| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.Cell |
|
|
| namespace OAI |
|
|
| section |
|
|
| open scoped BigOperators |
| namespace DirectionalTransience |
|
|
| def cellWidth (w m i : ℕ) : ℕ := 16^(m-i)*w |
|
|
| def cellOpportunity (v w m i : ℕ) : ℕ := |
| if i < m then v-2*cellWidth w m i else v+w |
|
|
| lemma cellWidth_zero (w m : ℕ) : cellWidth w m 0=16^m*w := by simp [cellWidth] |
| lemma cellWidth_last (w m : ℕ) : cellWidth w m m=w := by simp [cellWidth] |
| lemma cellWidth_antitone (w m : ℕ) : Antitone (cellWidth w m) := by |
| intro i j hij |
| apply Nat.mul_le_mul_right |
| exact Nat.pow_le_pow_right (by norm_num) (Nat.sub_le_sub_left hij m) |
| lemma cellWidth_pos {w : ℕ} (hw : 0 < w) (m i : ℕ) : 0 < cellWidth w m i := by |
| unfold cellWidth; positivity |
| lemma cellWidth_min (w m i : ℕ) : w ≤ cellWidth w m i := by |
| unfold cellWidth |
| have h : 1 ≤ 16^(m-i) := one_le_pow₀ (by norm_num) |
| simpa only [one_mul] using Nat.mul_le_mul_right w h |
| lemma cellWidth_step (w m i : ℕ) (hi : i < m) : |
| cellWidth w m i=16*cellWidth w m (i+1) := by |
| have he : m-i=(m-(i+1))+1 := by omega |
| unfold cellWidth |
| rw [he,pow_succ] |
| ring |
| lemma cellWidth_shift (w m i j : ℕ) (hij : i+j ≤ m) : |
| cellWidth w m i=16^j*cellWidth w m (i+j) := by |
| have he : m-i=j+(m-(i+j)) := by omega |
| unfold cellWidth |
| rw [he,pow_add] |
| ring |
| lemma cellOpportunity_last (v w m : ℕ) : cellOpportunity v w m m=v+w := by |
| simp [cellOpportunity] |
| lemma cellOpportunity_le_end (v w m i : ℕ) : cellOpportunity v w m i ≤ v+w := by |
| unfold cellOpportunity |
| split |
| · omega |
| · rfl |
| lemma cellOpportunity_monotone (v w m : ℕ) : Monotone (cellOpportunity v w m) := by |
| intro i j hij |
| by_cases hj : j < m |
| · have hi : i < m := lt_of_le_of_lt hij hj |
| simp only [cellOpportunity,ite_eq_left hi,ite_eq_left hj] |
| exact Nat.sub_le_sub_left (Nat.mul_le_mul_left 2 (cellWidth_antitone w m hij)) v |
| · simp only [cellOpportunity,ite_eq_right hj] |
| exact cellOpportunity_le_end v w m i |
|
|
| lemma cell_seed_endpoint (v w m i : ℕ) (hv : 2*cellWidth w m 0 ≤ v) (hi : i < m) : |
| cellOpportunity v w m i+cellWidth w m i=v-cellWidth w m i := by |
| have hw := Nat.mul_le_mul_left 2 (cellWidth_antitone w m (Nat.zero_le i)) |
| simp only [cellOpportunity,ite_eq_left hi] |
| omega |
|
|
| lemma cell_seed_before_next (v w m i : ℕ) (hv : 2*cellWidth w m 0 ≤ v) (hi : i < m) : |
| cellOpportunity v w m i+cellWidth w m i ≤ cellOpportunity v w m (i+1) := by |
| rw [cell_seed_endpoint v w m i hv hi] |
| by_cases hn : i+1 < m |
| · simp only [cellOpportunity,ite_eq_left hn] |
| apply Nat.sub_le_sub_left |
| rw [cellWidth_step w m i hi] |
| omega |
| · simp only [cellOpportunity,ite_eq_right hn] |
| omega |
|
|
| lemma cell_step_length (v w m i : ℕ) (hv : 2*cellWidth w m 0 ≤ v) (hi : i < m) : |
| cellOpportunity v w m (i+1)-cellOpportunity v w m i ≤ 3*cellWidth w m i := by |
| have hw := Nat.mul_le_mul_left 2 (cellWidth_antitone w m (Nat.zero_le i)) |
| have hmin := cellWidth_min w m i |
| have hend := cellOpportunity_le_end v w m (i+1) |
| simp only [cellOpportunity,ite_eq_left hi] |
| split <;> omega |
|
|
| def cellEarlyHeight (v w m i s : ℕ) : ℕ := |
| cellOpportunity v w m (i+min s (m-i))-(cellOpportunity v w m i+cellWidth w m i) |
|
|
| def cellLateHeight (v w m i s j : ℕ) : ℕ := |
| if i+s+j < m then |
