| import Mathlib |
| import OAI.Analysis.HardSphere.Paths.LiftedSkeleton |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal BigOperators |
| namespace HardSphere.DyadicSkeleton |
| open HardSphere.Lattice |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma measurable_simpleEncounter {K : ℕ} [NeZero K] (R : ℕ) |
| (x y : DiscreteTorus K) : |
| Measurable (fun p : SimplePath x y × SimplePath x y ↦ simpleEncounter R p.1 p.2) := |
| measurable_of_countable _ |
|
|
| lemma measurable_simpleEncounter_comp {Ω : Type*} [MeasurableSpace Ω] |
| {K : ℕ} [NeZero K] (R : ℕ) {x y : DiscreteTorus K} |
| (f : Ω → SimplePath x y) (hf : Measurable f) : |
| Measurable (fun z : Ω×Ω ↦ simpleEncounter (K:=K) R (x:=x) (y:=y) (f z.1) (f z.2)) := |
| by |
| have hf1 : Measurable (fun z : Ω×Ω ↦ f z.1) := hf.comp measurable_fst |
| have hf2 : Measurable (fun z : Ω×Ω ↦ f z.2) := hf.comp measurable_snd |
| have hpair : Measurable (fun z : Ω×Ω ↦ (f z.1,f z.2)) := hf1.prodMk hf2 |
| exact Measurable.comp |
| (g:=fun p : SimplePath x y × SimplePath x y ↦ simpleEncounter (K:=K) R p.1 p.2) |
| (f:=fun z : Ω×Ω ↦ (f z.1,f z.2)) |
| (measurable_simpleEncounter (K:=K) R x y) hpair |
|
|
| @[fun_prop] lemma measurable_dyadicEncounter {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) |
| (hn : 0<n) (d : Fin 2 → ℤ) (v : Vertex) (B : Finset (DiscreteTorus K)) |
| (hB : acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1])) : |
| Measurable (fun z : Noise×Noise ↦ simpleEncounter R |
| (dyadicDetour hK R hn d v B hB z.1) (dyadicDetour hK R hn d v B hB z.2)) := by |
| exact measurable_simpleEncounter_comp R (dyadicDetour hK R hn d v B hB) |
| (measurable_dyadicDetour hK R hn d v B hB) |
|
|
| def acceptedIntegrand {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) (hn : 0<n) |
| (d : Fin 2 → ℤ) (v : Vertex) (κ : ℝ) (B : Finset (DiscreteTorus K)) (z : Noise×Noise) : ℝ≥0∞ := |
| if hB : acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1]) then |
| ENNReal.ofReal (Real.exp (κ*(simpleEncounter R (dyadicDetour hK R hn d v B hB z.1) |
| (dyadicDetour hK R hn d v B hB z.2):ℝ))) else 0 |
|
|
| @[fun_prop] lemma measurable_acceptedIntegrand {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) (hn : 0<n) |
| (d : Fin 2 → ℤ) (v : Vertex) (κ : ℝ) (B : Finset (DiscreteTorus K)) : |
| Measurable (acceptedIntegrand hK R hn d v κ B) := by |
| unfold acceptedIntegrand |
| split_ifs <;> fun_prop |
|
|
| lemma dyadic_environment_moment {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) |
| (hn : 0<n) (hnK : 8*n≤K) (d : Fin 2 → ℤ) (hd : ∀j, |d j|≤n) |
| (v : Vertex) (μ : PMF (Finset (DiscreteTorus K))) (p κ : ℝ) |
| (hp : 0≤p) (hκ : 0≤κ) (hbad : JointBadLaw μ (ENNReal.ofReal p)) |
| (hq : ((candidateConstant (R+2):ℝ)*(p*Real.exp (κ*(encounterConstant R:ℝ))))*Real.exp 1≤1/2) |
| (z : Noise×Noise) : |
| (∑' B, μ B*acceptedIntegrand hK R hn d v κ B z)≤ |
