i*(D i)^2 := Finset.sum_nonneg fun i _ => mul_nonneg (hM0 i) (sq_nonneg _) |
| have hi (z : ℝ → ι → ℝ≥0) (hz : ∀ i, MonotoneOn (fun u => z u i) (Icc (0:ℝ) 1)) (i : ι) : |
| IntervalIntegrable (fun u => ((r u i:ℝ)-(q u i:ℝ))*pathPairMoment w φ z (coordinatePairTest φ i) u) volume 0 1 := by |
| simpa only [mul_comm] using intervalIntegrable_mul_bounded |
| (pathPairMoment_intervalIntegrable_all hw φ z hz _) (clockDirection_intervalIntegrable q r hq hr i) |
| (fun u hu => hC u ⟨hu.1.le,hu.2⟩ i) |
| have hI (i : ι) : |(∫ u in (0:ℝ)..1, ((r u i:ℝ)-(q u i:ℝ))*pathPairMoment w φ p (coordinatePairTest φ i) u)- |
| (∫ u in (0:ℝ)..1, ((r u i:ℝ)-(q u i:ℝ))*pathPairMoment w φ p' (coordinatePairTest φ i) u)|≤ |
| C i*(ClockTree.errorJetVariation 2 0 0*(D i)^2*(3*∑ j, M j*(D j)^2)) := by |
| rw [←intervalIntegral.integral_sub (hi p hp i) (hi p' hp' i)] |
| have he : (fun u => ((r u i:ℝ)-(q u i:ℝ))*pathPairMoment w φ p (coordinatePairTest φ i) u- |
| ((r u i:ℝ)-(q u i:ℝ))*pathPairMoment w φ p' (coordinatePairTest φ i) u)= |
| fun u => ((r u i:ℝ)-(q u i:ℝ))*(pathPairMoment w φ p (coordinatePairTest φ i) u- |
| pathPairMoment w φ p' (coordinatePairTest φ i) u) := by funext u; ring |
| rw [he] |
| apply intervalIntegral_mul_abs_le (clockDirection_intervalIntegrable q r hq hr i) |
| ((pathPairMoment_intervalIntegrable_all hw φ p hp _).sub |
| (pathPairMoment_intervalIntegrable_all hw φ p' hp' _)) (hC0 i) |
| (by apply mul_nonneg (mul_nonneg (ClockTree.errorJetVariation_nonneg _ _ _) (sq_nonneg _)); positivity) |
| (fun u hu => hC u ⟨hu.1.le,hu.2⟩ i) |
| intro u hu |
| exact pathPairMoment_sup_comparison hw φ D M hD hφ p p' hp hp' hM hu |
| (coordinatePairTest φ i) (sq_nonneg _) (coordinatePairTest_bound φ D hφ i) |
| unfold pathPressureRate |
| rw [←mul_sub,abs_mul,←Finset.sum_sub_distrib] |
| norm_num only [abs_neg,abs_div,abs_one,abs_of_nonneg (by norm_num : (0:ℝ)≤2)] |
| apply (mul_le_mul_of_nonneg_left (Finset.abs_sum_le_sum_abs _ _) (by norm_num : (0:ℝ)≤1/2)).trans |
| calc |
| _ ≤ (1/2:ℝ)*∑ i, C i*(ClockTree.errorJetVariation 2 0 0*(D i)^2*(3*∑ j, M j*(D j)^2)) := by |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| exact Finset.sum_le_sum fun i _ => hI i |
| _ = _ := by |
| conv_lhs => rw [Finset.mul_sum] |
| conv_rhs => rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| |
| end SKFluctuations.Hierarchy |
| |
| |
| |
| |
| |
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
| |
| theorem pathDisorderPressure_taylor {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (C D B : ι → ℝ) |
| (hself : ∀ σ i, (φ σ (Pi.single i 1))^2=C i) |
| (hD : ∀ i, 0≤D i) (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (q r : ℝ → ι → ℝ≥0) |
| (hq : ∀ i, MonotoneOn (fun u => q u i) (Icc (0:ℝ) 1)) |
| (hr : ∀ i, MonotoneOn (fun u => r u i) (Icc (0:ℝ) 1)) |
| (hB : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |(r u i:ℝ)-(q u i:ℝ)|≤B i) : |
| |pathDisorderPressure w φ C r 0-pathDisorderPressure w φ C q 0- |
| pathPressureRate w φ q (fun u i => (r u i:ℝ)-(q u i:ℝ))| ≤ |
