| import Mathlib |
| import OAI.Analysis.HardSphere.Allocation.CertificateSupply |
|
|
| namespace OAI |
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| section |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| namespace HardSphere.Groups |
| open Wiener PathCutoff CellOrbital BathSlice HardSphere.Lattice |
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|
| theorem ground_certificate_supply_ge_two : |
| ∃ α₀ ell : ℝ, 2≤α₀ ∧ 0<ell ∧ ∀ θ : ℝ, 0<θ → |
| ∃ b H t₀ : ℝ, 0<b ∧ b≤1/4 ∧ 0<H ∧ 0<t₀ ∧ |
| ∀ p : ℝ, 0<p → p≤1 → ∀ t : ℝ, 0<t → t≤t₀ → |
| ∃ D₀ : ℝ, 1≤D₀ ∧ |
| ∀ (n K : ℕ) [Nontrivial (Fin n)] [NeZero K] (L : Length), L.val=(K:ℝ) → |
| ∀ D α : ℝ, 100≤L.val → D=(n:ℝ)/L.val^3 → D₀≤D → α₀/2≤α → α≤2*α₀ → |
| ∃ (hD : 0<D) (hh : D⁻¹<2*t), ∀ (g : Jet n L), IsGroundVector n L (α/D) g → |
| ∀ A : Finset (Fin 3→Fin K), |
| groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈A, |
| ¬Certificate (sourceMesh t (α/D)) (α/D) (D^(1/12:ℝ)) (meanCost+1) |
| (spatialCloseScale D b/8) (spatialCloseScale D b) ell θ H D |
| (splitTime D⁻¹ (2*t-D⁻¹) (inv_pos.mpr hD) (sub_pos.mpr hh)) v ω}≤ |
| (6*ENNReal.ofReal p)^A.card := by |
| classical |
| obtain ⟨αE,hαE,hex⟩ := ShiftVisits.ground_excessVisitors_tail |
| obtain ⟨αL,hαL,hlong⟩ := ShiftVisits.ground_longVisitors_tail |
| obtain ⟨αR,hαR,hreg⟩ := ShiftVisits.ground_regularVisitors_uniform |
| let Amin := 2*max 1 (max αE (max αL αR)) |
| obtain ⟨α₀,ell,Dlow,Qlow,hA,hα₀,hell,hDlow,hQlow,hlow⟩ := grid_stationary_low_above Amin |
| have htwo : 2≤α₀ := le_trans (by dsimp [Amin]; linarith [le_max_left (1:ℝ) (max αE (max αL αR))]) hA |
| refine ⟨α₀,ell,htwo,hell,?_⟩ |
| intro θ hθ |
| obtain ⟨Qc,hQc,hclose⟩ := grid_close_tail |
| obtain ⟨b,Gc,hb,hb1,hGc,hclim,hclose⟩ := hclose θ hθ |
| obtain ⟨Dc,hDc,hclose⟩ := hclose (2*α₀) |
| obtain ⟨H,hH,hex⟩ := hex (2*α₀) (by positivity) |
| obtain ⟨tL,htL,hlong⟩ := hlong (2*α₀) θ (by positivity) hθ |
| let t₀ := min 1 (tL/2) |
| have ht₀ : 0<t₀ := lt_min zero_lt_one (by positivity) |
| refine ⟨b,H,t₀,hb,hb1,hH,ht₀,?_⟩ |
| intro p hp hp1 t ht htt |
| have ht1 : t≤1 := htt.trans (min_le_left _ _) |
| have htl : 2*t≤ min 2 tL := le_min (by linarith) (by have hh := htt.trans (min_le_right 1 (tL/2)); linarith) |
| obtain ⟨η,hη,hηp⟩ := exp_parameter hp |
| let C := 2*t/(α₀/2)^2+2 |
| let a := 2*t/(2*α₀)^2 |
| have hC : 0<C := by dsimp [C]; positivity |
| have ha : 0<a := by dsimp [a]; positivity |
| obtain ⟨Dr,hDr,hreg⟩ := hreg η hη (2*α₀) C a b (by positivity) hC ha hb |
| have hllim : Tendsto (fun D : ℝ ↦ Qlow/D^2) atTop (𝓝 0) := |
| tendsto_const_nhds.div_atTop (tendsto_pow_atTop (by norm_num : (2:ℕ)≠0)) |
| have hexlim : Tendsto (fun D : ℝ ↦ Real.exp (-D)) atTop (𝓝 0) := Real.tendsto_exp_neg_atTop_nhds_zero |
| have hlonglim : Tendsto (fun D : ℝ ↦ Real.exp (-(θ*D/2))) atTop (𝓝 0) := by |
