| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.CellCertificates |
|
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| namespace OAI |
|
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| section |
|
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| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| lemma cell_bad_event_subset {d k : ℕ} (ν : Measure (Row d)) |
| (e f : Direction d) (hef : e.1 ≠ f.1) (a C c g₀ g₁ A G : ℝ) (hc : 0 < c) |
| (hA : 0 ≤ A) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (v w m s : ℕ) (hv : 2*cellWidth w m 0 ≤ v) (hs : 0 < s) |
| (hG : ∀ i < m, G ≤ c*cellRadius ν e f C w m i/2) (q : ℝ≥0∞) : |
| {ω | ∃ t : ℕ, v ≤ t ∧ t ≤ v+w ∧ |
| rawTupleMixture (realPosition (step e)) t (π ω).val ω {y | TupleSeparated f G y} < |
| q*(ENNReal.ofReal g₀*ENNReal.ofReal g₁*ENNReal.ofReal (Real.exp (-A*k*m)))} ⊆ |
| {ω | ∃ h : ℕ, rawTupleMixture (realPosition (step e)) h (π ω).val ω Set.univ < q} ∪ |
| (⋃ i, scheduledCellSuccess ν e f hef a C c g₀ g₁ A hc π v w m s i)ᶜ := by |
| intro ω hω |
| by_cases hmass : ω ∈ {ω | ∃ h : ℕ, rawTupleMixture (realPosition (step e)) h (π ω).val ω Set.univ < q} |
| · exact Or.inl hmass |
| · refine Or.inr ?_ |
| intro hcert |
| obtain ⟨i,hi⟩ := Set.mem_iUnion.mp hcert |
| have him : i < m := by |
| by_contra hh |
| simp only [scheduledCellSuccess,ite_eq_right hh,Set.mem_empty_iff_false] at hi |
| obtain ⟨t,hvt,ht,hbad⟩ := hω |
| have hq : q ≤ rawTupleMixture (realPosition (step e)) (cellOpportunity v w m i) (π ω).val ω Set.univ := by |
| by_contra hh |
| exact hmass ⟨_,not_le.mp hh⟩ |
| have hp := successful_cell_certificate ν e f hef a C c g₀ g₁ A hc hA π v w m s i hv hs ω hi t hvt ht |
| have hmono : rawTupleMixture (realPosition (step e)) t (π ω).val ω |
| {y | TupleSeparated f (c*cellRadius ν e f C w m i/2) y} ≤ |
| rawTupleMixture (realPosition (step e)) t (π ω).val ω {y | TupleSeparated f G y} := by |
| apply measure_mono |
| intro y hy j l hjl |
| exact (hG i him).trans (hy j l hjl) |
| exact (not_lt_of_ge ((mul_le_mul_left hq _).trans (hp.trans hmono))) hbad |
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| lemma cells_bad_probability {d k n : ℕ} (ν : Measure (Row d)) |
| (e f : Direction d) (hef : e.1 ≠ f.1) (a C c g₀ g₁ A : ℝ) (hc : 0 < c) |
| (hA : 0 ≤ A) (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (v w : Fin n → ℕ) (m s : ℕ) (hs : 0 < s) (G : Fin n → ℝ) |
| (hv : ∀ j, 2*cellWidth (w j) m 0 ≤ v j) |
| (hG : ∀ j i, i < m → G j ≤ c*cellRadius ν e f C (w j) m i/2) |
| (q δ ε : ℝ≥0∞) |
| (hmass : environmentLaw ν {ω | ∃ h : ℕ, rawTupleMixture (realPosition (step e)) h (π ω).val ω Set.univ < q} ≤ δ) |
| (hcell : ∀ j, environmentLaw ν (⋃ i, scheduledCellSuccess ν e f hef a C c g₀ g₁ A hc π (v j) (w j) m s i)ᶜ ≤ ε) : |
| environmentLaw ν {ω | ∃ j : Fin n, ∃ t : ℕ, v j ≤ t ∧ t ≤ v j+w j ∧ |
| rawTupleMixture (realPosition (step e)) t (π ω).val ω {y | TupleSeparated f (G j) y} < |
| q*(ENNReal.ofReal g₀*ENNReal.ofReal g₁*ENNReal.ofReal (Real.exp (-A*k*m)))} ≤ δ+n*ε := by |
| let E := {ω | ∃ h : ℕ, rawTupleMixture (realPosition (step e)) h (π ω).val ω Set.univ < q} |
| let F := fun j : Fin n => (⋃ i, scheduledCellSuccess ν e f hef a C c g₀ g₁ A hc π (v j) (w j) m s i)ᶜ |
| have hsub : {ω | ∃ j : Fin n, ∃ t : ℕ, v j ≤ t ∧ t ≤ v j+w j ∧ |
| rawTupleMixture (realPosition (step e)) t (π ω).val ω {y | TupleSeparated f (G j) y} < |
| q*(ENNReal.ofReal g₀*ENNReal.ofReal g₁*ENNReal.ofReal (Real.exp (-A*k*m)))} ⊆ E ∪ ⋃ j, F j := by |
