| import Mathlib | |
| import OAI.Analysis.HardSphere.Geometry.LargeParentMarked | |
| import OAI.Analysis.HardSphere.Geometry.TorusNear | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped ENNReal Topology BigOperators | |
| open scoped BigOperators Classical | |
| open scoped BigOperators Classical | |
| open scoped BigOperators Classical ENNReal | |
| namespace HardSphere.BathSlice | |
| open HardSphere.Lattice | |
| def boxWeight (m : ℕ) : ℝ≥0∞ := ENNReal.ofReal ((2*((m:ℝ)+1))^3) | |
| lemma volume_largeBoxUnion_le {K : ℕ} [NeZero K] (L : Length) (B : Finset (DiscreteTorus K)) : | |
| (volume : Measure (Torus L)) (largeBoxUnion L B) ≤ animalCost (torusRangeGraph K 1) B boxWeight := by | |
| classical | |
| unfold largeBoxUnion animalCost | |
| apply (measure_iUnion_le _).trans | |
| apply ENNReal.tsum_le_tsum | |
| intro m | |
| apply (measure_iUnion_le _).trans | |
| rw [tsum_fintype] | |
| apply Finset.sum_le_sum | |
| intro v _ | |
| apply (measure_biUnion_finset_le _ _).trans | |
| apply Finset.sum_le_sum | |
| intro A _ | |
| by_cases hA : A⊆B | |
| · simp only [hA,Set.iUnion_true,ite_true] | |
| simpa only [boxWeight,Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using | |
| volume_rootBox_le L (siteRep v) (by positivity : (0:ℝ)≤(m:ℝ)+2) | |
| · simp only [hA,Set.iUnion_of_empty,measure_empty,ite_false,le_refl] | |
| attribute [local irreducible] torusDistance | |
| lemma pairCloseLabels_measurable {m : ℕ} (L : Length) (s : ℝ) (v : Torus L) : | |
| Measurable (fun Y : Configuration m L ↦ ((pairCloseLabels L s v Y).card:ℝ)) := by | |
| classical | |
| simp only [pairCloseLabels,Finset.card_filter,Nat.cast_sum,Nat.cast_ite,Nat.cast_one,Nat.cast_zero] | |
| apply Finset.measurable_sum | |
| intro i _ | |
| have hd : Continuous (fun Y : Configuration m L ↦ torusDistance L (Y i) v) := | |
| (continuous_torusDistance L).comp | |
| ((continuous_apply i).prodMk (continuous_const : Continuous (fun _ : Configuration m L ↦ v))) | |
| have hp : MeasurableSet {Y : Configuration m L | ∃ j, j≠i ∧ torusDistance L (Y i) (Y j)≤ s} := by | |
| simp only [Set.ofPred_exists] | |
| apply MeasurableSet.iUnion | |
| intro j | |
| have hc : Continuous (fun Y : Configuration m L ↦ torusDistance L (Y i) (Y j)) := | |
| (continuous_torusDistance L).comp ((show Continuous (fun Y : Configuration m L ↦ Y i) from continuous_apply i).prodMk | |
| (show Continuous (fun Y : Configuration m L ↦ Y j) from continuous_apply j)) | |
| by_cases hji : j≠i | |
| · convert measurableSet_le hc.measurable (show Measurable (fun _ : Configuration m L ↦ s) from measurable_const) using 1 | |
| ext Y | |
| exact and_iff_right hji | |
| · convert (MeasurableSet.empty : MeasurableSet (∅ : Set (Configuration m L))) using 1 | |
| ext Y | |
| exact iff_false_intro (fun h ↦ hji h.1) | |
| exact Measurable.ite ((measurableSet_le hd.measurable measurable_const).inter hp) | |
| measurable_const measurable_const | |
| lemma measurable_closeSet {n K : ℕ} [NeZero K] (L : Length) (s T : ℝ) : | |
| Measurable (closeSet (n := n) (K := K) L s T) := by | |
| classical | |
| apply measurable_finset_iff.mpr | |
| intro v | |
| simp only [closeSet,Finset.mem_filter,Finset.mem_univ,true_and] | |
| apply measurableSet_setOfPred.mp | |
| exact measurableSet_le measurable_const (pairCloseLabels_measurable L s (siteCenter L v)) | |
| lemma volume_torus (L : Length) : (volume : Measure (Torus L)) Set.univ = ENNReal.ofReal (L.val^3) := by | |