| (3/2:ℝ)*ClockTree.errorJetVariation 2 0 0*(∑ i, B i*(D i)^2)^2 := by |
| obtain ⟨t,ht,he⟩ := exists_hasDerivAt_eq_slope |
| (fun t => pathDisorderPressure w φ C (affineClockPath q r t) 0) |
| (fun t => pathPressureRate w φ (affineClockPath q r t) (fun u i => (r u i:ℝ)-(q u i:ℝ))) |
| (by norm_num : (0:ℝ)<1) |
| (pathDisorderPressure_affine_continuousOn hw φ C D hself hφ q r hq hr 0) |
| (fun t ht => pathDisorderPressure_hasDerivAt hw φ C hself q r hq hr ht) |
| simp only [affineClockPath_zero,affineClockPath_one,sub_zero,div_one] at he |
| rw [←he] |
| have hb : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |
| |(affineClockPath q r t u i:ℝ)-(q u i:ℝ)|≤B i := by |
| intro u hu i |
| have hd := affineClockPath_difference q r ⟨ht.1.le,ht.2.le⟩ |
| (show (0:ℝ) ∈ Icc (0:ℝ) 1 by constructor <;> norm_num) u i |
| simp only [affineClockPath_zero,sub_zero,abs_of_pos ht.1] at hd |
| rw [hd] |
| exact (mul_le_of_le_one_left (abs_nonneg _) ht.2.le).trans (hB u hu i) |
| have hbnd := pathPressureRate_sup_comparison hw φ D B B hD hφ |
| (affineClockPath q r t) q q r (affineClockPath_monotone q r hq hr t) hq hq hr hB hb |
| convert hbnd using 1; ring |
| |
| end SKFluctuations.Hierarchy |
| |
| |
| |
| |
| |
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
| |
| omit [Nonempty S] [Nonempty ι] in |
| theorem pathPressureRate_const_mul (w : S → ℝ) (φ : S → (ι → ℝ) →L[ℝ] ℝ) |
| (p : ℝ → ι → ℝ≥0) (K : ℝ → ι → ℝ) (a : ℝ) : |
| pathPressureRate w φ p (fun u i => a*K u i)=a*pathPressureRate w φ p K := by |
| unfold pathPressureRate |
| simp only [mul_assoc,intervalIntegral.integral_const_mul,←Finset.mul_sum] |
| ring |
| |
| theorem pathPressureRate_direction_integrable {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p : ℝ → ι → ℝ≥0) (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| (K : ℝ → ι → ℝ) (hK : ∀ i, IntervalIntegrable (fun u => K u i) volume 0 1) (i : ι) : |
| IntervalIntegrable (fun u => K u i*pathPairMoment w φ p (coordinatePairTest φ i) u) volume 0 1 := |
| intervalIntegrable_mul_bounded (hK i) (pathPairMoment_intervalIntegrable_all hw φ p hp _) |
| (fun _ hu => pathPairMoment_bound hw φ D hD hφ p hp _ (sq_nonneg _) (coordinatePairTest_bound φ D hφ i) hu) |
| |
| theorem pathPressureRate_sub {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p : ℝ → ι → ℝ≥0) (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| (K L : ℝ → ι → ℝ) |
| (hK : ∀ i, IntervalIntegrable (fun u => K u i) volume 0 1) |
| (hL : ∀ i, IntervalIntegrable (fun u => L u i) volume 0 1) : |
| pathPressureRate w φ p (fun u i => K u i-L u i)= |
| pathPressureRate w φ p K-pathPressureRate w φ p L := by |
| unfold pathPressureRate |
| simp only [sub_mul] |
| simp_rw [intervalIntegral.integral_sub (pathPressureRate_direction_integrable hw φ D hD hφ p hp K hK _) |
| (pathPressureRate_direction_integrable hw φ D hD hφ p hp L hL _)] |
| rw [Finset.sum_sub_distrib,mul_sub] |
| |
| theorem pathPressureRate_bound {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D B : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p : ℝ → ι → ℝ≥0) (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| (K : ℝ → ι → ℝ) (hK : ∀ i, IntervalIntegrable (fun u => K u i) volume 0 1) |