| have h := Real.tendsto_exp_neg_atTop_nhds_zero.comp |
| (tendsto_id.const_mul_atTop (by positivity : 0<θ/2)) |
| convert h using 1 |
| funext D |
| simp only [Function.comp_apply,id_eq] |
| congr 1 |
| ring |
| have hinvlim : Tendsto (fun D : ℝ ↦ D⁻¹) atTop (𝓝 (0:ℝ)) := tendsto_inv_atTop_zero |
| have hp6 : 0<p^6 := pow_pos hp _ |
| have hev : ∀ᶠ D : ℝ in atTop, |
| 1≤D ∧ Dlow≤D ∧ Dc≤D ∧ Dr≤D ∧ 2*(2*Real.pi*(2*α₀))^3≤D ∧ |
| Qlow/D^2≤p^6 ∧ (((Gc/b^2)*D^(-(4/3):ℝ))^(1/(Qc:ℝ)))≤p^6 ∧ |
| Real.exp (-D)≤p^6 ∧ Real.exp (-(θ*D/2))≤p^6 ∧ D⁻¹<2*t := by |
| filter_upwards [eventually_ge_atTop (1:ℝ),eventually_ge_atTop Dlow,eventually_ge_atTop Dc, |
| eventually_ge_atTop Dr,eventually_ge_atTop (2*(2*Real.pi*(2*α₀))^3), |
| hllim.eventually (gt_mem_nhds hp6),hclim.eventually (gt_mem_nhds hp6), |
| hexlim.eventually (gt_mem_nhds hp6),hlonglim.eventually (gt_mem_nhds hp6), |
| hinvlim.eventually (gt_mem_nhds (by positivity : (0:ℝ)<2*t))] with D h1 h2 h3 h4 h5 h6 h7 h8 h9 h10 |
| exact ⟨h1,h2,h3,h4,h5,h6.le,h7.le,h8.le,h9.le,h10⟩ |
| obtain ⟨D₀,hD₀⟩ := eventually_atTop.1 hev |
| refine ⟨max 1 D₀,le_max_left _ _,?_⟩ |
| intro n K inst instK L hLK D α hL hDeq hDD hαlo hαup |
| obtain ⟨hD1,hDlow',hDc',hDr',hsmall',hlp,hcp,hep,hlgp,hinv⟩ := hD₀ D ((le_max_right _ _).trans hDD) |
| have hD : 0<D := zero_lt_one.trans_le hD1 |
| have hα1 : 1≤α := by |
| have hh := le_max_left 1 (max αE (max αL αR)) |
| dsimp [Amin] at hA |
| linarith |
| have hαe : αE≤α := by |
| have hh := (le_max_left αE (max αL αR)).trans (le_max_right 1 (max αE (max αL αR))) |
| dsimp [Amin] at hA |
| linarith |
| have hαl : αL≤α := by |
| have hh := (le_max_left αL αR).trans ((le_max_right αE (max αL αR)).trans (le_max_right 1 (max αE (max αL αR)))) |
| dsimp [Amin] at hA |
| linarith |
| have hαr : αR≤α := by |
| have hh := (le_max_right αL αR).trans ((le_max_right αE (max αL αR)).trans (le_max_right 1 (max αE (max αL αR)))) |
| dsimp [Amin] at hA |
| linarith |
| have hr : 0<α/D := div_pos (zero_lt_one.trans_le hα1) hD |
| have hαeq : D*(α/D)=α := by field_simp |
| have hsmall := smallness_of_large_density L hDeq hD1 (zero_le_one.trans hα1) hαup hsmall' |
| have hK : 1≤K := Nat.one_le_iff_ne_zero.mpr (NeZero.ne K) |
| obtain ⟨hm,hτ,hτr,hspan⟩ := sourceMesh_properties ht hr |
| obtain ⟨hmlo,hmhi⟩ := sourceMesh_bounds (αmax:=2*α₀) ht hr hD1 (by positivity : 0<α₀/2) |
| (by rw [hαeq]; exact hαlo) (by rw [hαeq]; exact hαup) |
| refine ⟨hD,hinv,?_⟩ |
| intro g hg A |
| let u := splitTime D⁻¹ (2*t-D⁻¹) (inv_pos.mpr hD) (sub_pos.mpr hinv) |
| apply certificate_joint (sourceMesh t (α/D)) (α/D) (D^(1/12:ℝ)) (meanCost+1) |
| (spatialCloseScale D b/8) (spatialCloseScale D b) ell θ H D u |
| (groundPathLaw n L (α/D) (2*t) g) (ENNReal.ofReal_le_one.mpr hp1) |
| intro i B |
| fin_cases i |
