| import OAI.Probability.GammaPassage.RareFastEvents |
| import OAI.Probability.GammaPassage.GridAttachments |
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|
| namespace OAI |
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| namespace GammaFPP.CrossingBridge |
| open PlanarFPP Set Finset MeasureTheory ProbabilityTheory |
| noncomputable def crossingEvent (b m K s : ℝ) (n : ℤ) (t : Vertex) : Set (Edge → ℝ) := |
| strictLocalizedEvent (rareFastVertices b 1 8 s n t) |
| (fun p ds => p.1=t.1 ∧ (p+displacement ds).1=t.1+n) |
| (fun _ _ => m*n+K*s) |
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| theorem measurableSet_crossingEvent (b m K s : ℝ) (n : ℤ) (t : Vertex) : |
| MeasurableSet (crossingEvent b m K s n t) := measurableSet_strictLocalizedEvent _ _ _ |
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| theorem crossingEvent_depends (b m K s : ℝ) (n : ℤ) (t : Vertex) : |
| DependsOnEdges (crossingEvent b m K s n t) (internalEdges (rareFastVertices b 1 8 s n t)) := |
| strictLocalizedEvent_depends _ _ _ |
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| lemma crossingEvent_eq_rareFastEvent (b m K s : ℝ) (n : ℤ) (t : Vertex) : |
| crossingEvent b m K s n t = rareFastEvent b m 0 1 8 (-K) s n t := by |
| ext w |
| constructor <;> rintro ⟨p,ds,hcols,hin,hcost⟩ |
| all_goals |
| have hd : (displacement ds).1=n := by |
| have hh := hcols.2 |
| simp only [Prod.fst_add,hcols.1] at hh |
| omega |
| refine ⟨p,ds,hcols,hin,?_⟩ |
| simpa only [support,zero_mul,add_zero,hd,neg_mul,sub_neg_eq_add] using hcost |
|
|
| lemma crossingEvent_translate (b m K s : ℝ) (n : ℤ) (t : Vertex) (w : Edge → ℝ) : |
| (fun e => w (shiftEdge t e)) ∈ crossingEvent b m K s n 0 ↔ |
| w ∈ crossingEvent b m K s n t := by |
| rw [crossingEvent_eq_rareFastEvent,crossingEvent_eq_rareFastEvent] |
| exact rareFastEvent_translate b m 0 1 8 (-K) s n t w |
|
|
| noncomputable def gridTrialTranslate (b : ℝ) (X h w : ℤ) (t : ℕ) : Vertex := |
| (X,⌊b*X⌋+(h+30*(t+1 : ℕ))*w) |
|
|
| lemma gridTrialTranslate_center (b : ℝ) (X h w : ℤ) (t : ℕ) : |
| transverse b (gridTrialTranslate b X h w t) = |
| (⌊b*X⌋ : ℝ)-b*X+((h : ℝ)+30*(t+1))*w := by |
| simp [gridTrialTranslate,transverse] |
| ring |
|
|
| lemma gridTrialTranslate_separation (b : ℝ) (X h w : ℤ) (hw : 1 ≤ w) |
| {t u : ℕ} (htu : t ≠ u) : |
| 2*(8*(w : ℝ)) < |transverse b (gridTrialTranslate b X h w t)- |
| transverse b (gridTrialTranslate b X h w u)| := by |
| have hw' : (1 : ℝ) ≤ w := by exact_mod_cast hw |
| have hsep : 1 ≤ |(t : ℝ)-u| := by |
| have hn : (1 : ℤ) ≤ |(t : ℤ)-u| := by |
| rw [abs_eq_max_neg] |
| omega |
| exact_mod_cast hn |
| rw [gridTrialTranslate_center,gridTrialTranslate_center, |
| show ((⌊b*X⌋ : ℝ)-b*X+((h : ℝ)+30*(t+1))*w)- |
| ((⌊b*X⌋ : ℝ)-b*X+((h : ℝ)+30*(u+1))*w) = |
| 30*(w : ℝ)*((t : ℝ)-u) by ring,abs_mul,abs_of_nonneg (by positivity)] |
| nlinarith |
|
|
| lemma gridTrials_disjoint (b : ℝ) (X h w l : ℤ) (hw : 1 ≤ w) (j : ℕ) : |
| Pairwise (fun t u : Fin (j^2) => Disjoint |
| (rareFastVertices b 1 8 w l (gridTrialTranslate b X h w t)) |
| (rareFastVertices b 1 8 w l (gridTrialTranslate b X h w u))) := by |
| intro t u htu |
| apply strip_vertices_disjoint _ _ b |
| (transverse b (gridTrialTranslate b X h w t)) |
| (transverse b (gridTrialTranslate b X h w u)) (8*w) |
| · intro p hp |
| have hh := ((mem_rareFastVertices ..).mp hp).2.2 |
