openAI-math-dataset / lean /OAI /Probability /Ballisticity /Geometry /SuccessfulAverageTubeTransfer.lean
| import Mathlib | |
| import OAI.Probability.Ballisticity.Estimates.ConvolutionCentersClose | |
| import OAI.Probability.Ballisticity.Coupling.QuenchedConditionedTubeTransfer | |
| import OAI.Probability.Ballisticity.Estimates.SuccessfulAverage | |
| namespace OAI | |
| section | |
| section | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Pointwise Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical BoundedContinuousFunction | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| namespace DirectionalTransience | |
| theorem successfulAverage_tube_transfer {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] | |
| (hue : UniformElliptic ν) (e f : Direction d) | |
| (htrans : DirectionallyTransient ν (realPosition (step e))) | |
| (b : ℕ → ℝ) (H : ℕ → ℕ) (hH : Tendsto H atTop atTop) | |
| (z : ℕ → ℝ) (hz : Tendsto z atTop atTop) : | |
| limsup (fun i => (successfulAverage ν (realPosition (step e)) (H i)).real | |
| (MedianTubeFailure (realPosition (step e)) f b (H i) (z i))) atTop ≤ | |
| 10*limsup (fun i => (conditionedLaw ν (realPosition (step e))).real | |
| (MedianTubeFailure (realPosition (step e)) f b (H i) (z i/10))) atTop := by | |
| let ℓ := realPosition (step e) | |
| let μ := fun i => successfulAverage ν ℓ (H i) | |
| let η := fun i => rawConditionedWeight ν ℓ (fun ω => (crossingQuenched ℓ 0 (H i) ω)⁻¹) | |
| let : ∀ i, IsProbabilityMeasure (μ i) := fun i => successfulAverage_probability ν hue ℓ (signed_direction_unit e) htrans (H i) | |
| let : IsProbabilityMeasure (quenchedConditionedAverage ν ℓ) := | |
| quenchedConditionedAverage_probability ν hue ℓ (signed_direction_unit e) htrans | |
| let a := fun i => (μ i).real (MedianTubeFailure ℓ f b (H i) (z i)) | |
| let q := fun i => (quenchedConditionedAverage ν ℓ).real (MedianTubeFailure ℓ f b (H i) (z i)) | |
| have hab : IsBoundedUnder (· ≤ ·) atTop a := ⟨1, | |
| (show ∀ᶠ i in atTop, a i ≤ 1 from Eventually.of_forall fun _ => measureReal_le_one)⟩ | |
| have hqb : IsBoundedUnder (· ≤ ·) atTop q := ⟨1, | |
| (show ∀ᶠ i in atTop, q i ≤ 1 from Eventually.of_forall fun _ => measureReal_le_one)⟩ | |
| have hac : IsCoboundedUnder (· ≤ ·) atTop a := | |
| (show IsBoundedUnder (· ≥ ·) atTop a from ⟨0, | |
| (show ∀ᶠ i in atTop, 0 ≤ a i from Eventually.of_forall fun _ => measureReal_nonneg)⟩).isCoboundedUnder_le | |
| have hm : Tendsto (fun i => (η i).real Set.univ) atTop (𝓝 1) := by | |
| have hh := (ENNReal.continuousAt_toReal (by simp : (1:ℝ≥0∞) ≠ ∞)).tendsto.comp | |
| ((raw_crossingDensity_mass_tendsto ν hue ℓ (signed_direction_unit e) htrans).comp hH) | |
| simpa only [Function.comp_def,Measure.real,ENNReal.toReal_one,η] using hh | |
| have hmain : limsup a atTop ≤ limsup q atTop := by | |
| apply (limsup_le_iff hac hab).mpr | |
| intro u hu | |
| have hqt := eventually_lt_of_limsup_lt | |
| (show limsup q atTop < limsup q atTop+(u-limsup q atTop)/3 by linarith) hqb | |
| have hmt := hm.eventually (lt_mem_nhds (show 1-(u-limsup q atTop)/3 < (1:ℝ) by linarith)) | |
| filter_upwards [hqt,hmt] with i hi hi' | |
| have hb := successfulAverage_comparison ν hue ℓ (signed_direction_unit e) htrans (H i) | |
| (MedianTubeFailure ℓ f b (H i) (z i)) (measurableSet_medianTubeFailure ℓ f b (H i) (z i)) | |
| change a i ≤ q i+1-(η i).real Set.univ at hb | |
| linarith | |
| exact hmain.trans (quenchedConditioned_tube_transfer ν hue e f htrans b H z hz) | |
| end DirectionalTransience | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| namespace DirectionalTransience | |
| lemma conditioned_median_tube_bound {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] | |
