| import OAI.Probability.DirectionalWalk.ExperimentContact |
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| namespace OAI |
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| open MeasureTheory ProbabilityTheory Filter Preorder |
| open scoped ENNReal BigOperators Topology |
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| namespace DirectionalZeroOne |
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| open scoped Classical |
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| def latePrefixEvent {d ι : ℕ} (e : Step d) (k r N : ℕ) |
| (E : Word d → Fin ι → Set (Path d)) (start : Word d → Fin ι → Site d) : |
| Set (Path d × Path d) := |
| ⋃ a : {a : Word d | AxisBridgeWord e k a}, ⋃ q : Fin ι, |
| ⋃ b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b}, |
| (wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹' (E a q ∩ wordCylinder b)) ×ˢ |
| (shiftPath (start a q) ⁻¹' contactAvoid b) |
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| lemma latePrefixEvent_bound {d ι : ℕ} [NeZero ι] |
| (μ : Measure (Row d)) [IsProbabilityMeasure μ] (hell : StrictEllipticity μ) |
| (e : Step d) (k r N m : ℕ) (hrN : r ≤ N) |
| (E : Word d → Fin ι → Set (Path d)) (hE : ∀ a q, MeasurableSet (E a q)) |
| (hdis : ∀ a, Pairwise (fun i j => Disjoint (E a i) (E a j))) |
| (start : Word d → Fin ι → Site d) |
| (hstart : ∀ a, AxisBridgeWord e k a → ∀ q, axisHeight e (start a q) = (N : ℤ)-1) |
| (hΔ : axisReachProb μ e r-axisReachProb μ e N ≤ ENNReal.ofReal ((1/2 : ℝ)^m)) : |
| ((annealed μ 0).prod (conditioned μ (axisDirection (oppositeStep e)))) |
| (latePrefixEvent e k r N E start) ≤ |
| (annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ * |
| ENNReal.ofReal ((24 / posteriorExponent)*Real.log (ι+1)*(m+1)^2*(1/2 : ℝ)^m) := by |
| have : IsFiniteMeasure (conditioned μ (axisDirection (oppositeStep e))) := by |
| unfold conditioned |
| infer_instance |
| unfold latePrefixEvent |
| calc |
| _ ≤ ∑' a : {a : Word d | AxisBridgeWord e k a}, ∑' q : Fin ι, |
| ∑' b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b}, |
| ((annealed μ 0).prod (conditioned μ (axisDirection (oppositeStep e)))) |
| ((wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹' (E a q ∩ wordCylinder b)) ×ˢ |
| (shiftPath (start a q) ⁻¹' contactAvoid b)) := by |
| apply (measure_iUnion_le _).trans |
| apply ENNReal.tsum_le_tsum |
| intro a |
| apply (measure_iUnion_le _).trans |
| exact ENNReal.tsum_le_tsum (fun q => measure_iUnion_le _) |
| _ = (annealed μ 0 (nonBacktracking (axisDirection (oppositeStep e))))⁻¹ * |
| (∑' a : {a : Word d | AxisBridgeWord e k a}, ∑ q, |
| ∑' b : {b | upperContactWord 0 (axisUpper e r) (axisLower e 0 ∪ axisUpper e N) b}, |
| annealed μ 0 (wordCylinder a ∩ (shiftPath (-wordEnd a) ∘ tailPath a.val.1) ⁻¹' (E a q ∩ wordCylinder b)) * |
| annealed μ (start a q) (avoids (axisUpper e N) ∩ contactAvoid b)) := by |
| rw [← ENNReal.tsum_mul_left] |
| apply tsum_congr |
| intro a |
| rw [tsum_fintype,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro q _ |
| rw [← ENNReal.tsum_mul_left] |
| apply tsum_congr |
| intro b |
| rw [Measure.prod_prod,conditioned_negative_shift μ e _ N (hstart _ a.property _) _ (measurableSet_contactAvoid _)] |
| ac_rfl |
| _ ≤ _ := mul_le_mul_right (prefix_contact_total_bound μ hell e k r N m hrN E hE hdis start hΔ) _ |
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| def prependRelative {d : ℕ} (a : Word d) (R : Path d) : Path d := |
| concatPath a.1 (wordPath a) (shiftPath (wordEnd a) R) |
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| lemma measurable_prependRelative {d : ℕ} (a : Word d) : Measurable (prependRelative a) := by |
| apply Measurable.of_eval |
| intro j |
| by_cases hj : j < a.1 |
| · simpa only [prependRelative,concatPath,ite_eq_left hj] using |