| cellOpportunity v w m (i+s+j+1)-cellOpportunity v w m (i+s+j) else 0 |
|
|
| lemma cellEarly_endpoint (v w m i s : ℕ) (hv : 2*cellWidth w m 0 ≤ v) |
| (hi : i < m) (hs : 0 < s) : |
| cellOpportunity v w m i+cellWidth w m i+cellEarlyHeight v w m i s= |
| cellOpportunity v w m (i+min s (m-i)) := by |
| unfold cellEarlyHeight |
| have hmin : 1 ≤ min s (m-i) := by omega |
| have hh := cellOpportunity_monotone v w m (show i+1 ≤ i+min s (m-i) by omega) |
| have ht := cell_seed_before_next v w m i hv hi |
| omega |
|
|
| lemma cellEarlyHeight_bound (v w m i s : ℕ) (hv : 2*cellWidth w m 0 ≤ v) |
| (hi : i < m) : cellEarlyHeight v w m i s ≤ 3*cellWidth w m i := by |
| unfold cellEarlyHeight |
| rw [cell_seed_endpoint v w m i hv hi] |
| have hmin := cellWidth_min w m i |
| have hh := cellOpportunity_le_end v w m (i+min s (m-i)) |
| have hw := Nat.mul_le_mul_left 2 (cellWidth_antitone w m (Nat.zero_le i)) |
| omega |
|
|
| lemma cellLateHeight_bound (v w m i s j : ℕ) (hv : 2*cellWidth w m 0 ≤ v) : |
| cellLateHeight v w m i s j*16^(s+j) ≤ 3*cellWidth w m i := by |
| unfold cellLateHeight |
| split |
| · rename_i h |
| have hh := Nat.mul_le_mul_right (16^(s+j)) (cell_step_length v w m (i+s+j) hv h) |
| have he := cellWidth_shift w m i (s+j) (by omega) |
| calc |
| _ ≤ 3*cellWidth w m (i+(s+j))*16^(s+j) := by simpa only [Nat.add_assoc] using hh |
| _ = 3*cellWidth w m i := by rw [he]; ring |
| · simp |
|
|
| lemma cellLate_endpoint (v w m i s n : ℕ) (_ : 2*cellWidth w m 0 ≤ v) |
| (hn : i+s+n ≤ m) : |
| cellOpportunity v w m (i+s)+(∑ j∈Finset.range n,cellLateHeight v w m i s j)= |
| cellOpportunity v w m (i+s+n) := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| have hin : i+s+n < m := by omega |
| have hp := ih (by omega) |
| rw [Finset.sum_range_succ,←Nat.add_assoc,hp] |
| simp only [cellLateHeight,ite_eq_left hin] |
| have hh := cellOpportunity_monotone v w m (show i+s+n ≤ i+s+n+1 by omega) |
| have he : i+s+(n+1)=i+s+n+1 := by omega |
| rw [he] |
| omega |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal BigOperators Classical |
| namespace DirectionalTransience |
|
|
| noncomputable def cellRadius {d : ℕ} (ν : Measure (Row d)) (e f : Direction d) |
| (C : ℝ) (w m i : ℕ) : ℝ := |
| fluctuationRadius (independentConditionedPairLaw ν (realPosition (step e))) |
| (commonIncrementProcess (realPosition (step e)) f 0) ((cellWidth w m i:ℝ)/C) |
|
|
| lemma cellRadius_pos {d : ℕ} (ν : Measure (Row d)) (e f : Direction d) |
| (C : ℝ) (w m i : ℕ) : 0 < cellRadius ν e f C w m i := fluctuationRadius_pos _ _ _ |
|
|
| noncomputable def cellRestart {d k : ℕ} (e : Direction d) (a : ℝ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) (v w m i : ℕ) |
| (ω : Environment d) : LayerTupleProfile (k:=k) e (a+cellOpportunity v w m i) := |
| fullTupleProfile e a (cellOpportunity v w m i) (π ω) ω |
|
|
| lemma cellRestart_measurable {d k : ℕ} (e : Direction d) (a : ℝ) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (v w m i : ℕ) : |
| @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) (a+cellOpportunity v w m i))) _ |
| (cellRestart e a π v w m i) := fullTupleProfile_measurable e a _ π hπ |
|
|
| noncomputable def scheduledCellFailure {d k : ℕ} (ν : Measure (Row d)) |
| (e f : Direction d) (hef : e.1 ≠ f.1) (a C c g₀ g₁ A : ℝ) (hc : 0 < c) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) (v w m s i n : ℕ) : Set (Environment d) := |
| if i < m ∧ 0 < n ∧ n ≤ m-i then |
