| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.BufferRestorationScript |
| import OAI.Probability.Ballisticity.Geometry.BufferStructuralHeight |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| structure AdaptedBufferNode {d : ℕ} (e : Direction d) where |
| level : ℝ |
| radius : ℝ |
| law : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) level) |
| law_rows : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) level)) _ law |
| active : Set (Environment d) |
| active_rows : MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) level)] active |
|
|
| noncomputable def bufferNextRadius (R₀ r ε α : ℝ) (b : Bool) : ℝ := |
| max R₀ ((if b then 1+α else 1-ε)*r) |
|
|
| noncomputable def bufferRestoreLength (R₀ r ε α : ℝ) (b : Bool) : ℕ := |
| ⌈max 0 (R₀-((if b then 1+α else 1-ε)*r))⌉₊ |
|
|
| lemma bufferRestoreLength_enough (R₀ r ε α : ℝ) (b : Bool) : |
| bufferNextRadius R₀ r ε α b ≤ |
| (if b then 1+α else 1-ε)*r + bufferRestoreLength R₀ r ε α b := by |
| have h := Nat.le_ceil (max 0 (R₀-((if b then 1+α else 1-ε)*r))) |
| unfold bufferNextRadius bufferRestoreLength |
| apply max_le |
| · have := le_max_right 0 (R₀-((if b then 1+α else 1-ε)*r)) |
| linarith |
| · exact le_add_of_nonneg_right (Nat.cast_nonneg _) |
|
|
| lemma bufferStage_branch_rows {d : ℕ} (e f : Direction d) |
| (a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a)) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| {H : ℕ} (hH : 0 < H) (k : ℕ) (b : Bool) : |
| MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a+k))] |
| {ω | bufferFirstFailure (realPosition (step e)) f a (z₀+(1-ε)*r) g π H ω = k ∧ |
| (BufferStageEvent (realPosition (step e)) f H z₀ r ε α g (π ω).val ω ↔ b=true)} := by |
| have hstop := bufferFirstFailure_eq_measurable (realPosition (step e)) f a (z₀+(1-ε)*r) g π hπ H k |
| have hpass := bufferStage_success_stopped_rows e f a z₀ r ε α g π hπ hH k |
| cases b |
| · convert hstop.diff hpass using 1 |
| ext ω |
| simp only [Bool.false_eq_true,iff_false,Set.mem_sdiff,Set.mem_ofPred_eq] |
| tauto |
| · convert hpass using 1 |
| ext ω |
| simp |
|
|
| noncomputable def restoredStoppedLaw {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a)) |
| (H : ℕ) (hH : 0 < H) (κ : ℝ≥0) (k m : ℕ) (ω : Environment d) : |
| SupportedPairMeasures (PairAtHeight (realPosition (step e)) (a+k+1)) := |
| ⟨(bufferStageStoppedProbability e f a z₀ r ε α g π H hH κ k ω).val.map (pairRestore e f m), by |
| apply ae_iff.mp |
| rw [ae_map_iff (measurable_of_countable _).aemeasurable (Set.to_countable _).measurableSet] |
| filter_upwards [ae_iff.mpr (bufferStageStoppedProbability e f a z₀ r ε α g π H hH κ k ω).property] with x hx |
| exact pairRestore_height e f hef m (a+k) x hx⟩ |
|
|
| lemma restoredStoppedLaw_rows {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (a z₀ r ε α g : ℝ) (π : Environment d → SupportedPairMeasures (PairAtHeight (realPosition (step e)) a)) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (H : ℕ) (hH : 0 < H) (κ : ℝ≥0) (k m : ℕ) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+k+1))) _ |
