| import Mathlib |
| import OAI.Probability.Ballisticity.Coupling.SeedOrder |
|
|
| namespace OAI |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma exists_atom_probability {A : Type*} [MeasurableSpace A] [Countable A] |
| [MeasurableSingletonClass A] (μ : Measure A) [IsProbabilityMeasure μ] : ∃ x, μ {x} ≠ 0 := by |
| by_contra! hn |
| have hz : μ=0 := Measure.ext_of_singleton fun x => by simp [hn x] |
| have h := measure_univ (μ:=μ) |
| simp only [hz,Measure.coe_zero,Pi.zero_apply] at h |
| exact zero_ne_one h |
|
|
| lemma seedOrdered_trans {d : ℕ} (f : Direction d) (s : ℤ) (g h : ℝ) |
| (μ ρ σ : Measure (Lattice d)) [IsProbabilityMeasure ρ] |
| (h₁ : SeedOrdered f s g μ ρ) (h₂ : SeedOrdered f s h ρ σ) : SeedOrdered f s (g+h) μ σ := by |
| obtain ⟨y,hy⟩ := exists_atom_probability ρ |
| intro x hx z hz |
| have hxy := h₁ x hx y hy |
| have hyz := h₂ y hy z hz |
| nlinarith |
|
|
| lemma seedStateStep_ordered {d : ℕ} (e f : Direction d) (a s : ℤ) (k H : ℕ) |
| (θ z b j c g g' : ℝ) (q : SeedState e a) (ω : Environment d) |
| (hs : s=1 ∨ s= -1) (hq : 0<q.1) (hj : 0<j) (hc : 0<c) |
| (hg' : 0≤g') (hgg' : g'≤g-2*z) (hbg' : g'≤b-z) |
| (ho : SeedStateOrdered f s g q) |
| (hp : SeedPreserved e f a s k H θ z c q ω) : |
| SeedStateOrdered f s g' (seedStateStep e f a s k H θ z b j q ω) := by |
| let P := fun i => seedAdvance e a H (SeedCentral f H θ z) (seedCentralInput e f a s k q i) ω |
| have hm (i : ℕ) (hi : i<q.1) : seedEndpointRaw e H (SeedCentral f H θ z) |
| (seedCentralInput e f a s k q i).measure ω Set.univ ≠ 0 := by |
| intro hz |
| have hh := hp i hi |
| rw [hz,ENNReal.toReal_zero] at hh |
| linarith |
| have hop (i t : ℕ) (hit : i<t) (ht : t<q.1) : SeedOrdered f s g' (P i).measure (P t).measure := by |
| apply seedOrdered_mono f s (g:=g-2*z) _ hgg' |
| apply seedOrdered_central e f a s H θ z g hs _ _ ω (hm i (hit.trans ht)) (hm t ht) |
| intro x hx y hy |
| exact ho i t hit ht x (seedCentralInput_atom e f a s k q i hx) y (seedCentralInput_atom e f a s k q t hy) |
| intro i t hit ht |
| by_cases hsplit : q.1<k ∧ seedSplitTest e f a s H θ b j q ω |
| · have htt : t<q.1+1 := by simpa [seedStateStep,hsplit] using ht |
| have hi : i<q.1 := by omega |
| have hqi : i≠q.1 := by omega |
| by_cases hqt : t=q.1 |
| · let J := seedAdvance e a H (SeedJump f H s θ b) (seedJumpInput e f a s q) ω |
| have hjm : seedEndpointRaw e H (SeedJump f H s θ b) (seedJumpInput e f a s q).measure ω Set.univ ≠ 0 := by |
| intro hz |
| have hh := hsplit.2 |
| change j ≤ _ at hh |
| rw [hz,ENNReal.toReal_zero] at hh |
| linarith |
| have hlast : SeedOrdered f s g' (P (q.1-1)).measure J.measure := by |
| have he : seedCentralInput e f a s k q (q.1-1)=seedHalfProfile e f a s false (q.2 (q.1-1)) := by |
| simp [seedCentralInput,hsplit.1] |
| have hh := seedOrdered_split e f a s H θ z b hs (q.2 (q.1-1)) ω |
| (by simpa only [←he] using hm (q.1-1) (by omega)) hjm |
| apply seedOrdered_mono f s _ hbg' |
| simpa only [P,J,seedJumpInput,he] using hh |
| have hip : SeedOrdered f s g' (P i).measure J.measure := by |
| by_cases he : i=q.1-1 |
| · simpa only [he] using hlast |
| · have hh := seedOrdered_trans f s g' g' (P i).measure (P (q.1-1)).measure J.measure |
| (hop i (q.1-1) (by omega) (by omega)) hlast |
| exact seedOrdered_mono f s hh (by linarith) |
| simpa [seedStateStep,hsplit,hqt,hqi,P,J] using hip |
| · have htt' : t<q.1 := by omega |
| simpa [seedStateStep,hsplit,hqt,hqi,P] using hop i t hit htt' |
| · have htt : t<q.1 := by simpa [seedStateStep,hsplit] using ht |