| consumedCellFailure |
| (cellSeedMass e f (a+cellOpportunity v w m i) (c*cellRadius ν e f C w m i) |
| (cellWidth w m i) (cellRestart e a π v w m i)) |
| (cellEarlyMass e f hef (a+cellOpportunity v w m i) (c*cellRadius ν e f C w m i) |
| (cellWidth w m i) (cellEarlyHeight v w m i s) (cellRestart e a π v w m i)) |
| (cellLateMass e f hef (a+cellOpportunity v w m i) c (cellRadius ν e f C w m i) |
| hc (cellRadius_pos ν e f C w m i) (cellWidth w m i) (cellEarlyHeight v w m i s) s |
| (cellLateHeight v w m i s) (cellRestart e a π v w m i)) g₀ g₁ (A*k) s (m-i) n |
| else ∅ |
|
|
| def cellFiltration {d : ℕ} (e : Direction d) (a : ℝ) (v w m : ℕ) : |
| Filtration ℕ (inferInstance : MeasurableSpace (Environment d)) where |
| seq i := rowSigma (BelowHeight (realPosition (step e)) (a+cellOpportunity v w m i)) |
| mono' i j hij := by |
| have hh : (cellOpportunity v w m i : ℝ) ≤ cellOpportunity v w m j := |
| by exact_mod_cast cellOpportunity_monotone v w m hij |
| apply rowSigma_mono |
| intro x hx |
| change dot (realPosition x) (realPosition (step e)) < a+cellOpportunity v w m i at hx |
| change dot (realPosition x) (realPosition (step e)) < a+cellOpportunity v w m j |
| linarith only [hx,hh] |
| le' _ := rowSigma_le _ |
|
|
| lemma scheduledCellFailure_measurable {d k : ℕ} (ν : Measure (Row d)) |
| (e f : Direction d) (hef : e.1 ≠ f.1) (a C c g₀ g₁ A : ℝ) (hc : 0 < c) |
| (π : Environment d → LayerTupleProfile (k:=k) e a) |
| (hπ : @Measurable _ _ (rowSigma (BelowHeight (realPosition (step e)) a)) _ π) |
| (v w m s : ℕ) (hv : 2*cellWidth w m 0 ≤ v) (hs : 0 < s) (i n : ℕ) : |
| MeasurableSet[cellFiltration e a v w m (i+n)] |
| (scheduledCellFailure ν e f hef a C c g₀ g₁ A hc π v w m s i n) := by |
| unfold scheduledCellFailure |
| split |
| · rename_i h |
| rcases h with ⟨hi,hn,hnm⟩ |
| change MeasurableSet[rowSigma (BelowHeight (realPosition (step e)) (a+cellOpportunity v w m (i+n)))] _ |
| apply actual_cell_failure_measurable e f hef _ c _ hc _ _ _ s (m-i) n _ _ |
| (cellRestart_measurable e a π hπ v w m i) g₀ g₁ A (a+cellOpportunity v w m (i+n)) |
| · intro _ |
| have hh := (cell_seed_before_next v w m i hv hi).trans |
| (cellOpportunity_monotone v w m (show i+1 ≤ i+n by omega)) |
| have hh' : (cellOpportunity v w m i:ℝ)+cellWidth w m i ≤ cellOpportunity v w m (i+n) := |
| by exact_mod_cast hh |
| linarith only [hh'] |
| · intro h |
| have hmin : min s (m-i) ≤ n := by rcases h with h|⟨h,_⟩ <;> omega |
| have hh := cellOpportunity_monotone v w m (show i+min s (m-i) ≤ i+n by omega) |
| rw [←cellEarly_endpoint v w m i s hv hi hs] at hh |
| have hh' : (cellOpportunity v w m i:ℝ)+cellWidth w m i+cellEarlyHeight v w m i s ≤ |
| cellOpportunity v w m (i+n) := by exact_mod_cast hh |
| linarith only [hh'] |
| · intro hsn hnm' |
| have hmin : min s (m-i)=s := min_eq_left (by omega) |
| have hearly := cellEarly_endpoint v w m i s hv hi hs |
| rw [hmin] at hearly |
| have hlate := cellLate_endpoint v w m i s (n-s) hv (by omega) |
| have heq : i+s+(n-s)=i+n := by omega |
| rw [heq,←hearly] at hlate |
| have hh : (cellOpportunity v w m i:ℝ)+cellWidth w m i+cellEarlyHeight v w m i s+ |
| ((∑ j∈Finset.range (n-s),cellLateHeight v w m i s j):ℕ)=cellOpportunity v w m (i+n) := |
| by exact_mod_cast hlate |
| linarith only [hh] |
| · exact @MeasurableSet.empty _ (cellFiltration e a v w m (i+n)) |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