| (hB : ∀ u ∈ Ioc (0:ℝ) 1, ∀ i, |K u i|≤B i) : |
| |pathPressureRate w φ p K|≤(1/2:ℝ)*∑ i, B i*(D i)^2 := by |
| have hB0 (i : ι) : 0≤B i := (abs_nonneg _).trans (hB 1 (by constructor <;> norm_num) i) |
| unfold pathPressureRate |
| rw [abs_mul] |
| norm_num only [abs_neg,abs_div,abs_one,abs_of_nonneg (by norm_num : (0:ℝ)≤2)] |
| apply (mul_le_mul_of_nonneg_left (Finset.abs_sum_le_sum_abs _ _) (by norm_num : (0:ℝ)≤1/2)).trans |
| apply mul_le_mul_of_nonneg_left _ (by norm_num) |
| apply Finset.sum_le_sum |
| intro i _ |
| exact intervalIntegral_mul_abs_le (hK i) (pathPairMoment_intervalIntegrable_all hw φ p hp _) |
| (hB0 i) (sq_nonneg _) (fun u hu => hB u hu i) |
| (fun _ hu => pathPairMoment_bound hw φ D hD hφ p hp _ (sq_nonneg _) (coordinatePairTest_bound φ D hφ i) hu) |
| |
| end SKFluctuations.Hierarchy |
| |
| |
| |
| |
| |
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
| |
| theorem pathPressureRate_tendsto {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D B : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p : ℝ → ι → ℝ≥0) (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| {A : Type*} {l : Filter A} [l.IsCountablyGenerated] |
| (K : A → ℝ → ι → ℝ) (L : ℝ → ι → ℝ) |
| (hK : ∀ᶠ a in l, ∀ i, IntervalIntegrable (fun u => K a u i) volume 0 1) |
| (hB : ∀ᶠ a in l, ∀ u ∈ Ioc (0:ℝ) 1, ∀ i, |K a u i|≤B i) |
| (hL : ∀ i, ∀ᵐ u ∂volume, u ∈ Ioc (0:ℝ) 1 → Tendsto (fun a => K a u i) l (𝓝 (L u i))) : |
| Tendsto (fun a => pathPressureRate w φ p (K a)) l (𝓝 (pathPressureRate w φ p L)) := by |
| unfold pathPressureRate |
| apply Tendsto.const_mul |
| apply tendsto_finsetSum |
| intro i _ |
| apply intervalIntegral.tendsto_integral_filter_of_dominated_convergence (fun _ => B i*(D i)^2) |
| · filter_upwards [hK] with a ha |
| simpa only [uIoc_of_le (by norm_num : (0:ℝ)≤1)] using |
| (pathPressureRate_direction_integrable hw φ D hD hφ p hp (K a) ha i).1.aestronglyMeasurable |
| · filter_upwards [hB] with a ha |
| filter_upwards [] with u hu |
| rw [uIoc_of_le (by norm_num : (0:ℝ)≤1)] at hu |
| rw [Real.norm_eq_abs,abs_mul] |
| exact mul_le_mul (ha u hu i) |
| (pathPairMoment_bound hw φ D hD hφ p hp _ (sq_nonneg _) (coordinatePairTest_bound φ D hφ i) hu) |
| (abs_nonneg _) ((abs_nonneg _).trans (ha u hu i)) |
| · exact intervalIntegrable_const |
| · filter_upwards [hL i] with u hu hu' |
| rw [uIoc_of_le (by norm_num : (0:ℝ)≤1)] at hu' |
| exact (hu hu').mul_const _ |
|
|
| end SKFluctuations.Hierarchy |
|
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| |
|
|
| |
|
|
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
|
|
| theorem pathDisorderPressure_slope_error {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (C D B : ι → ℝ) |
| (hself : ∀ σ i, (φ σ (Pi.single i 1))^2=C i) |
| (hD : ∀ i, 0≤D i) (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (q r : ℝ → ι → ℝ≥0) |
| (hq : ∀ i, MonotoneOn (fun u => q u i) (Icc (0:ℝ) 1)) |
