| import Mathlib |
| import OAI.Probability.Ballisticity.Estimates.RawPairRetention |
| import OAI.Probability.Ballisticity.Estimates.BufferFirstFailure |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma hitWord_dirac_le_hitKernel {d : ℕ} (ω : Environment d) (x : Lattice d) |
| {S T : Set (Lattice d)} (_ : Disjoint S T) (w : HitWord x S T) : |
| ENNReal.ofReal (wordWeight ω x w.val) • Measure.dirac (wordPath x w.val w.val.length) ≤ |
| hitKernel S T (ω,x) := by |
| intro U |
| have hU := (Set.to_countable U).measurableSet |
| rw [Measure.smul_apply,smul_eq_mul] |
| by_cases hu : wordPath x w.val w.val.length ∈ U |
| · rw [Measure.dirac_apply' _ hU,Set.indicator_of_mem hu,Pi.one_apply,mul_one] |
| rw [hitKernel_apply] |
| apply le_trans _ (ENNReal.le_tsum w.val.length) |
| rw [←quenched_wordCylinder] |
| apply measure_mono |
| intro X hX |
| exact ⟨⟨(hitWord_prefix x S T w hX).1,by simpa only [hX _ le_rfl] using hu⟩, |
| (hitWord_prefix x S T w hX).2⟩ |
| · rw [Measure.dirac_apply' _ hU,Set.indicator_of_notMem hu,mul_zero] |
| exact bot_le |
|
|
| lemma elliptic_hitWord_dirac_le_hitKernel {d : ℕ} (ω : Environment d) (x : Lattice d) |
| {κ : ℝ≥0} (hκ : ∀ y e, κ ≤ (ω y).1 e) |
| {S T : Set (Lattice d)} (hST : Disjoint S T) (w : HitWord x S T) : |
| (κ : ℝ≥0∞)^w.val.length • Measure.dirac (wordPath x w.val w.val.length) ≤ |
| hitKernel S T (ω,x) := by |
| have hw := ENNReal.ofReal_le_ofReal (wordWeight_lower ω hκ x w.val) |
| simp only [ENNReal.ofReal_pow κ.coe_nonneg,ENNReal.ofReal_coe_nnreal] at hw |
| intro U |
| exact (mul_le_mul_left hw ((Measure.dirac _) U)).trans ((hitWord_dirac_le_hitKernel ω x hST w) U) |
|
|
| lemma wordPath_lateral_script_end {d : ℕ} (e f : Direction d) (x : Lattice d) (m : ℕ) : |
| wordPath x (List.replicate m f ++ [e]) (m+1) = x+m • step f+step e := by |
| induction m generalizing x with |
| | zero => simp [wordPath] |
| | succ m ih => |
| simp only [List.replicate_succ,List.cons_append,wordPath,ih,succ_nsmul] |
| abel |
|
|
| lemma wordPath_lateral_script_prefix {d : ℕ} (e f : Direction d) (x : Lattice d) |
| (m j : ℕ) (hj : j ≤ m) : |
| wordPath x (List.replicate m f ++ [e]) j = x+j • step f := by |
| induction m generalizing x j with |
| | zero => |
| have : j=0 := by omega |
| subst j |
| simp |
| | succ m ih => |
| cases j with |
| | zero => simp |
| | succ j => |
| simp only [List.replicate_succ,List.cons_append,wordPath, |
| ih (x+step f) j (by omega),succ_nsmul] |
| abel |
|
|
| lemma signedHeight_add_nsmul_other {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (x : Lattice d) (m : ℕ) : signedHeight e (x+m • step f)=signedHeight e x := by |
| induction m with |
| | zero => simp |
| | succ m ih => rw [succ_nsmul,←add_assoc,signedHeight_add_step_other e f hef,ih] |
|
|
| noncomputable def bufferRestoreWord {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (x : Lattice d) (m : ℕ) : HitWord x |
| (Strip (realPosition (step e)) x 1) (Upper (realPosition (step e)) x 1) := by |
| refine ⟨List.replicate m f ++ [e],?_,?_⟩ |
| · simp only [List.length_append,List.length_replicate,List.length_cons,List.length_nil, |
| zero_add,wordPath_lateral_script_end] |
| change dot (realPosition x) (realPosition (step e))+1 ≤ |
| dot (realPosition (x+m • step f+step e)) (realPosition (step e)) |
| simp only [signedHeight_projection,signedHeight_add_step_self, |
| signedHeight_add_nsmul_other e f hef,Int.cast_add,Int.cast_one] |
| exact le_rfl |