| (e f : Direction d) (htrans : DirectionallyTransient ν (realPosition (step e))) | |
| (H : ℕ) {z : ℝ} (hz : 0 < z) : | |
| let ℓ := realPosition (step e) | |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) | |
| let p := (annealedLaw ν (NoDrop ℓ 0)).toReal | |
| p^2/2 * (conditionedLaw ν ℓ).real | |
| (MedianTubeFailure ℓ f (fun j => (recordMedian ν ℓ hp f j:ℝ)) H z) ≤ | |
| 5*(H:ℝ)/fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) z := by | |
| dsimp only | |
| let ℓ := realPosition (step e) | |
| have hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) | |
| have he : MedianTubeFailure ℓ f (fun j => (recordMedian ν ℓ hp f j:ℝ)) H z =ᵐ[conditionedLaw ν ℓ] | |
| {X | ∃ j, 0 < j ∧ j ≤ H ∧ z < |signedCoordinate f (recordIndexPosition ℓ j X)-(recordMedian ν ℓ hp f j:ℝ)|} := by | |
| filter_upwards [(conditionedLaw_absolutelyContinuous ν ℓ).ae_le (annealed_initial ν)] with X hX | |
| apply propext | |
| constructor | |
| · rintro ⟨j,hj,hbad⟩ | |
| have hj0 : 0 < j := by | |
| by_contra hn | |
| have hjz : j = 0 := by omega | |
| simp [hjz,recordIndexPosition,recordIndexTime_zero,hX,recordMedian_zero,signedCoordinate] at hbad | |
| linarith | |
| exact ⟨j,hj0,hj,hbad⟩ | |
| · rintro ⟨j,_,hj,hbad⟩ | |
| exact ⟨j,hj,hbad⟩ | |
| rw [measureReal_congr he] | |
| exact conditioned_median_max_bound ν ℓ htrans (signedHeight e) (signedHeight_projection e) | |
| (signedHeight_step_le e) f H hz | |
| lemma conditioned_median_tube_short_bound {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] | |
| (hue : UniformElliptic ν) (e f : Direction d) (hef : e.1 ≠ f.1) | |
| (htrans : DirectionallyTransient ν (realPosition (step e))) | |
| {r c t : ℝ} (hr : 0 < r) (hc : 0 < c) (hc1 : c ≤ 1) (ht : 0 ≤ t) | |
| (H : ℕ) (hH : (H:ℝ) ≤ t*fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) | |
| (commonIncrementProcess (realPosition (step e)) f 0) r) : | |
| let ℓ := realPosition (step e) | |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) | |
| let p := (annealedLaw ν (NoDrop ℓ 0)).toReal | |
| (conditionedLaw ν ℓ).real (MedianTubeFailure ℓ f (fun j => (recordMedian ν ℓ hp f j:ℝ)) H (c*r)) ≤ | |
| 10*t/(p^2*c^2) := by | |
| dsimp only | |
| let ℓ := realPosition (step e) | |
| let μ := independentConditionedPairLaw ν ℓ | |
| let S := commonIncrementProcess ℓ f 0 | |
| let p := (annealedLaw ν (NoDrop ℓ 0)).toReal | |
| have hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) | |
| have hp0 : 0 < p := ENNReal.toReal_pos hp (measure_ne_top _ _) | |
| let : IsProbabilityMeasure μ := independentConditionedPairLaw_probability ν ℓ hp | |
| have hs := measurable_commonIncrementProcess ℓ f 0 | |
| have hne := independent_commonWordIncrement_nonzero ν hue e f hef htrans | |
| have hnr : 0 < fluctuationScale μ S r := div_pos (sq_pos_of_pos hr) (truncatedVariance_pos μ S hs hne hr) | |
| have hnc : 0 < fluctuationScale μ S (c*r) := div_pos (sq_pos_of_pos (mul_pos hc hr)) | |
| (truncatedVariance_pos μ S hs hne (mul_pos hc hr)) | |
| have hsc := fluctuationScale_scaling μ S hs hne hr hc hc1 | |
| have hb := conditioned_median_tube_bound ν e f htrans H (mul_pos hc hr) | |
| have hfrac : 5*(H:ℝ)/fluctuationScale μ S (c*r) ≤ 5*t/c^2 := by | |
| apply (div_le_iff₀ hnc).mpr | |
| have hh := mul_le_mul_of_nonneg_left hsc (show 0 ≤ 5*t/c^2 by positivity) | |
| have heq : (5*t/c^2)*(c^2*fluctuationScale μ S r) = 5*t*fluctuationScale μ S r := by field_simp | |
| rw [heq] at hh | |
| nlinarith | |
| have hh := hb.trans hfrac | |
| have hd : 0 < p^2/2 := by positivity | |
| have heq : 5*t/c^2 = (p^2/2)*(10*t/(p^2*c^2)) := by field_simp; ring | |