| (hr : ∀ i, MonotoneOn (fun u => r u i) (Icc (0:ℝ) 1)) |
| {t : ℝ} (ht : t≠0) |
| (hB : ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |(r u i:ℝ)-(q u i:ℝ)|≤|t| *B i) : |
| |(pathDisorderPressure w φ C r 0-pathDisorderPressure w φ C q 0)/t- |
| pathPressureRate w φ q (fun u i => ((r u i:ℝ)-(q u i:ℝ))/t)| ≤ |
| (3/2:ℝ)*ClockTree.errorJetVariation 2 0 0*(∑ i, B i*(D i)^2)^2*|t| := by |
| have hd : pathPressureRate w φ q (fun u i => ((r u i:ℝ)-(q u i:ℝ))/t)= |
| pathPressureRate w φ q (fun u i => (r u i:ℝ)-(q u i:ℝ))/t := by |
| simp only [div_eq_mul_inv,mul_comm _ t⁻¹] |
| exact pathPressureRate_const_mul w φ q _ _ |
| rw [hd,←sub_div,abs_div] |
| apply (div_le_iff₀ (abs_pos.mpr ht)).mpr |
| have hbnd := pathDisorderPressure_taylor hw φ C D (fun i => |t| *B i) hself hD hφ q r hq hr hB |
| have hs : (∑ i, |t| *B i*(D i)^2)=|t| *(∑ i, B i*(D i)^2) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| apply hbnd.trans |
| rw [hs] |
| exact le_of_eq (by ring) |
|
|
| end SKFluctuations.Hierarchy |
|
|
| |
|
|
| |
|
|
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
|
|
| theorem pathDisorderPressure_hasDerivWithinAt_curve {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (C D B : ι → ℝ) |
| (hself : ∀ σ i, (φ σ (Pi.single i 1))^2=C i) |
| (hD : ∀ i, 0≤D i) (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (q : ℝ → ℝ → ι → ℝ≥0) (K : ℝ → ι → ℝ) (s : Set ℝ) |
| (hq0 : ∀ i, MonotoneOn (fun u => q 0 u i) (Icc (0:ℝ) 1)) |
| (hq : ∀ᶠ t in 𝓝[s \ {0}] 0, ∀ i, MonotoneOn (fun u => q t u i) (Icc (0:ℝ) 1)) |
| (hB : ∀ᶠ t in 𝓝[s \ {0}] 0, ∀ u ∈ Icc (0:ℝ) 1, ∀ i, |
| |(q t u i:ℝ)-(q 0 u i:ℝ)|≤|t| *B i) |
| (hK : ∀ i, ∀ᵐ u ∂volume, u ∈ Ioc (0:ℝ) 1 → |
| Tendsto (fun t => ((q t u i:ℝ)-(q 0 u i:ℝ))/t) (𝓝[s \ {0}] 0) (𝓝 (K u i))) : |
| HasDerivWithinAt (fun t => pathDisorderPressure w φ C (q t) 0) |
| (pathPressureRate w φ (q 0) K) s 0 := by |
| let L : ℝ → ℝ → ι → ℝ := fun t u i => ((q t u i:ℝ)-(q 0 u i:ℝ))/t |
| have hne : ∀ᶠ t in 𝓝[s \ {0}] 0, t≠0 := by |
| filter_upwards [self_mem_nhdsWithin] with t ht |
| exact ht.2 |
| have hlim := pathPressureRate_tendsto hw φ D B hD hφ (q 0) hq0 L K |
| (by |
| filter_upwards [hq] with t ht i |
| exact (clockDirection_intervalIntegrable (q 0) (q t) hq0 ht i).div_const t) |
| (by |
| filter_upwards [hB,hne] with t ht ht0 u hu i |
| dsimp [L] |
| rw [abs_div,div_le_iff₀ (abs_pos.mpr ht0)] |
| simpa only [mul_comm] using ht u ⟨hu.1.le,hu.2⟩ i) |
| hK |
| have herr : Tendsto (fun t => |
| (pathDisorderPressure w φ C (q t) 0-pathDisorderPressure w φ C (q 0) 0)/t- |
| pathPressureRate w φ (q 0) (L t)) (𝓝[s \ {0}] 0) (𝓝 0) := by |
| refine squeeze_zero_norm' (a := fun t => |
| (3/2:ℝ)*ClockTree.errorJetVariation 2 0 0*(∑ i, B i*(D i)^2)^2*|t|) ?_ ?_ |
| · filter_upwards [hq,hB,hne] with t ht hb ht0 |
| rw [Real.norm_eq_abs] |
| exact pathDisorderPressure_slope_error hw φ C D B hself hD hφ (q 0) (q t) hq0 ht ht0 hb |
| · simpa only [abs_zero,mul_zero] using |
| ((continuous_abs.tendsto (0:ℝ)).mono_left nhdsWithin_le_nhds).const_mul |
| ((3/2:ℝ)*ClockTree.errorJetVariation 2 0 0*(∑ i, B i*(D i)^2)^2) |
