import Mathlib import OAI.Probability.Ballisticity.Coupling.QuenchedPrefixDeterminedFuture import OAI.Probability.Ballisticity.Estimates.RecordOrPrefixAdd 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 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 namespace DirectionalTransience lemma tubePrefix_mono_radius {d : ℕ} (ℓ : Vector d) (f : Direction d) (x : Lattice d) (θ z z' : ℝ) (hzz : z ≤ z') (s : ℕ) : TubePrefix ℓ f x θ z s ⊆ TubePrefix ℓ f x θ z' s := by rintro X hX obtain ⟨n,hn⟩ := Set.mem_iUnion.mp hX apply Set.mem_iUnion.mpr refine ⟨n,hn.1,?_⟩ intro j hj obtain ⟨a,ha,hr,hg⟩ := hn.2 j hj exact ⟨a,ha,hr,hg.trans hzz⟩ lemma tubePrefixMass_mono_radius {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z z' : ℝ) (hzz : z ≤ z') (s : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : tubePrefixMass ℓ f θ z s π ω ≤ tubePrefixMass ℓ f θ z' s π ω := lintegral_mono fun x => measure_mono (tubePrefix_mono_radius ℓ f x θ z z' hzz s) lemma tubePrefixMass_comp_le {d : ℕ} (e f : Direction d) (θ z₁ z₂ : ℝ) (hz₂ : 0 ≤ z₂) (s k : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : tubePrefixMass (realPosition (step e)) f θ z₂ k (tubeEndpointMixture (realPosition (step e)) f θ z₁ s π ω) ω ≤ tubePrefixMass (realPosition (step e)) f θ (z₁+z₂) (s+k) π ω := by let ℓ := realPosition (step e) rw [tubePrefixMass,tubeEndpointMixture_future ℓ f θ z₁ s π ω (fun y => TubePrefix ℓ f y θ z₂ k) (fun y => measurableSet_tubePrefix ℓ f y θ z₂ k)] apply lintegral_mono; intro x apply measure_mono_ae filter_upwards [quenched_initial_ae (ω,x),quenched_nearest_neighbor (ω,x)] with X h0 hnn intro hX obtain ⟨n,hn⟩ := Set.mem_iUnion.mp hX obtain ⟨m,hm⟩ := Set.mem_iUnion.mp hn.2 exact Set.mem_iUnion.mpr ⟨n+m,tubePrefixAt_concatenate e f x θ z₁ z₂ hz₂ s n k m X h0 hnn hn.1 hm⟩ lemma directBridge_lintegral {d : ℕ} (e : Direction d) (π : Measure (Lattice d)) (ω : Environment d) (G : Lattice d → ℝ≥0∞) : (∫⁻ y, G y ∂directBridge e π ω) = ∫⁻ x, ((ω x).1 e : ℝ≥0∞)*G (x+step e) ∂π := by rw [directBridge,lintegral_sum_measure,lintegral_countable'] apply tsum_congr; intro x rw [lintegral_smul_measure,lintegral_dirac',smul_eq_mul] · ring · exact measurable_of_countable G lemma single_step_tubePrefixAt {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) (x : Lattice d) (θ : ℝ) (X : Path d) (h0 : X 0=x) (h1 : X 1=x+step e) : X ∈ TubePrefixAt (realPosition (step e)) f x θ |θ| 1 1 := by let ℓ := realPosition (step e) have hheight : dot (realPosition (X 1)) ℓ = dot (realPosition x) ℓ+1 := by rw [h1,signedHeight_projection,signedHeight_add_step_self,Int.cast_add,Int.cast_one,signedHeight_projection] have hzero : dot (realPosition (0:Lattice d)) ℓ=0 := by simp [dot,realPosition] have hstrict : StrictRecord ℓ (fun k => X k-x) 1 := by intro j hj have hj0 : j=0 := by omega simp only [hj0,h0,sub_self,hzero] rw [dot_realPosition_sub,hheight] linarith refine ⟨⟨?