| ENNReal.ofReal (Real.exp (Real.exp 1*((encounterConstant R:ℝ)*κ+ |
| 2*((candidateConstant (R+2):ℝ)*(p*Real.exp (κ*(encounterConstant R:ℝ)))))* |
| ∑ j∈Finset.range (n+1), Real.exp |
| (-twoCopyDistance (min j (n-j)) z/(dyadicRadius R:ℝ)))) := by |
| let E (B : Finset (DiscreteTorus K)) := acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1]) |
| let J (B : Finset (DiscreteTorus K)) : ℝ := if hB : E B then |
| (simpleEncounter R (dyadicDetour hK R hn d v B hB z.1) |
| (dyadicDetour hK R hn d v B hB z.2):ℝ) else 0 |
| have he (B : Finset (DiscreteTorus K)) : acceptedIntegrand hK R hn d v κ B z= |
| if E B then ENNReal.ofReal (Real.exp (κ*J B)) else 0 := by |
| change (if hB : E B then ENNReal.ofReal (Real.exp (κ* |
| (simpleEncounter R (dyadicDetour hK R hn d v B hB z.1) |
| (dyadicDetour hK R hn d v B hB z.2):ℝ))) else 0)=_ |
| by_cases hB : E B |
| · simp only [dite_eq_left hB,ite_eq_left hB,J] |
| · simp only [dite_eq_right hB,ite_eq_right hB] |
| simp_rw [he] |
| refine (accepted_charge_environment_bound R μ p κ (encounterConstant R) |
| ((encounterSet (torusNear K R) (projectedSkeleton K n d v z.1) |
| (projectedSkeleton K n d v z.2)).card:ℝ) hp hκ hbad |
| (projectedSkeleton K n d v z.1) (projectedSkeleton K n d v z.2) E J ?_).trans ?_ |
| · intro B hB |
| simp only [J,dite_eq_left hB] |
| exact dyadicDetour_charge hK R hn d v B hB z.1 z.2 |
| · exact ENNReal.ofReal_le_ofReal (Real.exp_le_exp.mpr |
| (dyadic_nominal_potential R hn hnK d hd v z.1 z.2 κ |
| (p*Real.exp (κ*(encounterConstant R:ℝ))) hκ (by positivity) hq)) |
| |
| lemma dyadic_annealed_moment {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) |
| (hn : 0<n) (hnK : 8*n≤K) (d : Fin 2 → ℤ) (hd : ∀j, |d j|≤n) |
| (v : Vertex) (μ : PMF (Finset (DiscreteTorus K))) (p κ ξ : ℝ) |
| (hp : 0≤p) (hκ : 0≤κ) (hbad : JointBadLaw μ (ENNReal.ofReal p)) |
| (hq : ((candidateConstant (R+2):ℝ)*(p*Real.exp (κ*(encounterConstant R:ℝ))))*Real.exp 1≤1/2) |
| (hc : 2*(Real.exp 1*((encounterConstant R:ℝ)*κ+ |
| 2*((candidateConstant (R+2):ℝ)*(p*Real.exp (κ*(encounterConstant R:ℝ))))))≤ξ) |
| (hξ : ∀ T : ℕ, (∫⁻ z, ENNReal.ofReal (Real.exp (ξ* |
| ∑ u∈Finset.range (T+1),Real.exp (-twoCopyDistance u z/(dyadicRadius R:ℝ)))) ∂law.prod law)≤2) : |
| (∑' B, μ B*∫⁻ z, acceptedIntegrand hK R hn d v κ B z ∂law.prod law)≤2 := by |
| have hme (B : Finset (DiscreteTorus K)) : Measurable (fun z ↦ μ B*acceptedIntegrand hK R hn d v κ B z) := by fun_prop |
| simp_rw [← lintegral_const_mul' _ _ (PMF.apply_ne_top μ _)] |
| rw [← lintegral_tsum (fun B ↦ (hme B).aemeasurable)] |
| refine le_trans (lintegral_mono (fun z ↦ ?_)) (hξ (n/2)) |
| refine (dyadic_environment_moment hK R hn hnK d hd v μ p κ hp hκ hbad hq z).trans ?_ |
| apply ENNReal.ofReal_le_ofReal |
| apply Real.exp_le_exp.mpr |
| let c : ℝ := Real.exp 1*((encounterConstant R:ℝ)*κ+ |
| 2*((candidateConstant (R+2):ℝ)*(p*Real.exp (κ*(encounterConstant R:ℝ))))) |