| rw [hasDerivWithinAt_iff_tendsto_slope] |
| have hs : slope (fun t => pathDisorderPressure w φ C (q t) 0) 0 = |
| fun t => (pathDisorderPressure w φ C (q t) 0-pathDisorderPressure w φ C (q 0) 0)/(t-0) := |
| funext fun t => slope_def_field _ _ _ |
| rw [hs] |
| simpa only [sub_zero,sub_add_cancel,zero_add] using herr.add hlim |
| |
| end SKFluctuations.Hierarchy |
| |
| |
| |
| |
| |
| namespace SKFluctuations.Hierarchy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| variable {S ι : Type*} [Fintype S] [Nonempty S] [Fintype ι] [Nonempty ι] |
| |
| theorem pathPressureRate_integral_direction {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p : ℝ → ι → ℝ≥0) (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| (K : ℝ → ι → ℝ) (hK : ∀ i, IntervalIntegrable (fun u => K u i) volume 0 1) : |
| pathPressureRate w φ p K=-(1/2:ℝ)*∫ u in (0:ℝ)..1, |
| pathPairMoment w φ p (featureKernel φ (K u)) u := by |
| unfold pathPressureRate |
| rw [←intervalIntegral.integral_finsetSum (fun i _ => |
| pathPressureRate_direction_integrable hw φ D hD hφ p hp K hK i)] |
| congr 1 |
| apply intervalIntegral.integral_congr_ae |
| filter_upwards [] with u hu |
| rw [uIoc_of_le (by norm_num : (0:ℝ)≤1)] at hu |
| exact (pathPairMoment_featureKernel hw φ p hp _ hu).symm |
| |
| theorem pathPairMoment_mul_intervalIntegrable {w : S → ℝ} (hw : ∀ σ, 0<w σ) |
| (φ : S → (ι → ℝ) →L[ℝ] ℝ) (D : ι → ℝ) (hD : ∀ i, 0≤D i) |
| (hφ : ∀ σ i, |φ σ (Pi.single i 1)|≤D i) |
| (p : ℝ → ι → ℝ≥0) (hp : ∀ i, MonotoneOn (fun u => p u i) (Icc (0:ℝ) 1)) |
| (P : S → S → ℝ) {B : ℝ} (hB : 0≤B) (hP : ∀ σ τ, |P σ τ|≤B) |
| (K : ℝ → ℝ) (hK : IntervalIntegrable K volume 0 1) : |
| IntervalIntegrable (fun u => K u*pathPairMoment w φ p P u) volume 0 1 := |
| intervalIntegrable_mul_bounded hK (pathPairMoment_intervalIntegrable_all hw φ p hp _) |
| (fun _ hu => pathPairMoment_bound hw φ D hD hφ p hp P hB hP hu) |
| |
| end SKFluctuations.Hierarchy |
| |
| |
| |
| |
| |
| namespace SKFluctuations |
| open MeasureTheory ProbabilityTheory Filter Set Hierarchy |
| open scoped Topology NNReal BigOperators |
| |
| def orderedPathClock (n : ℕ) (b h : ℝ → ℝ≥0) (u : ℝ) : OrderedFeatureIndex n → ℝ≥0 := |
| Sum.elim (fun _ => h u) (fun _ => b u) |
| |
| theorem orderedPathClock_monotone (n : ℕ) {b h : ℝ → ℝ≥0} |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hh : MonotoneOn h (Icc (0:ℝ) 1)) : |
| ∀ i, MonotoneOn (fun u => orderedPathClock n b h u i) (Icc (0:ℝ) 1) := by |
| intro i |
| cases i with |
| | inl i => exact hh |
| | inr i => exact hb |
| |
| def orderedSKPathPressure (n : ℕ) (b h : ℝ → ℝ≥0) |
| (x : OrderedFeatureIndex n → ℝ) : ℝ := |
| pathDisorderPressure (uniformWeight n) (orderedFeature n) (fun i => (orderedAmplitude n i)^2) |
| (orderedPathClock n b h) x/(n:ℝ) |
| |
| theorem orderedPathClock_l1 (n : ℕ) (hn : 0<n) (b h b' h' : ℝ → ℝ≥0) : |
| clockL1Distance (orderedAmplitude n) (orderedPathClock n b h) (orderedPathClock n b' h') = |
| (n:ℝ)*((1/4:ℝ)*(∫ u in (0:ℝ)..1, |(b u:ℝ)-(b' u:ℝ)|)+ |
