| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.FreshClipping |
| import OAI.Probability.Ballisticity.Estimates.CellFailure |
| import OAI.Probability.Ballisticity.Coupling.CellSeedParameters |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| lemma belowHeight_add_nat_mono {d : ℕ} (e : Direction d) (a : ℝ) (n : ℕ) : |
| rowSigma (BelowHeight (realPosition (step e)) a) ≤ |
| rowSigma (BelowHeight (realPosition (step e)) (a+n)) := by |
| apply rowSigma_mono |
| intro x hx |
| change dot (realPosition x) (realPosition (step e)) < a+n |
| exact lt_of_lt_of_le hx (le_add_of_nonneg_right (Nat.cast_nonneg n)) |
|
|
| noncomputable def cellLateMass {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a c ρ : ℝ) (hc : 0 < c) (hρ : 0 < ρ) (Hs He s : ℕ) (H : ℕ → ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) : ℕ → Environment d → ℝ := |
| protectionMass e f (a+Hs+He) (3*c*ρ/4) H (lateBudget c ρ s) |
| (fun i => (lateBudget_pos hc hρ s i).le) |
| (cellLateInitial e f hef a c ρ hc.le hρ.le Hs He π) |
|
|
| theorem actual_consumed_cell_failure {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))) : |
| ∃ A : ℝ, 1 ≤ A ∧ ∀ k : ℕ, 2 ≤ k → |
| ∃ C c g₀ g₁ : ℝ, 0 < C ∧ ∃ hc : 0 < c, c ≤ 1 ∧ 0 < g₀ ∧ 0 < g₁ ∧ |
| ∃ s H₀ : ℕ, 0 < s ∧ ∀ Bstar : ℝ, 0 < Bstar → ∃ R : ℝ, 0 < R ∧ |
| ∀ Hs ≥ H₀, ∀ (ρ : ℝ) (hρ : 0 < ρ), R ≤ ρ → |
| ρ = fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) ((Hs:ℝ)/C) → |
| ∀ He : ℕ, (He:ℝ) ≤ 3*C*fluctuationScale |
| (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) ρ → |
| ∀ H : ℕ → ℕ, (∀ i, (H i:ℝ) ≤ 3*C*fluctuationScale |
| (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) ρ/(16:ℝ)^(s+i)) → |
| ∀ m : ℕ, (2*A*k)*m ≤ Bstar → ∀ n : ℕ, 1 ≤ n → n ≤ m → |
| ∀ (a : ℝ) (π : Environment d → LayerTupleProfile (k:=k) e a), |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π → |
| ∀ E : Set (Environment d), MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) a)] E → |
| environmentLaw ν (E∩consumedCellFailure |
| (cellSeedMass e f a (c*ρ) Hs π) (cellEarlyMass e f hef a (c*ρ) Hs He π) |
| (cellLateMass e f hef a c ρ hc hρ Hs He s H π) g₀ g₁ (A*k) s m n) ≤ |
| ENNReal.ofReal (Real.exp (-2*k*n))*environmentLaw ν E := by |
| obtain ⟨lam,Cmg,hlam,hlam1,hCmg,hlate⟩ := |
| actual_late_protection_first_failure ν hue e f hef htrans |
| let A := max 1 (4*(Cmg+8)/lam) |
| have hA : 1 ≤ A := le_max_left _ _ |
| have hchoice : Cmg+8 ≤ (lam/4)*A := by |
| have h := le_max_right 1 (4*(Cmg+8)/lam) |
| have hh := (div_le_iff₀ hlam).mp h |
| dsimp [A] |
| nlinarith only [hh] |
| refine ⟨A,hA,fun k hk => ?_⟩ |
| obtain ⟨C,c,g₀,hC,hc,hc1,hg₀,H₀,hseed⟩ := adapted_separated_seeds_capped |
| ν hue e f hef htrans k hk (Real.exp (-6*k)) (Real.exp_pos _) |
| obtain ⟨s,hs,hsbound⟩ := hlate A hA hchoice k hk c C hc hc1 hC |
| obtain ⟨g₁,hg₁,R₁,hR₁,hearly⟩ := adapted_initial_clipping ν hue e f hef htrans k |
| (show 0 < 3*C by positivity) (show 0 < c/4 by positivity) |
| (Real.exp_pos (-6*k*s)) |
| refine ⟨C,c,g₀,g₁,hC,hc,hc1,hg₀,hg₁,s,H₀,hs,fun Bstar hB => ?_⟩ |
| obtain ⟨R₂,hR₂,hb⟩ := hsbound Bstar hB |
| refine ⟨max R₁ R₂,hR₁.trans_le (le_max_left _ _),?_⟩ |