| have hc0 : 0≤c := by dsimp [c]; positivity |
| have hf := folded_sum_le n (fun u ↦ Real.exp (-twoCopyDistance u z/(dyadicRadius R:ℝ))) |
| (fun _ ↦ (Real.exp_pos _).le) |
| calc |
| _ ≤ c*(2*∑ u∈Finset.range (n/2+1),Real.exp (-twoCopyDistance u z/(dyadicRadius R:ℝ))) := |
| mul_le_mul_of_nonneg_left hf hc0 |
| _ ≤ _ := by |
| rw [← mul_assoc] |
| apply mul_le_mul_of_nonneg_right (by simpa [mul_comm,c] using hc) |
| exact Finset.sum_nonneg fun u _ ↦ (Real.exp_pos _).le |
| |
| lemma small_route_parameters (C A ξ : ℝ) (hC : 0<C) (hA : 0<A) (hξ : 0<ξ) : |
| ∃ p₀ κ : ℝ, 0<p₀ ∧ 0<κ ∧ ∀ p : ℝ, 0≤p → p≤p₀ → |
| (A*(p*Real.exp (κ*C)))*Real.exp 1≤1/2 ∧ |
| 2*(Real.exp 1*(C*κ+2*(A*(p*Real.exp (κ*C)))))≤ξ := by |
| let e := Real.exp 1 |
| let s := min 1 ξ |
| let κ := min 1 (s/(16*e*C)) |
| let p₀ := s/(16*e*A*Real.exp C) |
| have he : 0<e := Real.exp_pos _ |
| have hs : 0<s := lt_min (by norm_num) hξ |
| have hs1 : s≤1 := min_le_left _ _ |
| have hsξ : s≤ξ := min_le_right _ _ |
| have hκ : 0<κ := lt_min (by norm_num) (div_pos hs (by positivity)) |
| have hp₀ : 0<p₀ := div_pos hs (by positivity) |
| have hκ1 : κ≤1 := min_le_left _ _ |
| have hκs : e*(C*κ)≤ s/16 := by |
| have h := (le_div_iff₀ (by positivity : 0<16*e*C)).mp |
| (show κ≤ s/(16*e*C) from min_le_right _ _) |
| nlinarith |
| refine ⟨p₀,κ,hp₀,hκ,?_⟩ |
| intro p hp hpp |
| have hpbound := (le_div_iff₀ (by positivity : 0<16*e*A*Real.exp C)).mp hpp |
| have hexp : Real.exp (κ*C)≤Real.exp C := Real.exp_le_exp.mpr (by nlinarith) |
| have hqmon := mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left hexp hp) hA.le |
| have hqe : e*(A*(p*Real.exp (κ*C)))≤ s/16 := by |
| have hh := mul_le_mul_of_nonneg_left hqmon he.le |
| nlinarith |
| constructor |
| · change _*e≤_ |
| nlinarith |
| · change 2*(e*(C*κ+2*(A*(p*Real.exp (κ*C)))))≤ξ |
| nlinarith |
| |
| lemma dyadic_path_constants (R : ℕ) : |
| ∃ p₀ κ : ℝ, 0<p₀ ∧ 0<κ ∧ ∃ K₀ : ℕ, ∀ K : ℕ, K₀≤K → ∀ _ : NeZero K, |
| ∀ n : ℕ, ∀ hn : 0<n, 8*n≤K → ∀ d : Fin 2 → ℤ, (∀j, |d j|≤n) → |
| ∀ v : Vertex, ∀ μ : PMF (Finset (DiscreteTorus K)), ∀ p : ℝ, |
| 0≤p → p≤p₀ → JointBadLaw μ (ENNReal.ofReal p) → |
| (1/2:ℝ≥0∞)≤(∑' B, if acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1]) then μ B else 0) ∧ |
| ∃ hK : 2<K, (∑' B, μ B*∫⁻ z, acceptedIntegrand hK R hn d v κ B z ∂law.prod law)≤2 := by |
| obtain ⟨ξ,hξ,hsoft⟩ := two_copy_soft_moment (dyadicRadius R) (by exact_mod_cast dyadicRadius_pos R) |
| obtain ⟨p₁,κ,hp₁,hκ,hparam⟩ := small_route_parameters (encounterConstant R) (candidateConstant (R+2)) ξ |
| (by unfold encounterConstant; positivity) (by unfold candidateConstant; positivity) hξ |
| obtain ⟨p₂,hp₂,K₀,haccept⟩ := acceptedEnvironment_mass (R+2) (by omega) |
| refine ⟨min p₁ p₂,κ,lt_min hp₁ hp₂,hκ,max K₀ 3,?_⟩ |
| intro K hK inst n hn hnK d hd v μ p hp hpp hbad |
| have hK3 : 2<K := by omega |
| have hb := hparam p hp (hpp.trans (min_le_left _ _)) |