| simpa [seedStateStep,hsplit,P] using hop i t hit htt |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| section |
|
|
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal BigOperators Topology Classical |
| namespace DirectionalTransience |
|
|
| lemma seedProfileRaw_joint_rows {d : ℕ} (e : Direction d) (a : ℤ) {H : ℕ} (hH : 0<H) |
| (R : Lattice d → Lattice d → Prop) (S : Set (Lattice d)) |
| (hS : ∀ x, signedHeight e x=a → Strip (realPosition (step e)) x H ⊆ S) : |
| @Measurable (SeedProfile e a × Environment d) ℝ |
| (@Prod.instMeasurableSpace _ _ inferInstance (rowSigma S)) _ |
| (fun p => (seedEndpointRaw e H R p.1.measure p.2 Set.univ).toReal) := by |
| let : MeasurableSpace (Environment d) := rowSigma S |
| exact ((Measure.measurable_coe MeasurableSet.univ).comp |
| ((seedEndpointRaw_joint_rows e hH R {x | signedHeight e x=a} S hS).comp |
| (((SeedProfile.measurable_supported e a).comp measurable_fst).prodMk measurable_snd))).ennreal_toReal |
|
|
| lemma seedSplitTest_joint_rows {d : ℕ} (e f : Direction d) (a s : ℤ) {H : ℕ} (hH : 0<H) |
| (θ b j : ℝ) (S : Set (Lattice d)) |
| (hS : ∀ x, signedHeight e x=a → Strip (realPosition (step e)) x H ⊆ S) : |
| @MeasurableSet (SeedState e a × Environment d) |
| (@Prod.instMeasurableSpace _ _ inferInstance (rowSigma S)) |
| {p | seedSplitTest e f a s H θ b j p.1 p.2} := by |
| let : MeasurableSpace (Environment d) := rowSigma S |
| change MeasurableSet {p : SeedState e a × Environment d | j ≤ |
| (seedEndpointRaw e H (SeedJump f H s θ b) (seedJumpInput e f a s p.1).measure p.2 Set.univ).toReal} |
| have hm := (seedProfileRaw_joint_rows e a hH (SeedJump f H s θ b) S hS).comp |
| (((measurable_seedJumpInput e f a s).comp measurable_fst).prodMk measurable_snd) |
| exact measurableSet_le measurable_const hm |
|
|
| lemma seedStateStep_joint_rows {d : ℕ} (e f : Direction d) (a s : ℤ) (k : ℕ) {H : ℕ} (hH : 0<H) |
| (θ z b j : ℝ) (S : Set (Lattice d)) |
| (hS : ∀ x, signedHeight e x=a → Strip (realPosition (step e)) x H ⊆ S) : |
| @Measurable (SeedState e a × Environment d) (SeedState e (a+H)) |
| (@Prod.instMeasurableSpace _ _ inferInstance (rowSigma S)) _ |
| (fun p => seedStateStep e f a s k H θ z b j p.1 p.2) := by |
| let : MeasurableSpace (Environment d) := rowSigma S |
| have ht : MeasurableSet {p : SeedState e a × Environment d | p.1.1<k ∧ seedSplitTest e f a s H θ b j p.1 p.2} := (measurableSet_lt (measurable_fst.comp measurable_fst) measurable_const).inter |
| (seedSplitTest_joint_rows e f a s hH θ b j S hS) |
| apply Measurable.prodMk |
| · exact ((measurable_fst.comp measurable_fst).add_const 1).ite ht (measurable_fst.comp measurable_fst) |
| · apply measurable_pi_iff.mpr |
| intro i |
| have hjump := (seedAdvance_joint_rows e a hH (SeedJump f H s θ b) S hS).comp |
| (((measurable_seedJumpInput e f a s).comp measurable_fst).prodMk measurable_snd) |
| have hcentral := (seedAdvance_joint_rows e a hH (SeedCentral f H θ z) S hS).comp |
| (((measurable_seedCentralInput e f a s k i).comp measurable_fst).prodMk measurable_snd) |
| exact hjump.ite (ht.inter (measurableSet_eq_fun measurable_const (measurable_fst.comp measurable_fst))) hcentral |
|
|
| lemma seedPreserved_joint_rows {d : ℕ} (e f : Direction d) (a s : ℤ) (k : ℕ) {H : ℕ} (hH : 0<H) |
| (θ z c : ℝ) (S : Set (Lattice d)) |
| (hS : ∀ x, signedHeight e x=a → Strip (realPosition (step e)) x H ⊆ S) : |
| @MeasurableSet (SeedState e a × Environment d) |
| (@Prod.instMeasurableSpace _ _ inferInstance (rowSigma S)) |