| intro Hs hHs ρ hρ hρR hρeq He hHe H hH m hm n hn hnm a π hπ E hE |
| have hr1 : R₁ ≤ ρ := (le_max_left _ _).trans hρR |
| have hr2 : R₂ ≤ ρ := (le_max_right _ _).trans hρR |
| apply consumedCellFailure_bound (environmentLaw ν) _ _ _ g₀ g₁ (A*k) s m n k (by omega) hn E |
| · simpa only [cellSeedMass,hρeq] using hseed Hs hHs a π hπ E hE |
| · have hEs : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a+Hs))] E := |
| belowHeight_add_nat_mono e a Hs E hE |
| have hh := hearly ρ hr1 He hHe (a+Hs) (c*ρ) |
| (fun ω => seedSeparatedProfile e f hef a (c*ρ) Hs (π ω) ω) |
| (seedSeparatedProfile_measurable e f hef a (c*ρ) Hs π hπ) E hEs |
| have heq : (c/4)*ρ=(c*ρ)/4 := by ring |
| simpa only [heq,cellEarlyMass] using hh |
| · intro hsn hnm' |
| have heq : s+(n-s-1)+1=n := by omega |
| have heq' : (s:ℝ)+((n-s-1:ℕ):ℝ)+1=n := by exact_mod_cast heq |
| have ht : (2*A*k)*((s:ℝ)+((n-s-1:ℕ):ℝ)+1) ≤ Bstar := by |
| rw [heq'] |
| exact (mul_le_mul_of_nonneg_left (by exact_mod_cast hnm') (by positivity)).trans hm |
| have hEl : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He))] E := |
| belowHeight_add_nat_mono e (a+Hs) He E (belowHeight_add_nat_mono e a Hs E hE) |
| have hh := hb ρ hr2 (n-s-1) ht H (fun i _ => hH i) (a+Hs+He) |
| (cellLateInitial e f hef a c ρ hc.le hρ.le Hs He π) |
| (cellLateInitial_measurable e f hef a c ρ hc.le hρ.le Hs He π hπ) E hEl |
| simpa only [cellLateMass,heq'] using hh |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| lemma firstProtectionFailure_measurable {Ω : Type*} [m : MeasurableSpace Ω] |
| (q : ℕ → Ω → ℝ) (a s : ℝ) (n : ℕ) (hq : ∀ i ≤ n, Measurable (q i)) : |
| MeasurableSet (firstProtectionFailure q a s n) := by |
| unfold firstProtectionFailure |
| rw [Set.ofPred_and] |
| apply MeasurableSet.inter |
| · rw [Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro j |
| rw [Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro hj |
| apply measurableSet_le measurable_const |
| apply Finset.measurable_prod |
| intro i hi |
| exact hq i (by simp only [Finset.mem_range] at hi; omega) |
| · apply measurableSet_lt _ measurable_const |
| apply Finset.measurable_prod |
| intro i hi |
| exact hq i (by simpa using hi) |
|
|
| lemma consumedCellFailure_measurable {Ω : Type*} [mΩ : MeasurableSpace Ω] |
| (seed : Ω → ℝ≥0∞) (early : Ω → ℝ) (q : ℕ → Ω → ℝ) |
| (g₀ g₁ kA : ℝ) (s m n : ℕ) |
| (hseed : n=1 ∨ n=min s m ∨ (s < n ∧ n ≤ m) → Measurable seed) |
| (hearly : n=min s m ∨ (s < n ∧ n ≤ m) → Measurable early) |
| (hq : s < n → n ≤ m → ∀ i ≤ n-s-1, Measurable (q i)) : |
| MeasurableSet (consumedCellFailure seed early q g₀ g₁ kA s m n) := by |
| unfold consumedCellFailure |
| refine (MeasurableSet.union ?_ ?_).union ?_ |
| · by_cases hn : n=1 |
| · simpa only [hn,true_and] using measurableSet_lt (hseed (Or.inl hn)) measurable_const |
| · simp only [hn,false_and,Set.ofPred_false]; exact MeasurableSet.empty |
| · by_cases hn : n=min s m |
| · simpa only [hn,true_and,Set.ofPred_and] using |
| (measurableSet_le measurable_const (hseed (Or.inr (Or.inl hn)))).inter |
| (measurableSet_lt (hearly (Or.inl hn)) measurable_const) |