_,?_⟩,?_⟩ · simpa only [Upper,Set.mem_ofPred_eq,Nat.cast_one] using le_of_eq hheight.symm · intro j hj have hj0 : j=0 := by omega simp only [Strip,Set.mem_ofPred_eq,Nat.cast_one,hj0,h0] exact ⟨le_rfl,lt_add_one _⟩ · intro j hj rcases show j=0 ∨ j=1 by omega with rfl | rfl · refine ⟨0,by omega,by simp [RecordOrZeroPrefix],?_⟩ simp [h0,signedCoordinate] · refine ⟨1,le_rfl,?_,?_⟩ · rw [RecordOrZeroPrefix,ite_eq_right (by decide : (1:ℕ) ≠ 0)] refine ⟨by decide,hstrict,?_,?_⟩ · rw [show (1:ℕ)=0+1 from rfl,recordCount_succ,recordCount_zero,ite_eq_left hstrict] · intro j hj rcases show j=0 ∨ j=1 by omega with rfl | rfl · change dot (realPosition (0:Lattice d)) ℓ ≤ dot (realPosition (X 0-x)) ℓ rw [h0,sub_self] · change dot (realPosition (0:Lattice d)) ℓ ≤ dot (realPosition (X 1-x)) ℓ rw [dot_realPosition_sub,hheight,hzero]; linarith · have he : X 1-x = step e := by rw [h1]; abel have hz : signedCoordinate f (step e)=0 := by have hh := signedCoordinate_add_distinct_step e f hef (0:Lattice d) simpa [signedCoordinate] using hh simp only [he,hz,Nat.cast_one,one_mul,zero_sub,abs_neg,le_refl] lemma directBridge_tubeMass_le {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) (θ z : ℝ) (hz : 0 ≤ z) (k : ℕ) (π : Measure (Lattice d)) (ω : Environment d) : tubePrefixMass (realPosition (step e)) f θ z k (directBridge e π ω) ω ≤ tubePrefixMass (realPosition (step e)) f θ (|θ|+z) (1+k) π ω := by let ℓ := realPosition (step e) rw [tubePrefixMass,directBridge_lintegral] apply lintegral_mono; intro x let v := wordPath x [e] have hm := quenched_prefix_future ω x v (by simp [v]) 1 (measurableSet_tubePrefix ℓ f (x+step e) θ z k) have hv1 : v 1=x+step e := by simp [v,wordPath] rw [hv1,show pathCylinder v 1=wordCylinder x [e] from rfl,quenched_wordCylinder] at hm have hw : ENNReal.ofReal (wordWeight ω x [e])=((ω x).1 e:ℝ≥0∞) := by simp [wordWeight] rw [hw] at hm change ((ω x).1 e:ℝ≥0∞)*(quenchedKernel (ω,x+step e)) (TubePrefix ℓ f (x+step e) θ z k) ≤ _ rw [← hm] apply measure_mono_ae filter_upwards [quenched_nearest_neighbor (ω,x)] with X hnn rintro ⟨hfut,hcy⟩ have h0 : X 0=x := by simpa [v] using hcy 0 (by omega) have h1 : X 1=x+step e := by simpa [wordPath] using hcy 1 le_rfl obtain ⟨m,hm⟩ := Set.mem_iUnion.mp