| rw [volume_pi,Measure.pi_univ] | |
| simp only [AddCircle.measure_univ,Finset.prod_const,Finset.card_univ,Fintype.card_fin, | |
| ENNReal.ofReal_pow L.property.le] | |
| lemma microperiod_length {K : ℕ} {L : Length} {r : ℝ} (hr : 0 < r) | |
| (hLK : L.val=(K:ℝ)) : L.val = microMesh r * ((microPeriod r*K : ℕ):ℝ) := by | |
| have hq : (microPeriod r : ℝ) ≠ 0 := by exact_mod_cast (microPeriod_pos hr).ne' | |
| rw [hLK,microMesh,Nat.cast_mul] | |
| field_simp | |
| theorem omitted_volume_le {n K M : ℕ} [NeZero K] {L : Length} {r s T : ℝ} | |
| (hr : 0 < r) (hrsmall : r ≤ 1/4096) (hL : 10 < L.val) (hLK : L.val=(K:ℝ)) | |
| (hMK : M ≤ K) (hrs : 6*r ≤ s) (hT : T ≤ 1/(1372*r)) | |
| (Y : Configuration n L) (hY : Y ∈ allowed n L r) : | |
| (volume : Measure (Torus L)) (omitted (microMesh r) r Y) ≤ | |
| ENNReal.ofReal (1029*r^2*n) + | |
| animalCost (torusRangeGraph K 1) (closeSet L s T Y) boxWeight + | |
| (if LargeBadComponent (torusRangeGraph K 1) (closeSet L s T Y) M then | |
| ENNReal.ofReal (L.val^3) else 0) := by | |
| classical | |
| have hmesh := microMesh_bounds hr (by linarith : r ≤ 1) | |
| have hmark : LargeParentMarked (microPeriod r) (microMesh r) r Y (closeSet (K := K) L s T Y) := by | |
| simpa only [mul_one,Function.Embedding.refl_apply] using actual_large_parent_marked (K := K) (D := 1) (θ := T) | |
| hr hrsmall hL hLK hrs (by simpa using hT) (Function.Embedding.refl _) Y | |
| by_cases he : LargeBadComponent (torusRangeGraph K 1) (closeSet L s T Y) M | |
| · simp only [he,ite_true] | |
| calc | |
| _ ≤ (volume : Measure (Torus L)) Set.univ := measure_mono (Set.subset_univ _) | |
| _ = ENNReal.ofReal (L.val^3) := volume_torus L | |
| _ ≤ _ := le_add_self | |
| · simp only [he,ite_false,add_zero] | |
| have hK : 2 < K := by rw [hLK] at hL; exact_mod_cast (show (2:ℝ)<K by linarith) | |
| have hsub := omitted_subset_small_and_boxes (microPeriod_pos hr) hK hLK hMK Y hY | |
| (closeSet L s T Y) hmark he | |
| apply (measure_mono hsub).trans | |
| apply (measure_union_le _ _).trans | |
| apply add_le_add | |
| · apply (volume_smallFills_le hmesh.1 (hmesh.2.1.trans (by linarith)) | |
| (microperiod_length hr hLK) hmesh.2.2 Y).trans | |
| apply ENNReal.ofReal_le_ofReal | |
| have hsq : (microMesh r)^2 ≤ r^2 := sq_le_sq₀ hmesh.1.le hr.le |>.mpr hmesh.2.1 | |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hsq (by norm_num)) (Nat.cast_nonneg n) | |
| · exact volume_largeBoxUnion_le L _ | |
| theorem omitted_mean_marginal {m n : ℕ} (L : Length) (h r : ℝ) | |
| (e : Fin m ↪ Fin n) (μ : Measure (Configuration n L)) : | |
| (∫⁻ Y, (volume : Measure (Torus L)) (omitted h r Y) | |
| ∂(μ.map (fun X i ↦ X (e i)))) = | |
| ∫⁻ X, (volume : Measure (Torus L)) (omitted h r (fun i ↦ X (e i))) ∂μ := by | |
| exact lintegral_map (measurable_omitted_volume L h r) | |
| (Measurable.of_eval (fun i ↦ measurable_pi_apply (e i))) | |
| end HardSphere.BathSlice | |
| namespace HardSphere.BathSlice | |
| open HardSphere.Lattice | |
| lemma boxWeight_le (m : ℕ) : boxWeight (m+1) ≤ 64*(8:ℝ≥0∞)^m := by | |
| have hnat : m+2 ≤ 2^(m+1) := by have := (Nat.lt_two_pow_self (n := m+1)); omega | |
| have hbound : 8*(m+2)^3 ≤ 64*8^m := by | |
| calc | |
| _ ≤ 8*(2^(m+1))^3 := by gcongr | |
| _ = 64*8^m := by rw [←pow_mul,Nat.mul_comm (m+1) 3,pow_mul]; norm_num; rw [pow_succ]; ring | |
| have heq : boxWeight (m+1) = ((8*(m+2)^3:ℕ):ℝ≥0∞) := by | |
| unfold boxWeight | |
| norm_cast | |
| ring | |
| rw [heq] | |