| (1/2:ℝ)*(∫ u in (0:ℝ)..1, |(h u:ℝ)-(h' u:ℝ)|)) := by |
| unfold clockL1Distance |
| have he : (∑ i : OrderedFeatureIndex n, (orderedAmplitude n i)^2* |
| (∫ u in (0:ℝ)..1, |(orderedPathClock n b h u i:ℝ)-(orderedPathClock n b' h' u i:ℝ)|)) = |
| (n:ℝ)*((∫ u in (0:ℝ)..1, |(h u:ℝ)-(h' u:ℝ)|)+ |
| (∫ u in (0:ℝ)..1, |(b u:ℝ)-(b' u:ℝ)|)/2) := by |
| calc |
| _ = ∑ i : OrderedFeatureIndex n, Sum.elim |
| (fun _ => ∫ u in (0:ℝ)..1, |(h u:ℝ)-(h' u:ℝ)|) |
| (fun _ => ∫ u in (0:ℝ)..1, |(b u:ℝ)-(b' u:ℝ)|) i*(orderedAmplitude n i)^2 := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| cases i <;> exact mul_comm _ _ |
| _ = _ := orderedAmplitude_sum n hn _ _ |
| rw [he] |
| ring |
| |
| theorem orderedSKPathPressure_l1_comparison (n : ℕ) (hn : 0<n) |
| (b h b' h' : ℝ → ℝ≥0) |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hh : MonotoneOn h (Icc (0:ℝ) 1)) |
| (hb' : MonotoneOn b' (Icc (0:ℝ) 1)) (hh' : MonotoneOn h' (Icc (0:ℝ) 1)) |
| (x : OrderedFeatureIndex n → ℝ) : |
| |orderedSKPathPressure n b h x-orderedSKPathPressure n b' h' x| ≤ |
| (1/4:ℝ)*(∫ u in (0:ℝ)..1, |(b u:ℝ)-(b' u:ℝ)|)+ |
| (1/2:ℝ)*(∫ u in (0:ℝ)..1, |(h u:ℝ)-(h' u:ℝ)|) := by |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| have hnp : 0<(n:ℝ) := Nat.cast_pos.mpr hn |
| have hd := pathDisorderPressure_l1_comparison (uniformWeight_pos n) (orderedFeature n) |
| (fun i => (orderedAmplitude n i)^2) (orderedAmplitude n) (orderedFeature_square n) |
| (fun i σ => (orderedFeature_abs n σ i).le) (orderedPathClock n b h) (orderedPathClock n b' h') |
| (orderedPathClock_monotone n hb hh) (orderedPathClock_monotone n hb' hh') x |
| rw [orderedPathClock_l1 n hn] at hd |
| unfold orderedSKPathPressure |
| rw [← sub_div,abs_div,abs_of_pos hnp] |
| exact (div_le_iff₀ hnp).mpr (by simpa only [mul_comm] using hd) |
| |
| end SKFluctuations |
| |
| |
| |
| |
| |
| namespace SKFluctuations |
| open MeasureTheory ProbabilityTheory Filter Set Hierarchy |
| open scoped Topology NNReal BigOperators |
| attribute [local instance] Classical.propDecidable |
| local instance (n : ℕ) : DecidableEq (OrderedFeatureIndex n) := fun _ _ => Classical.propDecidable _ |
| |
| def orderedPathPair (n : ℕ) (b h : ℝ → ℝ≥0) (P : Spin n → Spin n → ℝ) (u : ℝ) : ℝ := |
| pathPairMoment (uniformWeight n) (orderedFeature n) (orderedPathClock n b h) P u |
| def orderedPathS (n : ℕ) (b h : ℝ → ℝ≥0) : ℝ → ℝ := |
| orderedPathPair n b h (orderedOverlap n) |
| def orderedPathB (n : ℕ) (b h : ℝ → ℝ≥0) : ℝ → ℝ := |
| orderedPathPair n b h (fun σ τ => (orderedOverlap n σ τ)^2) |
| |
| theorem orderedPathPair_linear (n : ℕ) (hn : 0<n) (b h : ℝ → ℝ≥0) |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hh : MonotoneOn h (Icc (0:ℝ) 1)) |
| (P Q : Spin n → Spin n → ℝ) (c d : ℝ) {u : ℝ} (hu : u ∈ Ioc (0:ℝ) 1) : |
| orderedPathPair n b h (fun σ τ => c*P σ τ+d*Q σ τ) u= |
| c*orderedPathPair n b h P u+d*orderedPathPair n b h Q u := by |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| unfold orderedPathPair |
| rw [pathPairMoment_add (uniformWeight_pos n) _ _ (orderedPathClock_monotone n hb hh) _ _ hu, |