| · intro j hj |
| have hjm : j ≤ m := by simpa only [List.length_append,List.length_replicate, |
| List.length_cons,List.length_nil,zero_add,Nat.lt_succ_iff] using hj |
| rw [wordPath_lateral_script_prefix e f x m j hjm] |
| change dot (realPosition x) (realPosition (step e)) ≤ |
| dot (realPosition (x+j • step f)) (realPosition (step e)) ∧ |
| dot (realPosition (x+j • step f)) (realPosition (step e)) < |
| dot (realPosition x) (realPosition (step e))+1 |
| simp only [signedHeight_projection,signedHeight_add_nsmul_other e f hef] |
| exact ⟨le_rfl,by norm_num⟩ |
|
|
| lemma elliptic_lateral_script {d : ℕ} (ω : Environment d) (e f : Direction d) |
| (hef : e.1 ≠ f.1) (x : Lattice d) (m : ℕ) {κ : ℝ≥0} |
| (hκ : ∀ y u, κ ≤ (ω y).1 u) : |
| (κ : ℝ≥0∞)^(m+1) • Measure.dirac (x+m • step f+step e) ≤ |
| hitKernel (Strip (realPosition (step e)) x 1) (Upper (realPosition (step e)) x 1) (ω,x) := by |
| have hh := elliptic_hitWord_dirac_le_hitKernel ω x hκ (disjoint_strip_upper _ _ _) |
| (bufferRestoreWord e f hef x m) |
| simpa only [bufferRestoreWord,List.length_append,List.length_replicate,List.length_cons, |
| List.length_nil,zero_add,wordPath_lateral_script_end] using hh |
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| def pairRestore {d : ℕ} (e f : Direction d) (m : ℕ) (x : Lattice d × Lattice d) : |
| Lattice d × Lattice d := (x.1+step e,x.2+m • step f+step e) |
|
|
| lemma pairRestore_gap {d : ℕ} (e f : Direction d) (m : ℕ) (x : Lattice d × Lattice d) : |
| signedCoordinate f ((pairRestore e f m x).2-(pairRestore e f m x).1) = |
| signedCoordinate f (x.2-x.1)+m := by |
| simp only [pairRestore,signedCoordinate_sub,signedCoordinate_add,signedCoordinate_nsmul, |
| signedCoordinate_step_self,mul_one] |
| ring |
|
|
| lemma pairRestore_height {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (m : ℕ) (a : ℝ) (x : Lattice d × Lattice d) (hx : x ∈ PairAtHeight (realPosition (step e)) a) : |
| pairRestore e f m x ∈ PairAtHeight (realPosition (step e)) (a+1) := by |
| have h1 := hx.1 |
| have h2 := hx.2 |
| simp only [signedHeight_projection] at h1 h2 |
| constructor |
| · change dot (realPosition (x.1+step e)) (realPosition (step e))=a+1 |
| simp only [signedHeight_projection,signedHeight_add_step_self,Int.cast_add,Int.cast_one,h1] |
| · change dot (realPosition (x.2+m • step f+step e)) (realPosition (step e))=a+1 |
| simp only [signedHeight_projection,signedHeight_add_step_self, |
| signedHeight_add_nsmul_other e f hef,Int.cast_add,Int.cast_one,h2] |
|
|
| lemma rawPairEndpointLaw_restoration {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (m : ℕ) (ω : Environment d) (x : Lattice d × Lattice d) |
| {κ : ℝ≥0} (hκ : ∀ y u, κ ≤ (ω y).1 u) : |
| (κ : ℝ≥0∞)^(m+2) • Measure.dirac (pairRestore e f m x) ≤ |
| rawPairEndpointLaw (realPosition (step e)) 1 ω x := by |
| have h1 := elliptic_lateral_script ω e f hef x.1 0 hκ |
| simp only [zero_nsmul,add_zero,zero_add,pow_one] at h1 |
| have h2 := elliptic_lateral_script ω e f hef x.2 m hκ |
| have hfinite (y : Lattice d) : IsFiniteMeasure |
| (hitKernel (Strip (realPosition (step e)) y 1) (Upper (realPosition (step e)) y 1) (ω,y)) := |
| ⟨lt_of_le_of_lt (hitKernel_total_le_one (disjoint_strip_upper _ _ _) _) ENNReal.one_lt_top⟩ |
| let := hfinite x.1 |
| let := hfinite x.2 |
| have hh := Measure.prod_mono h1 h2 |
| rw [Measure.prod_smul_left,Measure.prod_smul_right,smul_smul,Measure.dirac_prod_dirac] at hh |