| · change groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈B,(centeredCount L (gridCenter L v) (ω 0):ℝ)<ell*D}≤_ |
| exact (hlow K n L D α hK hLK hDeq hDlow' hαlo hαup g hg (2*t) (by positivity) 0 B).trans |
| (ofReal_power_tail (by positivity) hp.le hlp B.card) |
| · change groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈B,θ*D≤((localCloseLabels L (spatialCloseScale D b) (gridCenter L v) (ω 0)).card:ℝ)}≤_ |
| exact (hclose n K L hLK D α hL hDeq hDc' hα1 hαup g hg (2*t) (by positivity) 0 B).trans |
| (ofReal_power_tail (Real.rpow_nonneg (by positivity) _) hp.le hcp B.card) |
| · change groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈B,θ*D≤((localCloseLabels L (spatialCloseScale D b) (gridCenter L v) (ω 1)).card:ℝ)}≤_ |
| exact (hclose n K L hLK D α hL hDeq hDc' hα1 hαup g hg (2*t) (by positivity) 1 B).trans |
| (ofReal_power_tail (Real.rpow_nonneg (by positivity) _) hp.le hcp B.card) |
| · change groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈B,H*D<((visitors (gridCenter L v) ω).card:ℝ)}≤_ |
| have he := hex n K L hLK (α/D) D hDeq hD1 (by rwa [hαeq]) (by rwa [hαeq]) hr hsmall g hg |
| (gridShift L) (B.image gridSite) (by positivity : 0<2*t) (by linarith : 2*t≤2) |
| rw [Finset.card_image_of_injective _ gridSite_injective,←joint_event_image] at he |
| simp only [grid_visitors_eq] at he |
| apply (measure_mono (fun ω hω v hv ↦ (hω v hv).le)).trans |
| exact he.trans (by simpa only [neg_mul] using exp_power_tail hp.le hep B.card) |
| · change groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈B,¬∀j,pathVisitsBall L 12 (gridCenter L v) j ω→ |
| Regular (sourceMesh t (α/D)) (α/D) (D^(1/12:ℝ)) (meanCost+1) u (spatialCloseScale D b/8) (onePathAt j ω)}≤_ |
| have he := hreg n K L hLK (α/D) D hDeq hDr' (by rwa [hαeq]) (by rwa [hαeq]) hr hsmall g hg |
| (gridShift L) (B.image gridSite) (sourceMesh t (α/D)) (sourceStep t (α/D)) hm hτ hτr |
| hmlo hmhi (by rw [hspan]; linarith) hD (by rwa [hspan]) |
| rw [Finset.card_image_of_injective _ gridSite_injective,←joint_event_image] at he |
| simp only [hspan,←grid_nonregular_iff] at he |
| exact he.trans (by simpa only [neg_div] using exp_power_tail hp.le hηp B.card) |
| · change groundPathLaw n L (α/D) (2*t) g {ω | ∀v∈B,θ*D≤((longVisitors (gridCenter L v) ω).card:ℝ)}≤_ |
| have he := hlong n K L hLK (α/D) D hDeq hD1 (by rwa [hαeq]) (by rwa [hαeq]) hr hsmall g hg |
| (gridShift L) (B.image gridSite) (by positivity : 0<2*t) htl |
| rw [Finset.card_image_of_injective _ gridSite_injective,←joint_event_image] at he |
| simp only [grid_longVisitors_eq] at he |
| exact he.trans (exp_power_tail hp.le hlgp B.card) |
|
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| end HardSphere.Groups |
| end |
| end |
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| end OAI |
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