| exact_mod_cast hbound | |
| lemma box_series_le {q : ℝ≥0∞} (hq : 8*q ≤ (2:ℝ≥0∞)⁻¹) : | |
| (∑' m : ℕ, boxWeight (m+1)*q^(m+1)) ≤ 128*q := by | |
| calc | |
| _ ≤ ∑' m : ℕ, (64*q)*((2:ℝ≥0∞)⁻¹)^m := by | |
| apply ENNReal.tsum_le_tsum | |
| intro m | |
| calc | |
| _ ≤ (64*8^m)*q^(m+1) := mul_le_mul' (boxWeight_le m) le_rfl | |
| _ = (64*q)*(8*q)^m := by rw [mul_pow,pow_succ]; ring | |
| _ ≤ _ := mul_le_mul' le_rfl (pow_le_pow_left' hq m) | |
| _ = (64*q)*2 := by rw [ENNReal.tsum_mul_left,ENNReal.tsum_geometric_two] | |
| _ = _ := by ring | |
| theorem omitted_mean_le {n K M : ℕ} [NeZero K] {L : Length} {r s T : ℝ} | |
| (hr : 0 < r) (hrsmall : r ≤ 1/4096) (hL : 10 < L.val) (hLK : L.val=(K:ℝ)) | |
| (hMK : M ≤ K) (hM : 1 ≤ M) (hrs : 6*r ≤ s) (hT : T ≤ 1/(1372*r)) | |
| (μ : Measure (Configuration n L)) [IsProbabilityMeasure μ] | |
| (hallowed : ∀ᵐ Y ∂μ, Y ∈ allowed n L r) (p : ℝ≥0∞) | |
| (hbad : ∀ A : Finset (DiscreteTorus K), μ {Y | A ⊆ closeSet L s T Y} ≤ p^A.card) | |
| (hp : 8*((2^27:ℕ):ℝ≥0∞)*p ≤ (2:ℝ≥0∞)⁻¹) : | |
| (∫⁻ Y, (volume : Measure (Torus L)) (omitted (microMesh r) r Y) ∂μ) ≤ | |
| ENNReal.ofReal (1029*r^2*n) + | |
| (K:ℝ≥0∞)^3*(128*(((2^27:ℕ):ℝ≥0∞)*p)) + | |
| ENNReal.ofReal (L.val^3)*(K:ℝ≥0∞)^3*(((2^27:ℕ):ℝ≥0∞)*p)^M := by | |
| classical | |
| let B := closeSet (n := n) (K := K) L s T | |
| let E := {Y : Configuration n L | LargeBadComponent (torusRangeGraph K 1) (B Y) M} | |
| have hB : Measurable B := measurable_closeSet L s T | |
| have hE : MeasurableSet E := by | |
| exact (Set.to_countable {C : Finset (DiscreteTorus K) | | |
| LargeBadComponent (torusRangeGraph K 1) C M}).measurableSet.preimage hB | |
| have hdegree (v : DiscreteTorus K) : (torusRangeGraph K 1).degree v ≤ 27 := by | |
| simpa using torusRange_degree 1 v | |
| have hcard : Fintype.card (DiscreteTorus K) = K^3 := by | |
| simp only [DiscreteTorus,Fintype.card_fun,ZMod.card,Fintype.card_fin] | |
| have hcost := animalCost_mean_le (torusRangeGraph K 1) 27 hdegree μ B hB p hbad boxWeight | |
| rw [hcard,Nat.cast_pow] at hcost | |
| have hprob := largeBadComponent_measure (torusRangeGraph K 1) 27 hdegree μ B p hbad M hM | |
| rw [hcard,Nat.cast_pow] at hprob | |
| have hi : (fun Y ↦ if Y∈E then ENNReal.ofReal (L.val^3) else 0) = | |
| E.indicator (fun _ ↦ ENNReal.ofReal (L.val^3)) := by | |
| ext Y | |
| simp only [Set.indicator_apply] | |
| calc | |
| _ ≤ ∫⁻ Y, ENNReal.ofReal (1029*r^2*n) + animalCost (torusRangeGraph K 1) (B Y) boxWeight + | |
| (if Y∈E then ENNReal.ofReal (L.val^3) else 0) ∂μ := by | |
| apply lintegral_mono_ae | |
| filter_upwards [hallowed] with Y hY | |
| exact omitted_volume_le hr hrsmall hL hLK hMK hrs hT Y hY | |
| _ = ENNReal.ofReal (1029*r^2*n) + (∫⁻ Y, animalCost (torusRangeGraph K 1) (B Y) boxWeight ∂μ) + | |
| ENNReal.ofReal (L.val^3)*μ E := by | |
| have hm : Measurable (fun Y ↦ ENNReal.ofReal (1029*r^2*n) + | |
| animalCost (torusRangeGraph K 1) (B Y) boxWeight) := | |
| measurable_const.add (measurable_animalCost _ B hB _) | |
| rw [lintegral_add_left hm, | |
| lintegral_add_left measurable_const,hi,lintegral_indicator_const hE,lintegral_const] | |
| simp only [measure_univ,mul_one] | |
| _ ≤ _ := by | |
| apply add_le_add | |
| · apply add_le_add_right | |
| exact hcost.trans (mul_le_mul' le_rfl (box_series_le (by simpa only [mul_assoc] using hp))) | |
| · exact (mul_le_mul' le_rfl hprob).trans_eq (by ring) | |
| end HardSphere.BathSlice | |
| end | |
| end OAI | |