| exact ⟨haccept K ((le_max_left _ _).trans hK) inst μ p hp |
| (hpp.trans (min_le_right _ _)) hbad v _,hK3, |
| dyadic_annealed_moment hK3 R hn hnK d hd v μ p κ ξ hp hκ.le hbad hb.1 hb.2 hsoft⟩ |
| |
| lemma double_lintegral_map {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| (P : Measure Ω) [IsProbabilityMeasure P] (f : Ω → X) (hf : Measurable f) |
| (g : X×X → ℝ≥0∞) (hg : Measurable g) : |
| (∫⁻ x, ∫⁻ y, g (x,y) ∂P.map f ∂P.map f)= |
| ∫⁻ z, g (f z.1,f z.2) ∂P.prod P := by |
| let : IsProbabilityMeasure (P.map f) := |
| (Measure.isProbabilityMeasure_map_iff hf.aemeasurable).mpr inferInstance |
| rw [← lintegral_prod g hg.aemeasurable,Measure.map_prod_map P P hf hf, |
| lintegral_map hg (hf.prodMap hf)] |
| rfl |
| |
| lemma dyadicDetour_good_mass {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) (hn : 0<n) |
| (d : Fin 2 → ℤ) (v : Vertex) (B : Finset (DiscreteTorus K)) |
| (hB : acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1])) : |
| (law.map (dyadicDetour hK R hn d v B hB)) {q | simplePathGood B q}=1 := by |
| rw [Measure.map_apply (measurable_dyadicDetour hK R hn d v B hB) MeasurableSet.of_discrete] |
| have he : (dyadicDetour hK R hn d v B hB) ⁻¹' {q | simplePathGood B q}=Set.univ := by |
| apply Set.eq_univ_of_forall |
| intro ω x hx |
| exact (dyadicDetour_spec hK R hn d v B hB ω x hx).1 |
| rw [he,measure_univ] |
|
|
| lemma dyadicDetour_double_moment {K n : ℕ} [NeZero K] (hK : 2<K) (R : ℕ) (hn : 0<n) |
| (d : Fin 2 → ℤ) (v : Vertex) (κ : ℝ) (B : Finset (DiscreteTorus K)) |
| (hB : acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1])) : |
| (∫⁻ q, ∫⁻ q', ENNReal.ofReal (Real.exp (κ*(simpleEncounter R q q':ℝ))) |
| ∂law.map (dyadicDetour hK R hn d v B hB) ∂law.map (dyadicDetour hK R hn d v B hB))= |
| ∫⁻ z, acceptedIntegrand hK R hn d v κ B z ∂law.prod law := by |
| rw [double_lintegral_map law _ (measurable_dyadicDetour hK R hn d v B hB) |
| (fun p ↦ ENNReal.ofReal (Real.exp (κ*(simpleEncounter R p.1 p.2:ℝ)))) (measurable_of_countable _)] |
| apply lintegral_congr |
| intro z |
| simp only [acceptedIntegrand,dite_eq_left hB] |
|
|
| theorem dyadic_good_paths_lift (R : ℕ) : |
| ∃ p₀ κ : ℝ, 0<p₀ ∧ 0<κ ∧ ∃ K₀ : ℕ, ∀ K : ℕ, K₀≤K → ∀ _ : NeZero K, |
| ∀ {Ω : Type*} [MeasurableSpace Ω], ∀ P : Measure Ω, ∀ _ : IsProbabilityMeasure P, |
| ∀ b : Ω → Finset (DiscreteTorus K), Measurable b → |
| ∀ p : ℝ, 0≤p → p≤p₀ → |
| (∀ A : Finset (DiscreteTorus K), P {d | A⊆b d}≤(ENNReal.ofReal p)^A.card) → |
| ∀ n : ℕ, 0<n → 8*n≤K → ∀ d : Fin 2 → ℤ, (∀j, |d j|≤n) → ∀ v : Vertex, |
| ∃ E : Set Ω, MeasurableSet E ∧ (1/2:ℝ≥0∞)≤P E ∧ |
| ∃ ν : Kernel Ω (SimplePath (torusProjection K v) (torusProjection K (v+![(n:ℤ),d 0,d 1]))), |
| IsMarkovKernel ν ∧ |
| (∀ e∈E, ν e {q | simplePathGood (b e) q}=1) ∧ |
| (∫⁻ e, E.indicator (fun e ↦ ∫⁻ q, ∫⁻ q', |
| ENNReal.ofReal (Real.exp (κ*(simpleEncounter R q q':ℝ))) ∂ν e ∂ν e) e ∂P)≤2 := by |