hfut apply Set.mem_iUnion.mpr exact ⟨1+m,tubePrefixAt_concatenate e f x θ |θ| z hz 1 1 k m X h0 hnn (single_step_tubePrefixAt e f hef x θ X h0 h1) (by simpa only [h1] using hm)⟩ lemma bridged_continuation_le_tube {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) (θ z : ℝ) (hz : 0 ≤ z) (hθ : |θ| ≤ z) (H s : ℕ) (hs0 : 0 < s) (hsH : s ≤ H) (π : Measure (Lattice d)) (ω : Environment d) : tubePrefixMass (realPosition (step e)) f θ z (H-s) (directBridge e (stoppedTubeEndpoint (realPosition (step e)) f θ z (s-1) π ω) ω) ω ≤ tubePrefixMass (realPosition (step e)) f θ (3*z) H π ω := by let ℓ := realPosition (step e) have hb := directBridge_tubeMass_le e f hef θ z hz (H-s) (stoppedTubeEndpoint ℓ f θ z (s-1) π ω) ω by_cases hs : s-1=0 · have hs1 : s=1 := by omega have he : 1+(H-s)=H := by omega simp only [stoppedTubeEndpoint,hs,ite_eq_left,he] at hb simpa only [stoppedTubeEndpoint,hs,ite_eq_left] using hb.trans (tubePrefixMass_mono_radius ℓ f θ (|θ|+z) (3*z) (by linarith) H π ω) · simp only [stoppedTubeEndpoint,hs] at hb have hc := tubePrefixMass_comp_le e f θ z (|θ|+z) (by positivity) (s-1) (1+(H-s)) π ω have he : (s-1)+(1+(H-s))=H := by omega rw [he] at hc simpa only [stoppedTubeEndpoint,hs,ite_false] using (hb.trans hc).trans (tubePrefixMass_mono_radius ℓ f θ (z+(|θ|+z)) (3*z) (by linarith) H π ω) end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical namespace DirectionalTransience lemma tubePrefixMass_ne_top {d : ℕ} (ℓ : Vector d) (f : Direction d) (θ z : ℝ) (s : ℕ) (π : Measure (Lattice d)) [IsFiniteMeasure π] (ω : Environment d) : tubePrefixMass ℓ f θ z s π ω ≠ ⊤ := by rw [← tubeEndpointMixture_total] exact measure_ne_top _ _ theorem short_block_slope_survival {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 ≤ 20) : ∃ C : ℝ, 0 < C ∧ ∃ t₀ : ℝ, 0 < t₀ ∧ ∀ t : ℝ, 0 < t → t ≤ t₀ → ∃ δ : ℝ, 0 < δ ∧ ∃ R : ℝ, 0 < R ∧ ∀ r : ℝ, R ≤ r → ∀ H : ℕ, 0 < H → (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 θ := (recordMedian ν ℓ hp f H:ℝ)/H ∀ (a : ℤ) (π : Measure (Lattice d)), IsProbabilityMeasure π → (∀ᵐ x ∂π, signedHeight e x = a) → (environmentLaw ν).real {ω | (tubePrefixMass ℓ f θ (3*(w*r)) H π ω).toReal < δ} ≤ C*t^2 := by obtain ⟨C,hC,t₀,ht₀,hb⟩ := canonical_two_tube_threat ν hue e f hef htrans hw hw1 obtain ⟨κ,hκ,hUE⟩ := environment_uniform_elliptic ν hue refine ⟨C,hC,t₀,ht₀,?_⟩ intro t ht htt obtain ⟨δ₀,hδ₀,hδ₀1,R,hR,hb⟩ := hb t ht htt let δ := min δ₀ ((κ:ℝ)*δ₀^2) have hκreal : (0:ℝ) < κ := by exact_mod_cast hκ have hδ : 0 < δ := lt_min hδ₀ (by positivity) refine ⟨δ,hδ,R,hR,?