| {p | SeedPreserved e f a s k H θ z c p.1 p.2} := by |
| let : MeasurableSpace (Environment d) := rowSigma S |
| simp only [SeedPreserved,Set.ofPred_forall] |
| apply MeasurableSet.iInter |
| intro i |
| have hc := (seedProfileRaw_joint_rows e a hH (SeedCentral f H θ z) S hS).comp |
| (((measurable_seedCentralInput e f a s k i).comp measurable_fst).prodMk measurable_snd) |
| have hi : MeasurableSet {p : SeedState e a × Environment d | i<p.1.1} := |
| measurableSet_lt measurable_const (measurable_fst.comp measurable_fst) |
| have hmass : MeasurableSet {p : SeedState e a × Environment d | c ≤ |
| (seedEndpointRaw e H (SeedCentral f H θ z) (seedCentralInput e f a s k p.1 i).measure p.2 Set.univ).toReal} := |
| measurableSet_le measurable_const hc |
| convert hi.compl.union hmass using 1 |
| ext p |
| simp only [Set.mem_ofPred_eq,Set.mem_union,Set.mem_compl_iff,imp_iff_not_or] |
|
|
| def seedBelow {d : ℕ} (e : Direction d) (a : ℤ) := {x : Lattice d | signedHeight e x<a} |
| def seedAbove {d : ℕ} (e : Direction d) (a : ℤ) := {x : Lattice d | a ≤ signedHeight e x} |
|
|
| lemma seedBelow_disjoint_above {d : ℕ} (e : Direction d) (a : ℤ) : Disjoint (seedBelow e a) (seedAbove e a) := by |
| apply Set.disjoint_left.mpr |
| intro x hx hy |
| change signedHeight e x < a at hx |
| change a ≤ signedHeight e x at hy |
| omega |
|
|
| lemma strip_subset_seedAbove {d : ℕ} (e : Direction d) (a : ℤ) (H : ℕ) |
| (x : Lattice d) (hx : signedHeight e x=a) : Strip (realPosition (step e)) x H ⊆ seedAbove e a := by |
| intro y hy |
| have hh := hy.1 |
| simp only [signedHeight_projection,hx] at hh |
| exact_mod_cast hh |
|
|
| lemma strip_subset_seedBelow {d : ℕ} (e : Direction d) (a : ℤ) (H : ℕ) |
| (x : Lattice d) (hx : signedHeight e x=a) : Strip (realPosition (step e)) x H ⊆ seedBelow e (a+H) := by |
| intro y hy |
| have hh := hy.2 |
| simp only [signedHeight_projection,hx] at hh |
| change signedHeight e y<a+(H:ℤ) |
| exact_mod_cast hh |
|
|
| lemma seed_fresh_test_bound {d : ℕ} {A : Type*} [MeasurableSpace A] |
| (ν : Measure (Row d)) [IsProbabilityMeasure ν] (e : Direction d) (a : ℤ) |
| (U : Environment d → A) (hU : @Measurable _ _ (rowSigma (seedBelow e a)) _ U) |
| (E : Set (Environment d)) (hE : MeasurableSet[rowSigma (seedBelow e a)] E) |
| (G : Set (A × Environment d)) |
| (hG : @MeasurableSet _ (@Prod.instMeasurableSpace _ _ inferInstance (rowSigma (seedAbove e a))) G) |
| (c : ℝ≥0∞) (hb : ∀ η ∈ E, environmentLaw ν {ω | (U η,ω)∈G}≤c) : |
| environmentLaw ν {ω | ω∈E ∧ (U ω,ω)∈G} ≤ c*environmentLaw ν E := by |
| let S := seedBelow e a |
| let T := seedAbove e a |
| let K : Set (Environment d × Environment d) := {p | (U p.1,p.2)∈G} |
| have hK : @MeasurableSet _ (MeasurableSpace.prod (rowSigma S) (rowSigma T)) K := |
| hG.preimage ((hU.comp measurable_fst).prodMk measurable_snd) |
| have hV : @Measurable (Environment d) (Environment d) inferInstance (rowSigma T) id := |
| measurable_id.mono le_rfl (rowSigma_le T) |
| have hb' (η : Environment d) (hη : η∈E) : |
| (@Measure.map (Environment d) (Environment d) inferInstance (rowSigma T) id (environmentLaw ν)) |
| {ω | (η,ω)∈K} ≤ c := by |
| rw [show {ω | (η,ω)∈K}=Prod.mk η ⁻¹' K from rfl, |
| Measure.map_apply hV (hK.preimage measurable_prodMk_left)] |
| exact hb η hη |
| exact @fresh_rows_random_test_bound d (Environment d) (Environment d) (rowSigma S) (rowSigma T) |
| ν _ S T (seedBelow_disjoint_above e a) id id measurable_id measurable_id E hE K hK c hb' |
|
|
| end DirectionalTransience |
|
|
| end |
|
|
| end OAI |
|
|