| pathPairMoment_const_mul (uniformWeight_pos n) _ _ (orderedPathClock_monotone n hb hh) _ _ hu, |
| pathPairMoment_const_mul (uniformWeight_pos n) _ _ (orderedPathClock_monotone n hb hh) _ _ hu] |
| |
| theorem orderedPathPair_direction_integrable (n : ℕ) (hn : 0<n) (b h q r : ℝ → ℝ≥0) |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hh : MonotoneOn h (Icc (0:ℝ) 1)) |
| (hq : MonotoneOn q (Icc (0:ℝ) 1)) (hr : MonotoneOn r (Icc (0:ℝ) 1)) |
| (P : Spin n → Spin n → ℝ) : |
| IntervalIntegrable (fun u => ((r u:ℝ)-(q u:ℝ))*orderedPathPair n b h P u) volume 0 1 := by |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| have hi := clockDirection_intervalIntegrable (fun (u : ℝ) (_ : Unit) => q u) (fun (u : ℝ) (_ : Unit) => r u) |
| (fun _ => hq) (fun _ => hr) () |
| simpa only [mul_comm,orderedPathPair] using intervalIntegrable_mul_bounded |
| (pathPairMoment_intervalIntegrable_all (uniformWeight_pos n) (orderedFeature n) |
| (orderedPathClock n b h) (orderedPathClock_monotone n hb hh) P) hi |
| (fun u hu => clockDirection_bound (fun (u : ℝ) (_ : Unit) => q u) (fun (u : ℝ) (_ : Unit) => r u) |
| (fun _ => hq) (fun _ => hr) ⟨hu.1.le,hu.2⟩ ()) |
| |
| theorem pathPressureRate_ordered (n : ℕ) (hn : 0<n) (b h b₀ h₀ b₁ h₁ : ℝ → ℝ≥0) |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hh : MonotoneOn h (Icc (0:ℝ) 1)) |
| (hb₀ : MonotoneOn b₀ (Icc (0:ℝ) 1)) (hh₀ : MonotoneOn h₀ (Icc (0:ℝ) 1)) |
| (hb₁ : MonotoneOn b₁ (Icc (0:ℝ) 1)) (hh₁ : MonotoneOn h₁ (Icc (0:ℝ) 1)) : |
| pathPressureRate (uniformWeight n) (orderedFeature n) (orderedPathClock n b h) |
| (fun u i => (orderedPathClock n b₁ h₁ u i:ℝ)-(orderedPathClock n b₀ h₀ u i:ℝ))/(n:ℝ)= |
| -(1/4:ℝ)*(∫ u in (0:ℝ)..1, ((b₁ u:ℝ)-(b₀ u:ℝ))*orderedPathB n b h u)- |
| (1/2:ℝ)*(∫ u in (0:ℝ)..1, ((h₁ u:ℝ)-(h₀ u:ℝ))*orderedPathS n b h u) := by |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| have hk (u : ℝ) : featureKernel (orderedFeature n) |
| (fun i => (orderedPathClock n b₁ h₁ u i:ℝ)-(orderedPathClock n b₀ h₀ u i:ℝ))= |
| fun σ τ => ((n:ℝ)/2*((b₁ u:ℝ)-(b₀ u:ℝ)))*(orderedOverlap n σ τ)^2+ |
| ((n:ℝ)*((h₁ u:ℝ)-(h₀ u:ℝ)))*orderedOverlap n σ τ := by |
| funext σ τ |
| have he : (fun i => (orderedPathClock n b₁ h₁ u i:ℝ)-(orderedPathClock n b₀ h₀ u i:ℝ))= |
| Sum.elim (fun _ => (h₁ u:ℝ)-(h₀ u:ℝ)) (fun _ => (b₁ u:ℝ)-(b₀ u:ℝ)) := by |
| funext i; cases i <;> rfl |
| rw [he] |
| simp only [featureKernel] |
| have hs (σ : Spin n) (i : OrderedFeatureIndex n) : |
| orderedFeature n σ (Pi.single i 1)=orderedFeatureCoefficient n σ i := by |
| simp [orderedFeature_apply,Pi.single_apply] |
| simp only [hs] |
| rw [orderedCovariance n hn] |
| ring |
| rw [pathPressureRate_integral (uniformWeight_pos n) _ _ _ _ (orderedPathClock_monotone n hb hh) |
| (orderedPathClock_monotone n hb₀ hh₀) (orderedPathClock_monotone n hb₁ hh₁)] |
| have he : (∫ u in (0:ℝ)..1, pathPairMoment (uniformWeight n) (orderedFeature n) |
| (orderedPathClock n b h) (featureKernel (orderedFeature n) |
| (fun i => (orderedPathClock n b₁ h₁ u i:ℝ)-(orderedPathClock n b₀ h₀ u i:ℝ))) u)= |