_⟩ intro r hr H hH hscale obtain ⟨hθ,hb⟩ := hb r hr H hH hscale let ℓ := realPosition (step e) let hp := ne_of_gt (noDrop_positive_of_directionallyTransient ν ℓ htrans) let θ := (recordMedian ν ℓ hp f H:ℝ)/H dsimp only intro a π hπ hsupp let := hπ obtain ⟨x₀,hx₀⟩ := hsupp.exists have hb' := hb a x₀ hx₀ π hπ hsupp have hz : 0 ≤ w*r := mul_nonneg hw.le (hR.trans_le hr).le have hδ1 : δ₀ ≤ 1 := by linarith have hπ' : ∀ᵐ x ∂π, dot (realPosition x) ℓ = (a:ℝ) := hsupp.mono fun x hx => by rw [signedHeight_projection,hx] have hδEN : ENNReal.ofReal δ ≤ (κ:ℝ≥0∞)*ENNReal.ofReal δ₀*ENNReal.ofReal δ₀ := by have he : ENNReal.ofReal ((κ:ℝ)*δ₀^2) = (κ:ℝ≥0∞)*ENNReal.ofReal δ₀*ENNReal.ofReal δ₀ := by rw [pow_two,← mul_assoc,ENNReal.ofReal_mul (mul_nonneg hκreal.le hδ₀.le), ENNReal.ofReal_mul hκreal.le,ENNReal.ofReal_coe_nnreal] rw [← he] exact ENNReal.ofReal_le_ofReal (min_le_right _ _) have hle : environmentLaw ν {ω | (tubePrefixMass ℓ f θ (3*(w*r)) H π ω).toReal < δ} ≤ environmentLaw ν (⋃ s ∈ Finset.range (H+1), {ω | ω ∈ FirstTubeThreat ℓ f a θ (w*r) δ₀ s π ∧ (bridgedRestart e f x₀ θ (w*r) π s ω,ω) ∈ TubeThreat ℓ f ((a:ℝ)+s) θ (w*r) δ₀ H}) := by apply measure_mono_ae filter_upwards [hUE] with ω hω intro hbad have hcontra (h : ENNReal.ofReal δ ≤ tubePrefixMass ℓ f θ (3*(w*r)) H π ω) : False := by have hh := ENNReal.toReal_mono (tubePrefixMass_ne_top ℓ f θ (3*(w*r)) H π ω) h rw [ENNReal.toReal_ofReal hδ.le] at hh exact (not_lt_of_ge hh) hbad by_cases hfirst : (π,ω) ∈ TubeThreat ℓ f a θ (w*r) δ₀ H · have hh : ω ∈ ⋃ s ≤ H, FirstTubeThreat ℓ f a θ (w*r) δ₀ s π := by rw [firstTubeThreat_iUnion] exact hfirst obtain ⟨s,hs⟩ := Set.mem_iUnion.mp hh obtain ⟨hsH,hs⟩ := Set.mem_iUnion.mp hs by_cases hsecond : (bridgedRestart e f x₀ θ (w*r) π s ω,ω) ∈ TubeThreat ℓ f ((a:ℝ)+s) θ (w*r) δ₀ H · exact Set.mem_iUnion.mpr ⟨s,Set.mem_iUnion.mpr ⟨Finset.mem_range.mpr (Nat.lt_succ_of_le hsH),hs,hsecond⟩⟩ · exfalso apply hcontra exact (hδEN.trans (firstThreat_noSecond_retained_mass e f a x₀ hx₀ θ (w*r) δ₀ hz hδ1 H s hsH π hsupp ω κ (fun y => hω y e) hs hsecond)).trans (bridged_continuation_le_tube e f hef θ (w*r) hz hθ H s hs.1 hsH π ω) · exfalso apply hcontra exact ((ENNReal.ofReal_le_ofReal (min_le_left _ _)).trans (noThreat_mass_lower ℓ f a θ (w*r) δ₀ hz hδ1 H π hπ' ω hfirst le_rfl)).trans (tubePrefixMass_mono_radius ℓ f θ (w*r) (3*(w*r)) (by linarith) H π ω) exact (ENNReal.toReal_mono (measure_ne_top _ _) hle).trans hb' end DirectionalTransience open MeasureTheory ProbabilityTheory Filter open scoped ENNReal NNReal BigOperators Topology Classical end end end OAI