| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.ProtectionFirstFailure |
| import OAI.Probability.Ballisticity.Estimates.CellAttempt |
|
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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 |
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| noncomputable def cellLateInitial {d k : ℕ} (e f : Direction d) (hef : e.1≠f.1) |
| (a c ρ : ℝ) (hc : 0 ≤ c) (hρ : 0 ≤ ρ) (Hs He : ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) (ω : Environment d) : |
| BudgetProfile (k:=k) e f (a+Hs+He) (3*c*ρ/4) := by |
| let p := cellEarlyProfile e f hef a (c*ρ) (mul_nonneg hc hρ) Hs He π ω |
| refine ⟨p.val,p.property.1,?_⟩ |
| have heq : 3*(c*ρ)/4=3*c*ρ/4 := by ring |
| simpa only [heq] using p.property.2 |
|
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| lemma cellLateInitial_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 : ℕ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He))) _ |
| (cellLateInitial e f hef a c ρ hc hρ Hs He π) := by |
| let : MeasurableSpace (Environment d) := rowSigma (BelowHeight (realPosition (step e)) (a+Hs+He)) |
| exact (measurable_subtype_coe.comp |
| (cellEarlyProfile_measurable e f hef a (c*ρ) (mul_nonneg hc hρ) Hs He π hπ)).subtype_mk |
|
|
| def consumedCellFailure {Ω : Type*} (seed : Ω → ℝ≥0∞) (early : Ω → ℝ) |
| (q : ℕ → Ω → ℝ) (g₀ g₁ kA : ℝ) (s m n : ℕ) : Set Ω := |
| {ω | n=1 ∧ seed ω < ENNReal.ofReal g₀} ∪ |
| {ω | n=min s m ∧ ENNReal.ofReal g₀ ≤ seed ω ∧ early ω < g₁} ∪ |
| {ω | s < n ∧ n ≤ m ∧ ENNReal.ofReal g₀ ≤ seed ω ∧ g₁ ≤ early ω ∧ |
| ω∈firstProtectionFailure q kA s (n-s-1)} |
|
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| lemma triple_exp_failure {k n : ℕ} (hk : 1 ≤ k) (hn : 1 ≤ n) : |
| 3*Real.exp (-5*(k:ℝ)*n) ≤ Real.exp (-2*(k:ℝ)*n) := by |
| have hk' : (1:ℝ) ≤ k := by exact_mod_cast hk |
| have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn |
| have hkn : 1 ≤ (k:ℝ)*n := one_le_mul_of_one_le_of_one_le hk' hn' |
| have hh : 3 ≤ Real.exp (3*(k:ℝ)*n) := by |
| have ht := Real.add_one_le_exp (3*(k:ℝ)*n) |
| nlinarith only [hkn,ht] |
| calc |
| _ ≤ Real.exp (3*(k:ℝ)*n)*Real.exp (-5*(k:ℝ)*n) := |
| mul_le_mul_of_nonneg_right hh (Real.exp_pos _).le |
| _ = _ := by rw [←Real.exp_add]; congr 1; ring |
|
|
| lemma consumedCellFailure_bound {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| (seed : Ω → ℝ≥0∞) (early : Ω → ℝ) (q : ℕ → Ω → ℝ) |
| (g₀ g₁ kA : ℝ) (s m n k : ℕ) (hk : 1 ≤ k) (hn : 1 ≤ n) |
| (E : Set Ω) |
| (hseed : μ (E∩{ω | seed ω < ENNReal.ofReal g₀}) ≤ ENNReal.ofReal (Real.exp (-6*(k:ℝ)))*μ E) |
| (hearly : μ (E∩{ω | early ω < g₁}) ≤ ENNReal.ofReal (Real.exp (-6*(k:ℝ)*s))*μ E) |
| (hlate : s < n → n ≤ m → μ (E∩firstProtectionFailure q kA s (n-s-1)) ≤ |
| ENNReal.ofReal (Real.exp (-5*(k:ℝ)*n))*μ E) : |
| μ (E∩consumedCellFailure seed early q g₀ g₁ kA s m n) ≤ |
| ENNReal.ofReal (Real.exp (-2*(k:ℝ)*n))*μ E := by |
| let b := ENNReal.ofReal (Real.exp (-5*(k:ℝ)*n))*μ E |
| have hs : μ (E∩{ω | n=1 ∧ seed ω < ENNReal.ofReal g₀}) ≤ b := by |
| by_cases he : n=1 |
| · subst n |
| calc |
| _ ≤ μ (E∩{ω | seed ω < ENNReal.ofReal g₀}) := |
| measure_mono (fun ω hω => ⟨hω.1,hω.2.2⟩) |
| _ ≤ ENNReal.ofReal (Real.exp (-6*(k:ℝ)))*μ E := hseed |
| _ ≤ b := by |
| dsimp [b] |
| apply mul_le_mul_left |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| simp only [Nat.cast_one,mul_one] |
| linarith [Nat.cast_nonneg (α:=ℝ) k] |
| · simp [he,b] |
| have he : μ (E∩{ω | n=min s m ∧ ENNReal.ofReal g₀ ≤ seed ω ∧ early ω < g₁}) ≤ b := by |
| by_cases hnm : n=min s m |
| · have hns : n ≤ s := hnm ▸ min_le_left _ _ |
| have hns' : (n:ℝ) ≤ s := by exact_mod_cast hns |
| calc |
| _ ≤ μ (E∩{ω | early ω < g₁}) := |
| measure_mono (fun ω hω => ⟨hω.1,hω.2.2.2⟩) |
| _ ≤ ENNReal.ofReal (Real.exp (-6*(k:ℝ)*s))*μ E := hearly |
| _ ≤ b := by |
| dsimp [b] |
| apply mul_le_mul_left |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| have hn0 : (0:ℝ) ≤ n := Nat.cast_nonneg _ |
| nlinarith [mul_le_mul_of_nonneg_left hns' (Nat.cast_nonneg (α:=ℝ) k), |
| mul_nonneg (Nat.cast_nonneg (α:=ℝ) k) hn0] |
| · simp [hnm,b] |
| have hl : μ (E∩{ω | s < n ∧ n ≤ m ∧ ENNReal.ofReal g₀ ≤ seed ω ∧ g₁ ≤ early ω ∧ |
| ω∈firstProtectionFailure q kA s (n-s-1)}) ≤ b := by |
| by_cases hs : s < n |
| · by_cases hm : n ≤ m |
| · calc |
| _ ≤ μ (E∩firstProtectionFailure q kA s (n-s-1)) := |
| measure_mono (fun ω hω => ⟨hω.1,hω.2.2.2.2.2⟩) |
| _ ≤ b := hlate hs hm |
| · simp [hm,b] |
| · simp [hs,b] |
| have ht : μ (E∩consumedCellFailure seed early q g₀ g₁ kA s m n) ≤ b+b+b := by |
| unfold consumedCellFailure |
| rw [Set.inter_union_distrib_left,Set.inter_union_distrib_left] |
| exact (measure_union_le _ _).trans (add_le_add ((measure_union_le _ _).trans (add_le_add hs he)) hl) |
| apply ht.trans |
| dsimp [b] |
| rw [←add_mul,←add_mul,←ENNReal.ofReal_add (Real.exp_pos _).le (Real.exp_pos _).le, |
| ←ENNReal.ofReal_add (by positivity) (Real.exp_pos _).le] |
| apply mul_le_mul_left |
| apply ENNReal.ofReal_le_ofReal |
| convert triple_exp_failure hk hn using 1 |
| ring |
|
|
| end DirectionalTransience |
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| end |
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| end OAI |
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