| · simp only [hn,false_and,Set.ofPred_false]; exact MeasurableSet.empty |
| · by_cases hs : s < n |
| · by_cases hm : n ≤ m |
| · simpa only [hs,hm,true_and,Set.ofPred_and,Set.ofPred_mem_eq] using |
| (measurableSet_le measurable_const (hseed (Or.inr (Or.inr ⟨hs,hm⟩)))).inter |
| ((measurableSet_le measurable_const (hearly (Or.inr ⟨hs,hm⟩))).inter |
| (firstProtectionFailure_measurable q kA s (n-s-1) (hq hs hm))) |
| · simp only [hm,and_false,false_and,Set.ofPred_false]; exact MeasurableSet.empty |
| · simp only [hs,false_and,Set.ofPred_false]; exact MeasurableSet.empty |
|
|
| lemma cellLateMass_measurable {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a c ρ : ℝ) (hc : 0 < c) (hρ : 0 < ρ) (Hs He s : ℕ) (H : ℕ → ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) (i : ℕ) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) |
| (a+Hs+He+((∑ j∈Finset.range (i+1),H j):ℕ)))) _ |
| (cellLateMass e f hef a c ρ hc hρ Hs He s H π i) := by |
| have hm := protectionMass_measurable e f (a+Hs+He) (3*c*ρ/4) H (lateBudget c ρ s) |
| (fun j => (lateBudget_pos hc hρ s j).le) |
| (cellLateInitial e f hef a c ρ hc.le hρ.le Hs He π) |
| (cellLateInitial_measurable e f hef a c ρ hc.le hρ.le Hs He π hπ) i |
| change @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) |
| (protectionHeight (a+Hs+He) H (i+1)))) _ _ at hm |
| have ht : protectionHeight (a+Hs+He) H (i+1)= |
| a+Hs+He+((∑ j∈Finset.range (i+1),H j):ℕ) := by |
| rw [protectionHeight_eq,Nat.cast_sum] |
| rw [ht] at hm |
| exact hm |
|
|
| theorem actual_cell_failure_measurable {d k : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a c ρ : ℝ) (hc : 0 < c) (hρ : 0 < ρ) (Hs He s m n : ℕ) (H : ℕ → ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (g₀ g₁ A b : ℝ) |
| (hseed : n=1 ∨ n=min s m ∨ (s < n ∧ n ≤ m) → a+Hs ≤ b) |
| (hearly : n=min s m ∨ (s < n ∧ n ≤ m) → a+Hs+He ≤ b) |
| (hlate : s < n → n ≤ m → a+Hs+He+((∑ j∈Finset.range (n-s),H j):ℕ) ≤ b) : |
| MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) b)] |
| (consumedCellFailure (cellSeedMass e f a (c*ρ) Hs π) |
| (cellEarlyMass e f hef a (c*ρ) Hs He π) |
| (cellLateMass e f hef a c ρ hc hρ Hs He s H π) g₀ g₁ (A*k) s m n) := by |
| let mB := rowSigma (BelowHeight (realPosition (step e)) b) |
| have hmono {t : ℝ} (ht : t ≤ b) : rowSigma (BelowHeight (realPosition (step e)) t) ≤ mB := |
| rowSigma_mono (fun x hx => lt_of_lt_of_le hx ht) |
| apply @consumedCellFailure_measurable _ mB |
| · intro hn |
| exact (cellSeedMass_measurable e f a (c*ρ) Hs π hπ).mono (hmono (hseed hn)) le_rfl |
| · intro hn |
| exact (cellEarlyMass_measurable e f hef a (c*ρ) Hs He π hπ).mono (hmono (hearly hn)) le_rfl |
| · intro hsn hnm i hi |
| apply (cellLateMass_measurable e f hef a c ρ hc hρ Hs He s H π hπ i).mono _ le_rfl |
| apply hmono |
| apply le_trans _ (hlate hsn hnm) |
| have hsum : (∑ j∈Finset.range (i+1),H j) ≤ ∑ j∈Finset.range (n-s),H j := |
| Finset.sum_le_sum_of_subset (Finset.range_mono (by omega)) |
| have hsum' : (((∑ j∈Finset.range (i+1),H j):ℕ):ℝ) ≤ |
| (((∑ j∈Finset.range (n-s),H j):ℕ):ℝ) := by exact_mod_cast hsum |
| linarith only [hsum'] |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