| obtain ⟨p₀,κ,hp₀,hκ,K₀,hall⟩ := dyadic_path_constants R |
| refine ⟨p₀,κ,hp₀,hκ,K₀,?_⟩ |
| intro K hK inst Ω mΩ P hP b hb p hp hpp hbad n hn hnK d hd v |
| let := (Measure.isProbabilityMeasure_map_iff (μ:=P) hb.aemeasurable).mpr inferInstance |
| let μ := (P.map b).toPMF |
| obtain ⟨hmass,hK2,hmoment⟩ := hall K hK inst n hn hnK d hd v μ p hp hpp |
| (mapped_bad_law P b hb _ hbad) |
| let E (B : Finset (DiscreteTorus K)) := acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1]) |
| have hex : ∃ B, E B := by |
| by_contra hh |
| have hh' : ∀ B, ¬E B := not_exists.mp hh |
| simp only [show ∀ B, ¬acceptedEnvironment (R+2) B v (v+![(n:ℤ),d 0,d 1]) from hh', |
| ite_false,tsum_zero] at hmass |
| norm_num at hmass |
| obtain ⟨B₀,hB₀⟩ := hex |
| let ω₀ : Noise := fun _ ↦ ⟨0,by norm_num,by norm_num⟩ |
| let q₀ := dyadicDetour hK2 R hn d v B₀ hB₀ ω₀ |
| let η (B : Finset (DiscreteTorus K)) := if hB : E B then |
| law.map (dyadicDetour hK2 R hn d v B hB) else Measure.dirac q₀ |
| have hη (B : Finset (DiscreteTorus K)) : IsProbabilityMeasure (η B) := by |
| dsimp only [η] |
| split_ifs |
| · exact (Measure.isProbabilityMeasure_map_iff |
| (measurable_dyadicDetour hK2 R hn d v B _).aemeasurable).mpr inferInstance |
| · infer_instance |
| let ν₀ : Kernel (Finset (DiscreteTorus K)) |
| (SimplePath (torusProjection K v) (torusProjection K (v+![(n:ℤ),d 0,d 1]))) := |
| ⟨η,Measurable.of_discrete⟩ |
| let ν := ν₀.comap b hb |
| refine ⟨{e | E (b e)},hb (t := {B | E B}) MeasurableSet.of_discrete,?_,ν,?_,?_,?_⟩ |
| · change (1/2:ℝ≥0∞)≤P (b ⁻¹' {B | E B}) |
| rw [← Measure.map_apply (μ:=P) hb (s:={B | E B}) MeasurableSet.of_discrete, |
| ← Measure.toPMF_toMeasure (P.map b),pmf_event_discrete] |
| exact hmass |
| · exact ⟨fun e ↦ hη (b e)⟩ |
| · intro e he |
| change E (b e) at he |
| change η (b e) {q | simplePathGood (b e) q}=1 |
| simp only [η,dite_eq_left he] |
| exact dyadicDetour_good_mass hK2 R hn d v (b e) he |
| · have hpoint (e : Ω) : |
| {e | E (b e)}.indicator (fun e ↦ ∫⁻ q, ∫⁻ q', |
| ENNReal.ofReal (Real.exp (κ*(simpleEncounter R q q':ℝ))) ∂ν e ∂ν e) e = |
| ∫⁻ z, acceptedIntegrand hK2 R hn d v κ (b e) z ∂law.prod law := by |
| by_cases he : E (b e) |
| · rw [Set.indicator_of_mem (s:={e | E (b e)}) (a:=e) he] |
| change (∫⁻ q, ∫⁻ q', ENNReal.ofReal (Real.exp (κ*(simpleEncounter R q q':ℝ))) |
| ∂η (b e) ∂η (b e))=_ |
| simp only [η,dite_eq_left he] |
| exact dyadicDetour_double_moment hK2 R hn d v κ (b e) he |
| · rw [Set.indicator_of_notMem (s:={e | E (b e)}) (a:=e) he] |
| change ¬acceptedEnvironment (R+2) (b e) v (v+![(n:ℤ),d 0,d 1]) at he |
| simp only [acceptedIntegrand,dite_eq_right he,lintegral_zero] |
| simp_rw [hpoint] |
| rw [finite_datum_expectation P b hb |
| (fun B ↦ ∫⁻ z, acceptedIntegrand hK2 R hn d v κ B z ∂law.prod law)] |
| exact hmoment |
| |
| end HardSphere.DyadicSkeleton |
| end |
| end |
| |
| end OAI |
| |