| have hp : (κ : ℝ≥0∞)*(κ : ℝ≥0∞)^(m+1)=(κ : ℝ≥0∞)^(m+2) := by simpa only [Nat.add_assoc] using (pow_succ' (κ : ℝ≥0∞) (m+1)).symm |
| rw [hp] at hh |
| simpa only [rawPairEndpointLaw,pairRestore,Nat.cast_one] using hh |
|
|
| noncomputable def bufferRestorationLaw {d : ℕ} (e f : Direction d) (m : ℕ) |
| (π : Measure (Lattice d × Lattice d)) (κ : ℝ≥0) : Measure (Lattice d × Lattice d) := |
| (κ : ℝ≥0∞)^(m+2) • π.map (pairRestore e f m) |
|
|
| lemma bufferRestorationLaw_le {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (m : ℕ) (ω : Environment d) (π : Measure (Lattice d × Lattice d)) |
| {κ : ℝ≥0} (hκ : ∀ y u, κ ≤ (ω y).1 u) : |
| bufferRestorationLaw e f m π κ ≤ rawPairMixture (realPosition (step e)) 1 π ω := by |
| intro U |
| have hU := (Set.to_countable U).measurableSet |
| have hm : Measurable (pairRestore e f m) := measurable_of_countable _ |
| rw [bufferRestorationLaw,Measure.smul_apply,smul_eq_mul,Measure.map_apply hm hU, |
| rawPairMixture_apply] |
| have he : (κ : ℝ≥0∞)^(m+2)*π ((pairRestore e f m) ⁻¹' U) = |
| ∫⁻ x, ((κ : ℝ≥0∞)^(m+2) • Measure.dirac (pairRestore e f m x)) U ∂π := by |
| simp only [Measure.smul_apply,smul_eq_mul,Measure.dirac_apply' _ hU] |
| rw [lintegral_const_mul _ (measurable_of_countable _)] |
| congr 1 |
| rw [←lintegral_indicator_one (hm hU)] |
| apply lintegral_congr |
| intro x |
| by_cases hx : pairRestore e f m x ∈ U <;> simp [hx] |
| rw [he] |
| exact lintegral_mono fun x => (rawPairEndpointLaw_restoration e f hef m ω x hκ) U |
| |
| lemma bufferRestorationLaw_mass {d : ℕ} (e f : Direction d) (m : ℕ) |
| (π : Measure (Lattice d × Lattice d)) [IsProbabilityMeasure π] (κ : ℝ≥0) : |
| bufferRestorationLaw e f m π κ Set.univ = (κ : ℝ≥0∞)^(m+2) := by |
| simp [bufferRestorationLaw,Measure.map_apply (measurable_of_countable _)] |
|
|
| lemma bufferRestorationLaw_support {d : ℕ} (e f : Direction d) (m : ℕ) |
| (π : Measure (Lattice d × Lattice d)) (κ : ℝ≥0) (z : ℝ) |
| (hπ : ∀ᵐ x ∂π, z ≤ signedCoordinate f (x.2-x.1)) : |
| ∀ᵐ x ∂bufferRestorationLaw e f m π κ, z+m ≤ signedCoordinate f (x.2-x.1) := by |
| unfold bufferRestorationLaw |
| apply Measure.ae_smul_measure |
| rw [ae_map_iff (measurable_of_countable _).aemeasurable |
| (measurableSet_le measurable_const (measurable_of_countable _))] |
| filter_upwards [hπ] with x hx |
| rw [pairRestore_gap] |
| linarith |
|
|
| lemma bufferRestorationLaw_height {d : ℕ} (e f : Direction d) (hef : e.1 ≠ f.1) |
| (m : ℕ) (π : Measure (Lattice d × Lattice d)) (κ : ℝ≥0) (a : ℝ) |
| (hπ : ∀ᵐ x ∂π, x ∈ PairAtHeight (realPosition (step e)) a) : |
| ∀ᵐ x ∂bufferRestorationLaw e f m π κ, x ∈ PairAtHeight (realPosition (step e)) (a+1) := by |
| unfold bufferRestorationLaw |
| apply Measure.ae_smul_measure |
| rw [ae_map_iff (measurable_of_countable _).aemeasurable (Set.to_countable _).measurableSet] |
| filter_upwards [hπ] with x hx |
| exact pairRestore_height e f hef m a x hx |
|
|
| lemma bufferRestorationLaw_measurable {d : ℕ} (e f : Direction d) (m : ℕ) (κ : ℝ≥0) : |
| Measurable (fun π : Measure (Lattice d × Lattice d) => bufferRestorationLaw e f m π κ) := by |
| apply Measure.measurable_of_measurable_coe |
| intro U hU |
| simp only [bufferRestorationLaw,Measure.smul_apply,smul_eq_mul, |
| Measure.map_apply (measurable_of_countable _) hU] |
| exact measurable_const.mul (Measure.measurable_coe ((measurable_of_countable _) hU)) |
| end DirectionalTransience |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