| ∫ u in (0:ℝ)..1, (n:ℝ)/2*(((b₁ u:ℝ)-(b₀ u:ℝ))*orderedPathB n b h u)+ |
| (n:ℝ)*(((h₁ u:ℝ)-(h₀ u:ℝ))*orderedPathS n b h u) := by |
| apply intervalIntegral.integral_congr_ae |
| filter_upwards [] with u hu |
| rw [uIoc_of_le (by norm_num : (0:ℝ)≤1)] at hu |
| rw [hk] |
| change orderedPathPair n b h _ u=_ |
| rw [orderedPathPair_linear n hn b h hb hh _ _ _ _ hu] |
| dsimp [orderedPathB,orderedPathS] |
| ring |
| rw [he] |
| change (-(1/2:ℝ)*(∫ u in (0:ℝ)..1, (n:ℝ)/2*(((b₁ u:ℝ)-(b₀ u:ℝ))* |
| orderedPathPair n b h (fun σ τ => (orderedOverlap n σ τ)^2) u)+(n:ℝ)* |
| (((h₁ u:ℝ)-(h₀ u:ℝ))*orderedPathPair n b h (orderedOverlap n) u)))/(n:ℝ)=_ |
| rw [intervalIntegral.integral_add |
| ((orderedPathPair_direction_integrable n hn b h b₀ b₁ hb hh hb₀ hb₁ _).const_mul ((n:ℝ)/2)) |
| ((orderedPathPair_direction_integrable n hn b h h₀ h₁ hb hh hh₀ hh₁ _).const_mul (n:ℝ)), |
| intervalIntegral.integral_const_mul,intervalIntegral.integral_const_mul] |
| have hn0 : (n:ℝ)≠0 := ne_of_gt (Nat.cast_pos.mpr hn) |
| dsimp only [orderedPathB,orderedPathS] |
| field_simp |
| ring |
| |
| def affineNNPath (b c : ℝ → ℝ≥0) (s : ℝ) (u : ℝ) : ℝ≥0 := |
| Real.toNNReal (1-s)*b u+Real.toNNReal s*c u |
| |
| theorem affineNNPath_monotone (b c : ℝ → ℝ≥0) |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hc : MonotoneOn c (Icc (0:ℝ) 1)) (s : ℝ) : |
| MonotoneOn (affineNNPath b c s) (Icc (0:ℝ) 1) := by |
| intro u hu v hv huv |
| exact add_le_add (mul_le_mul_of_nonneg_left (hb hu hv huv) zero_le) |
| (mul_le_mul_of_nonneg_left (hc hu hv huv) zero_le) |
| |
| theorem orderedSKPathPressure_hasDerivAt (n : ℕ) (hn : 0<n) (b h b' h' : ℝ → ℝ≥0) |
| (hb : MonotoneOn b (Icc (0:ℝ) 1)) (hh : MonotoneOn h (Icc (0:ℝ) 1)) |
| (hb' : MonotoneOn b' (Icc (0:ℝ) 1)) (hh' : MonotoneOn h' (Icc (0:ℝ) 1)) |
| {t : ℝ} (ht : t ∈ Ioo (0:ℝ) 1) : |
| HasDerivAt (fun s => orderedSKPathPressure n (affineNNPath b b' s) (affineNNPath h h' s) 0) |
| (-(1/4:ℝ)*(∫ u in (0:ℝ)..1, ((b' u:ℝ)-(b u:ℝ))* |
| orderedPathB n (affineNNPath b b' t) (affineNNPath h h' t) u)- |
| (1/2:ℝ)*(∫ u in (0:ℝ)..1, ((h' u:ℝ)-(h u:ℝ))* |
| orderedPathS n (affineNNPath b b' t) (affineNNPath h h' t) u)) t := by |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| have he (s : ℝ) : affineClockPath (orderedPathClock n b h) (orderedPathClock n b' h') s= |
| orderedPathClock n (affineNNPath b b' s) (affineNNPath h h' s) := by |
| funext u i; cases i <;> rfl |
| have hd := (pathDisorderPressure_hasDerivAt (uniformWeight_pos n) (orderedFeature n) |
| (fun i => (orderedAmplitude n i)^2) (fun σ i => by |
| simpa [orderedFeature_apply,Pi.single_apply] using orderedFeature_square n σ i) |
| (orderedPathClock n b h) (orderedPathClock n b' h') |
| (orderedPathClock_monotone n hb hh) (orderedPathClock_monotone n hb' hh') ht).div_const (n:ℝ) |
| simp_rw [he] at hd |
| rw [pathPressureRate_ordered n hn _ _ b h b' h' |
| (affineNNPath_monotone b b' hb hb' t) (affineNNPath_monotone h h' hh hh' t) hb hh hb' hh'] at hd |
| exact hd |
| |
| end SKFluctuations |
| |
| end |
| |
| end OAI |
| |