| convert hh using 1 |
| simp [transverse] |
| congr 1 |
| ring |
| · intro p hp |
| have hh := ((mem_rareFastVertices ..).mp hp).2.2 |
| convert hh using 1 |
| simp [transverse] |
| congr 1 |
| ring |
| · exact gridTrialTranslate_separation b X h w hw (fun he => htu (Fin.ext he)) |
|
|
| variable {shape rate : ℝ} |
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} {τ : Edge → Ω → ℝ} |
|
|
| lemma crossingEvent_probability_translate (h : GammaEnvironment shape rate P τ) |
| (b m K s : ℝ) (n : ℤ) (t : Vertex) : |
| P {ω | (fun e => τ e ω) ∈ crossingEvent b m K s n t} = |
| P {ω | (fun e => τ e ω) ∈ crossingEvent b m K s n 0} := by |
| simp only [crossingEvent_eq_rareFastEvent] |
| exact rareFastEvent_probability_translate h b m 0 1 8 (-K) s n t |
|
|
| / |
| theorem gridTrials_failure_bound (henv : GammaEnvironment shape rate P τ) |
| (b m K q₀ : ℝ) (X h w l : ℤ) (hw : 1 ≤ w) (j : ℕ) |
| (hq : q₀ ≤ P.real {ω | (fun e => τ e ω) ∈ crossingEvent b m K w l 0}) : |
| P.real {ω | ∀ t : Fin (j^2), (fun e => τ e ω) ∉ |
| crossingEvent b m K w l (gridTrialTranslate b X h w t)} ≤ (1-q₀)^(j^2) := by |
| have hh := Grid.disjoint_trials_failure_bound henv (Finset.univ : Finset (Fin (j^2))) |
| (fun t => crossingEvent b m K w l (gridTrialTranslate b X h w t)) |
| (fun t => internalEdges (rareFastVertices b 1 8 w l (gridTrialTranslate b X h w t))) |
| (fun _ _ => measurableSet_crossingEvent ..) |
| (fun t _ => crossingEvent_depends b m K w l (gridTrialTranslate b X h w t)) |
| (fun _ _ _ _ htu => internalEdges_disjoint (gridTrials_disjoint b X h w l hw j htu)) |
| q₀ (fun t _ => ?_) |
| · simpa using hh |
| · unfold Measure.real |
| rw [crossingEvent_probability_translate henv] |
| exact hq |
|
|
| / |
| probability. -/ |
| noncomputable def gridBinGood (shape rate b m K : ℝ) (X h w l : ℤ) (j : ℕ) : |
| Set (Edge → ℝ) := |
| Grid.binGood Finset.univ |
| (fun t : Fin (j^2) => crossingEvent b m K w l (gridTrialTranslate b X h w t)) |
| (Grid.transverseIntervalEdges b X (Grid.gridIntervalLower h w) (Grid.gridIntervalUpper j h w)) |
| (Grid.transverseIntervalEdges b (X+l) (Grid.gridIntervalLower h w) (Grid.gridIntervalUpper j h w)) |
| shape rate ((j : ℝ)^2*w) |
|
|
| theorem gridBinGood_failure_bound (henv : GammaEnvironment shape rate P τ) |
| (hs : 0 < shape) (hr : 0 < rate) (b m K q₀ : ℝ) |
| (X h w l : ℤ) (j : ℕ) (hj : 1 ≤ j) (hw : 1 ≤ w) |
| (hq : q₀ ≤ P.real {ω | (fun e => τ e ω) ∈ crossingEvent b m K w l 0}) : |
| P.real {ω | (fun e => τ e ω) ∉ gridBinGood shape rate b m K X h w l j} ≤ |
| (1-q₀)^(j^2)+2*Real.exp (-Grid.verticalExponent shape*(j : ℝ)^2*w) := by |
| refine Grid.source_bin_failure_bound henv hs hr b X h w l j hj hw |
| (fun t => crossingEvent b m K w l (gridTrialTranslate b X h w t)) |
| (fun t => internalEdges (rareFastVertices b 1 8 w l (gridTrialTranslate b X h w t))) |
| ?_ ?_ ?_ q₀ ?_ |
| · intro t |
| exact measurableSet_crossingEvent .. |
| · intro t |
| exact crossingEvent_depends b m K w l (gridTrialTranslate b X h w t) |
| · intro t u htu |
| exact internalEdges_disjoint (gridTrials_disjoint b X h w l hw j htu) |
| · intro t |
| unfold Measure.real |
| rw [crossingEvent_probability_translate henv] |
| exact hq |
|
|
| end GammaFPP.CrossingBridge |
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| end OAI |
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