| rintro ω ⟨j,hj⟩ |
| rcases cell_bad_event_subset ν e f hef a C c g₀ g₁ A (G j) hc hA π (v j) (w j) m s (hv j) hs (hG j) q hj with he|hf |
| · exact Or.inl he |
| · exact Or.inr (Set.mem_iUnion.mpr ⟨j,hf⟩) |
| apply (measure_mono hsub).trans |
| apply (measure_union_le _ _).trans |
| apply add_le_add hmass |
| apply (measure_iUnion_fintype_le _ _).trans |
| calc |
| ∑ j : Fin n, environmentLaw ν (F j) ≤ ∑ _j : Fin n, ε := Finset.sum_le_sum (fun j _ => hcell j) |
| _ = n*ε := by simp [nsmul_eq_mul] |
|
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| end DirectionalTransience |
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| end |
|
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| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| def stageBadEvent {d k : ℕ} (e f : Direction d) (H : ℕ) (Gminus Gplus : ℝ) |
| (π : Measure (Fin k → Lattice d)) (p : ℝ≥0∞) : Set (Environment d) := |
| {ω | (∃ h : ℕ, 1 ≤ h ∧ h ≤ H ∧ |
| rawTupleMixture (realPosition (step e)) h π ω {y | TupleSeparated f Gminus y} < p) ∨ |
| rawTupleMixture (realPosition (step e)) H π ω {y | TupleSeparated f Gplus y} < p} |
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| theorem finite_stage_assembly {d k n : ℕ} (ν : Measure (Row d)) |
| (e f : Direction d) (hef : e.1 ≠ f.1) (a C c g₀ g₁ A G Gminus Gplus r : ℝ) |
| (hc : 0 < c) (hA : 0 ≤ A) (π : BudgetProfile (k:=k) e f a G) |
| (He H : ℕ) (v w : Fin n → ℕ) (m s : ℕ) (hs : 0 < s) (Gs : Fin n → ℝ) |
| (hv : ∀ j, 2*cellWidth (w j) m 0 ≤ v j) |
| (hGs : ∀ j i, i < m → Gs j ≤ c*cellRadius ν e f C (w j) m i/2) |
| (hcover : ∀ h : ℕ, He < h → h ≤ H → ∃ j, v j ≤ h ∧ h ≤ v j+w j ∧ Gminus ≤ Gs j) |
| (hgrowth : ∃ j, v j ≤ H ∧ H ≤ v j+w j ∧ Gplus ≤ Gs j) |
| (hshortgap : Gminus ≤ G-r) (p q δ₀ δ₁ ε : ℝ≥0∞) |
| (hp : p ≤ q*(ENNReal.ofReal g₀*ENNReal.ofReal g₁*ENNReal.ofReal (Real.exp (-A*k*m)))) |
| (hshort : environmentLaw ν {ω | relativeBudgetMassENN e f He r π.val ω < p} ≤ δ₀) |
| (hmass : environmentLaw ν {ω | ∃ h : ℕ, rawTupleMixture (realPosition (step e)) h π.val ω Set.univ < q} ≤ δ₁) |
| (hcell : ∀ j, environmentLaw ν (⋃ i, scheduledCellSuccess ν e f hef a C c g₀ g₁ A hc |
| (fun _ => π.toLayerProfile) (v j) (w j) m s i)ᶜ ≤ ε) : |
| environmentLaw ν (stageBadEvent e f H Gminus Gplus π.val p) ≤ δ₀+δ₁+n*ε := by |
| let E := {ω | relativeBudgetMassENN e f He r π.val ω < p} |
| let F := {ω | ∃ j : Fin n, ∃ t : ℕ, v j ≤ t ∧ t ≤ v j+w j ∧ |
| rawTupleMixture (realPosition (step e)) t π.val ω {y | TupleSeparated f (Gs j) y} < |
| q*(ENNReal.ofReal g₀*ENNReal.ofReal g₁*ENNReal.ofReal (Real.exp (-A*k*m)))} |
| have hF : environmentLaw ν F ≤ δ₁+n*ε := |
| cells_bad_probability ν e f hef a C c g₀ g₁ A hc hA (fun _ => π.toLayerProfile) |
| v w m s hs Gs hv hGs q δ₁ ε hmass hcell |
| have hsub : stageBadEvent e f H Gminus Gplus π.val p ⊆ E ∪ F := by |
| intro ω hω |
| rcases hω with ⟨h,hh,hH,hbad⟩|hbad |
| · by_cases he : h ≤ He |
| · apply Or.inl |
| apply lt_of_le_of_lt _ hbad |
| have hb := budget_profile_prefix_certificate e f a G r π ω h He he |
| apply hb.trans |
| exact measure_mono (fun y hy j l hjl => hshortgap.trans (hy j l hjl)) |
| · obtain ⟨j,hv,hw,hg⟩ := hcover h (not_le.mp he) hH |
| refine Or.inr ⟨j,h,hv,hw,lt_of_le_of_lt ?_ (hbad.trans_le hp)⟩ |
| exact measure_mono (fun y hy i l hil => hg.trans (hy i l hil)) |
| · obtain ⟨j,hv,hw,hg⟩ := hgrowth |
| refine Or.inr ⟨j,H,hv,hw,lt_of_le_of_lt ?_ (hbad.trans_le hp)⟩ |