| (restoredStoppedLaw e f hef a z₀ r ε α g π H hH κ k m) := by |
| apply Measurable.subtype_mk |
| have hm := (bufferStageStoppedProbability_rows e f a z₀ r ε α g π hπ H hH κ k).subtype_val |
| have hu : rowSigma (BelowHeight (realPosition (step e)) (a+k)) ≤ |
| rowSigma (BelowHeight (realPosition (step e)) (a+k+1)) := |
| by simpa only [Nat.cast_one] using rowSigma_below_height_mono (realPosition (step e)) (a+k) 1 |
| apply Measure.measurable_of_measurable_coe |
| intro U hU |
| simp only [Measure.map_apply (measurable_of_countable _) hU] |
| exact ((Measure.measurable_coe ((measurable_of_countable _) hU)).comp hm).mono hu le_rfl |
|
|
| noncomputable def bufferChildNode {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (R₀ z₀ ε α g : ℝ) (κ : ℝ≥0) (N : AdaptedBufferNode e) |
| (H : ℕ) (hH : 0 < H) (k : ℕ) (b : Bool) : AdaptedBufferNode e where |
| level := N.level+k+1 |
| radius := bufferNextRadius R₀ N.radius ε α b |
| law := restoredStoppedLaw e f hef N.level z₀ N.radius ε α g N.law H hH κ k |
| (bufferRestoreLength R₀ N.radius ε α b) |
| law_rows := restoredStoppedLaw_rows e f hef N.level z₀ N.radius ε α g N.law N.law_rows H hH κ k _ |
| active := N.active ∩ {ω | bufferFirstFailure (realPosition (step e)) f N.level (z₀+(1-ε)*N.radius) g N.law H ω = k ∧ |
| (BufferStageEvent (realPosition (step e)) f H z₀ N.radius ε α g (N.law ω).val ω ↔ b=true)} |
| active_rows := by |
| have ha := N.active_rows |
| have hb := bufferStage_branch_rows e f N.level z₀ N.radius ε α g N.law N.law_rows hH k b |
| have h1 := rowSigma_below_height_mono (realPosition (step e)) N.level k |
| have h2 : rowSigma (BelowHeight (realPosition (step e)) (N.level+k)) ≤ |
| rowSigma (BelowHeight (realPosition (step e)) (N.level+k+1)) := by |
| simpa only [Nat.cast_one] using rowSigma_below_height_mono (realPosition (step e)) (N.level+k) 1 |
| exact (h2 _ (h1 _ ha)).inter (h2 _ hb) |
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter Function |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| def BufferNodeValid {d : ℕ} (e f : Direction d) (z₀ : ℝ) (κ : ℝ≥0) (N : AdaptedBufferNode e) : Prop := |
| ∀ ω : Environment d, (∀ y u, κ ≤ (ω y).1 u) → ω ∈ N.active → |
| IsProbabilityMeasure (N.law ω).val ∧ |
| ∀ᵐ x ∂(N.law ω).val, z₀+N.radius ≤ signedCoordinate f (x.2-x.1) |
|
|
| lemma bufferChildNode_valid {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (R₀ z₀ ε α g : ℝ) {κ : ℝ≥0} (hκpos : 0 < κ) (hκ1 : κ ≤ 1) |
| (hg : 0 < g) (hg1 : g ≤ 1) (hε : 0 ≤ ε) (N : AdaptedBufferNode e) |
| (hr : 0 ≤ N.radius) (hN : BufferNodeValid e f z₀ κ N) |
| (H : ℕ) (hH : 0 < H) (k : ℕ) (b : Bool) : |
| BufferNodeValid e f z₀ κ (bufferChildNode e f hef R₀ z₀ ε α g κ N H hH k b) := by |
| intro ω hκ hω |
| have hparent : ω ∈ N.active := hω.1 |
| have hk : bufferFirstFailure (realPosition (step e)) f N.level (z₀+(1-ε)*N.radius) g N.law H ω = k := hω.2.1 |