| rw [heq] at hh | |
| exact (mul_le_mul_iff_right₀ hd).mp hh | |
| theorem quenched_median_tube_short_uniform {d : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] | |
| (hue : UniformElliptic ν) (e f : Direction d) (hef : e.1 ≠ f.1) | |
| (htrans : DirectionallyTransient ν (realPosition (step e))) | |
| {w : ℝ} (hw : 0 < w) (hw1 : w ≤ 10) : | |
| ∃ C : ℝ, 0 < C ∧ ∀ t : ℝ, 0 < t → ∃ R : ℝ, 0 < R ∧ ∀ r : ℝ, R ≤ r → | |
| ∀ H : ℕ, (H:ℝ) ≤ t*fluctuationScale (independentConditionedPairLaw ν (realPosition (step e))) | |
| (commonIncrementProcess (realPosition (step e)) f 0) r → | |
| let ℓ := realPosition (step e) | |
| let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) | |
| (quenchedConditionedAverage ν ℓ).real | |
| (MedianTubeFailure ℓ f (fun j => (recordMedian ν ℓ hp f j:ℝ)) H (w*r)) ≤ C*t := by | |
| let ℓ := realPosition (step e) | |
| have hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) | |
| let p := (annealedLaw ν (NoDrop ℓ 0)).toReal | |
| have hp0 : 0 < p := ENNReal.toReal_pos hp (measure_ne_top _ _) | |
| let C := 100/(p^2*(w/10)^2) | |
| have hC : 0 < C := by dsimp [C]; positivity | |
| let μ := quenchedConditionedAverage ν ℓ | |
| let : IsProbabilityMeasure μ := quenchedConditionedAverage_probability ν hue ℓ (signed_direction_unit e) htrans | |
| let n := fluctuationScale (independentConditionedPairLaw ν ℓ) (commonIncrementProcess ℓ f 0) | |
| let b := fun j => (recordMedian ν ℓ hp f j:ℝ) | |
| refine ⟨2*C,by positivity,?_⟩ | |
| intro t ht | |
| by_contra hn | |
| push Not at hn | |
| have hbad (i : ℕ) : ∃ r : ℝ, (max (i:ℝ) 1) ≤ r ∧ ∃ H : ℕ, (H:ℝ) ≤ t*n r ∧ | |
| 2*C*t < μ.real (MedianTubeFailure ℓ f b H (w*r)) := by | |
| obtain ⟨r,hr,H,hH,hbad⟩ := hn (max (i:ℝ) 1) (lt_of_lt_of_le zero_lt_one (le_max_right _ _)) | |
| exact ⟨r,hr,H,hH,lt_of_not_ge hbad⟩ | |
| choose r hr H hH hbad using hbad | |
| have hrpos (i : ℕ) : 0 < r i := lt_of_lt_of_le zero_lt_one ((le_max_right _ _).trans (hr i)) | |
| have hrinf : Tendsto r atTop atTop := tendsto_atTop_mono (fun i => (le_max_left _ _).trans (hr i)) tendsto_natCast_atTop_atTop | |
| let q := fun i => (conditionedLaw ν ℓ).real (MedianTubeFailure ℓ f b (H i) ((w*r i)/10)) | |
| let a := fun i => μ.real (MedianTubeFailure ℓ f b (H i) (w*r i)) | |
| have hqb : ∀ i, q i ≤ 10*t/(p^2*(w/10)^2) := by | |
| intro i | |
| have hh := conditioned_median_tube_short_bound ν hue e f hef htrans (hrpos i) | |
| (show 0 < w/10 by positivity) (show w/10 ≤ 1 by linarith) ht.le (H i) (hH i) | |
| simpa only [show w/10*r i = w*r i/10 by ring] using hh | |
| have hqc : IsCoboundedUnder (· ≤ ·) atTop q := | |
| (show IsBoundedUnder (· ≥ ·) atTop q from ⟨0, | |
| (show ∀ᶠ i in atTop, 0 ≤ q i from Eventually.of_forall fun _ => measureReal_nonneg)⟩).isCoboundedUnder_le | |
| have hlimq := limsup_le_of_le hqc (Eventually.of_forall hqb) | |
| have htransfer := quenchedConditioned_tube_transfer ν hue e f htrans b H (fun i => w*r i) (hrinf.const_mul_atTop hw) | |
| have hlim : limsup a atTop ≤ C*t := by | |
| have hh := htransfer.trans (mul_le_mul_of_nonneg_left hlimq (by norm_num : (0:ℝ) ≤ 10)) | |
| have heq : 10*(10*t/(p^2*(w/10)^2)) = C*t := by dsimp [C]; ring | |
| rwa [heq] at hh | |
| have hab : IsBoundedUnder (· ≤ ·) atTop a := ⟨1, | |
| (show ∀ᶠ i in atTop, a i ≤ 1 from Eventually.of_forall fun _ => measureReal_le_one)⟩ | |
| have hlo : 2*C*t ≤ limsup a atTop := le_limsup_of_frequently_le | |
| (Frequently.of_forall fun i => (hbad i).le) hab | |
| nlinarith | |
| end DirectionalTransience | |
| open MeasureTheory ProbabilityTheory Filter | |
| open scoped ENNReal NNReal BigOperators Topology Classical | |
| end | |
| end | |
| end OAI | |