| exact measure_mono (fun y hy i l hil => hg.trans (hy i l hil)) |
| exact (measure_mono hsub).trans ((measure_union_le E F).trans |
| (by simpa only [add_assoc] using add_le_add hshort hF)) |
|
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| end DirectionalTransience |
|
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| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| lemma noDropProduct_le_raw {d k : ℕ} (ℓ : Vector d) (ω : Environment d) |
| (hω : ∀ x H, NoDrop ℓ x ≤ᵐ[quenchedKernel (ω,x)] Cross ℓ x H) |
| (x : Fin k → Lattice d) (H : ℕ) : |
| ENNReal.ofReal (noDropProduct ℓ x ω) ≤ rawTupleEndpointLaw ℓ H ω x Set.univ := by |
| rw [noDropProduct,ENNReal.ofReal_prod_of_nonneg (fun j _ => ENNReal.toReal_nonneg), |
| rawTupleEndpointLaw,Measure.pi_univ] |
| apply Finset.prod_le_prod |
| intro j _ |
| rw [ENNReal.ofReal_toReal (show noDropQuenched ℓ (x j) ω ≠ ⊤ from measure_ne_top _ _)] |
| have hh : noDropQuenched ℓ (x j) ω ≤ crossingQuenched ℓ (x j) H ω := |
| measure_mono_ae (hω (x j) H) |
| rw [crossingQuenched_eq_hitKernel] at hh |
| exact hh |
|
|
| lemma fullStream_noDrop_lower {d k : ℕ} (ν : Measure (Row d)) [IsProbabilityMeasure ν] |
| (ℓ : Vector d) (htrans : DirectionallyTransient ν ℓ) : |
| ∀ᵐ ω ∂environmentLaw ν, ∀ (π : Measure (Fin k → Lattice d)) [IsProbabilityMeasure π], |
| ∀ H : ℕ, ENNReal.ofReal (∫ x, noDropProduct ℓ x ω ∂π) ≤ |
| rawTupleMixture ℓ H π ω Set.univ := by |
| filter_upwards [quenched_noDrop_subset_cross_ae ν ℓ htrans] with ω hω π hπ H |
| have hI : Integrable (fun x => noDropProduct ℓ x ω) π := |
| integrable_of_nonneg_bound π (measurable_of_countable _) (fun x => noDropProduct_bounds ℓ x ω) |
| rw [ofReal_integral_eq_lintegral_ofReal hI (Filter.Eventually.of_forall |
| (fun x => (noDropProduct_bounds ℓ x ω).1)),rawTupleMixture_apply] |
| exact lintegral_mono fun x => noDropProduct_le_raw ℓ ω hω x H |
|
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| theorem fullStream_small_mass_probability {d : ℕ} (ν : Measure (Row d)) |
| [IsProbabilityMeasure ν] (hue : UniformElliptic ν) (ℓ : Vector d) |
| (hℓ : dot ℓ ℓ = 1) (htrans : DirectionallyTransient ν ℓ) (e : Fin d) : |
| ∃ lam C : ℝ, 0 < lam ∧ lam ≤ 1 ∧ 0 < C ∧ |
| ∀ Bstar : ℝ, 0 < Bstar → ∀ k : ℕ, 1 ≤ k → ∃ G : ℕ, |
| ∀ (π : Measure (Fin k → Lattice d)) [IsProbabilityMeasure π], |
| (∀ᵐ x ∂π, Pairwise fun i j => (G:ℤ) ≤ |x i e-x j e|) → |
| ∀ s : ℝ, Real.exp (-Bstar) ≤ s → |
| environmentLaw ν {ω | ∃ H : ℕ, rawTupleMixture ℓ H π ω Set.univ < ENNReal.ofReal s} ≤ |
| ENNReal.ofReal (Real.exp (C*k)*s^lam) := by |
| obtain ⟨lam,C,hlam,hlam1,hC,hb⟩ := separated_initial_inverse_moment ν hue ℓ hℓ htrans e |
| refine ⟨lam,C,hlam,hlam1,hC,fun Bstar hB k hk => ?_⟩ |
| obtain ⟨G,hG⟩ := hb Bstar hB k hk |
| refine ⟨G,fun π hπ hsep s hs => ?_⟩ |
| apply le_trans _ (hG π hsep s hs) |
| apply measure_mono_ae |
| filter_upwards [fullStream_noDrop_lower ν ℓ htrans] with ω hω |
| rintro ⟨H,hH⟩ |
| have hq := (hω π H).trans_lt hH |
| have hs0 : 0 < s := (Real.exp_pos _).trans_le hs |
| have hqs : (∫ x, noDropProduct ℓ x ω ∂π) < s := (ENNReal.ofReal_lt_ofReal_iff hs0).mp hq |
| change (∫ x, noDropProduct ℓ x ω ∂π) < 2*s |
| linarith |
|
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| end DirectionalTransience |
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| end |
|
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| end OAI |
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