| (measurable_const : Measurable (fun _ : Path d => wordPath a j)) |
| · have hm : Measurable (fun R : Path d => R (j-a.1)) := measurable_pi_apply _ |
| simpa only [prependRelative,concatPath,ite_eq_right hj,shiftPath] using hm.add_const (wordEnd a) |
|
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| lemma prependRelative_recover {d : ℕ} (a : Word d) (X : Path d) (hX : X ∈ wordCylinder a) : |
| prependRelative a (shiftPath (-wordEnd a) (tailPath a.1 X)) = X := by |
| funext j |
| by_cases hj : j < a.1 |
| · simpa only [prependRelative,concatPath,ite_eq_left hj] using (hX j hj.le).symm |
| · simp only [prependRelative,concatPath,ite_eq_right hj,shiftPath,tailPath, |
| Nat.add_sub_of_le (Nat.le_of_not_gt hj),neg_add_cancel_right] |
|
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| lemma latePrefixEvent_of_contact {d ι : ℕ} (e : Step d) |
| (k r N : ℕ) (hr : k ≤ r) (hN : k ≤ N) |
| (E : Fin ι → Set (Path d)) (z : Fin ι → Site d) |
| (X Y : Path d) (q : Fin ι) (hE : X ∈ E q) |
| (t τ : ℕ) (htτ : t ≤ τ) |
| (hold : AxisBridgeWord e k (prefixWord t X)) |
| (hband : ∀ j, t ≤ j → j ≤ τ → (k : ℤ) ≤ axisHeight e (X j) ∧ axisHeight e (X j) < (N : ℤ)) |
| (hreached : ∃ j, t ≤ j ∧ j ≤ τ ∧ (r : ℤ) ≤ axisHeight e (X j)) |
| (hhit : ∃ j, Y j+z q = X τ) |
| (hbefore : ∀ i, t ≤ i → i < τ → ∀ j, Y j+z q ≠ X i) : |
| (X,Y) ∈ latePrefixEvent e k (r-k) (N-k) |
| (fun a q => prependRelative a ⁻¹' E q) (fun a q => z q-wordEnd a) := by |
| let a := prefixWord t X |
| let R := shiftPath (-wordEnd a) (tailPath t X) |
| let b := prefixWord (τ-t) R |
| have ha : X ∈ wordCylinder a := fun j hj => (wordPath_prefixWord t X hj).symm |
| have hend : wordEnd a = X t := wordPath_prefixWord t X le_rfl |
| have hbpath (j : ℕ) (hj : j ≤ τ-t) : wordPath b j = X (t+j)-X t := by |
| rw [wordPath_prefixWord (τ-t) R hj] |
| change X (t+j)+(-wordEnd a) = _ |
| rw [hend,sub_eq_add_neg] |
| have hb : upperContactWord 0 (axisUpper e ((r-k : ℕ) : ℤ)) |
| (axisLower e 0 ∪ axisUpper e ((N-k : ℕ) : ℤ)) b := by |
| refine ⟨?_,?_,?_⟩ |
| · rw [hbpath 0 (Nat.zero_le _),Nat.add_zero,sub_self] |
| · obtain ⟨j,htj,hjτ,hjr⟩ := hreached |
| refine ⟨j-t,by change j-t ≤ τ-t;omega,?_⟩ |
| rw [hbpath (j-t) (by omega),Nat.add_sub_of_le htj] |
| change ((r-k : ℕ) : ℤ) ≤ axisHeight e (X j-X t) |
| rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,← hend,hold.1,Nat.cast_sub hr] |
| omega |
| · intro j hj |
| rw [hbpath j hj] |
| have hjτ : t+j ≤ τ := by change j ≤ τ-t at hj;omega |
| obtain ⟨hlo,hhi⟩ := hband (t+j) (by omega) hjτ |
| change ¬(axisHeight e (X (t+j)-X t) < (0 : ℤ) ∨ |
| ((N-k : ℕ) : ℤ) ≤ axisHeight e (X (t+j)-X t)) |
| rw [sub_eq_add_neg,axisHeight_add,axisHeight_neg,← hend,hold.1,Nat.cast_sub hN] |
| omega |
| have hca : shiftPath (z q-wordEnd a) Y ∈ contactAvoid b := by |
| rw [contactAvoid_iff_firstHit] |
| constructor |
| · obtain ⟨j,hj⟩ := hhit |
| refine ⟨j,?_⟩ |
| change Y j+(z q-wordEnd a) = wordPath b (τ-t) |
| rw [hbpath (τ-t) le_rfl,Nat.add_sub_of_le htτ,hend] |
| rw [sub_eq_add_neg,← add_assoc,hj,sub_eq_add_neg] |
| · intro i hi |
| rintro ⟨j,hj⟩ |
| apply hbefore (t+i) (by omega) (by change i < τ-t at hi;omega) j |
| rw [hbpath i hi.le] at hj |
| change Y j+(z q-wordEnd a) = X (t+i)-X t at hj |
| rw [hend,sub_eq_add_neg,sub_eq_add_neg,← add_assoc] at hj |
| exact add_right_cancel hj |
| refine Set.mem_iUnion.mpr ⟨⟨a,hold⟩,Set.mem_iUnion.mpr ⟨q,Set.mem_iUnion.mpr ⟨⟨b,hb⟩,?_⟩⟩⟩ |
| refine ⟨⟨ha,?_,?_⟩,hca⟩ |
| · change prependRelative a R ∈ E q |
| exact (prependRelative_recover a X ha).symm ▸ hE |
| · exact fun j hj => (wordPath_prefixWord (τ-t) R hj).symm |
|
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| end DirectionalZeroOne |
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| end OAI |
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