| have hb : BufferStageEvent (realPosition (step e)) f H z₀ N.radius ε α g (N.law ω).val ω ↔ b=true := hω.2.2 |
| obtain ⟨hp,hgap⟩ := hN ω hκ hparent |
| let := hp |
| have hp' : IsProbabilityMeasure (bufferStageStoppedProbability e f N.level z₀ N.radius ε α g N.law H hH κ k ω).val := by |
| simp only [bufferStageStoppedProbability,ite_eq_left hk] |
| exact bufferStageRetainedLaw_probability e f N.level z₀ N.radius ε α g N.law hH ω |
| (fun y => hκ y e) hκpos hκ1 hg hg1 hr hε hgap |
| let := hp' |
| have hmap : IsProbabilityMeasure ((bufferStageStoppedProbability e f N.level z₀ N.radius ε α g N.law H hH κ k ω).val.map |
| (pairRestore e f (bufferRestoreLength R₀ N.radius ε α b))) := by |
| infer_instance |
| have hgapmap : ∀ᵐ x ∂(bufferStageStoppedProbability e f N.level z₀ N.radius ε α g N.law H hH κ k ω).val.map |
| (pairRestore e f (bufferRestoreLength R₀ N.radius ε α b)), |
| z₀+bufferNextRadius R₀ N.radius ε α b ≤ signedCoordinate f (x.2-x.1) := by |
| rw [ae_map_iff (measurable_of_countable _).aemeasurable (measurableSet_le measurable_const (measurable_of_countable _))] |
| filter_upwards [bufferStageStoppedProbability_gap e f N.level z₀ N.radius ε α g N.law H hH κ k ω] with x hx |
| rw [pairRestore_gap] |
| have hR := bufferRestoreLength_enough R₀ N.radius ε α b |
| cases b |
| · have hn : ¬ BufferStageEvent (realPosition (step e)) f H z₀ N.radius ε α g (N.law ω).val ω := by simpa using hb |
| simp only [ite_eq_right hn] at hx |
| simp only [Bool.false_eq_true,↓reduceIte] at hR |
| linarith |
| · have hy : BufferStageEvent (realPosition (step e)) f H z₀ N.radius ε α g (N.law ω).val ω := hb.mpr rfl |
| simp only [ite_eq_left hy] at hx |
| simp only [↓reduceIte] at hR |
| linarith |
| exact ⟨hmap,hgapmap⟩ |
|
|
| lemma bufferChildNode_cover {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (R₀ z₀ ε α g : ℝ) (κ : ℝ≥0) (N : AdaptedBufferNode e) |
| (H : ℕ) (hH : 0 < H) : |
| (⋃ p : ℕ × Bool, (bufferChildNode e f hef R₀ z₀ ε α g κ N H hH p.1 p.2).active) = N.active := by |
| ext ω |
| constructor |
| · rintro ⟨_,⟨p,rfl⟩,hω⟩ |
| exact hω.1 |
| · intro hω |
| let k := bufferFirstFailure (realPosition (step e)) f N.level (z₀+(1-ε)*N.radius) g N.law H ω |
| let b := decide (BufferStageEvent (realPosition (step e)) f H z₀ N.radius ε α g (N.law ω).val ω) |
| apply Set.mem_iUnion.mpr |
| refine ⟨(k,b),hω,rfl,?_⟩ |
| simp [b] |
|
|
| lemma bufferChildNode_disjoint {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (R₀ z₀ ε α g : ℝ) (κ : ℝ≥0) (N : AdaptedBufferNode e) |
| (H : ℕ) (hH : 0 < H) : |
| Pairwise (Disjoint on fun p : ℕ × Bool => |
| (bufferChildNode e f hef R₀ z₀ ε α g κ N H hH p.1 p.2).active) := by |
| intro p q hpq |
| apply Set.disjoint_left.mpr |
| intro ω hp hq |
| apply hpq |
| apply Prod.ext |
| · exact hp.2.1.symm.trans hq.2.1 |
| · have hh : (p.2=true) ↔ (q.2=true) := hp.2.2.symm.trans hq.2.2 |
| exact Bool.eq_iff_iff.mpr hh |
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
