| import Mathlib |
|
|
| namespace OAI |
|
|
| universe cycleUniverse1 cycleUniverse2 cycleUniverse3 cycleUniverse4 cycleUniverse5 cycleUniverse6 cycleUniverse7 cycleUniverse8 cycleUniverse9 cycleUniverse10 cycleUniverse11 cycleUniverse12 cycleUniverse13 cycleUniverse14 cycleUniverse15 |
|
|
| section |
|
|
| namespace ErdosGallai.Sampling |
| open MeasureTheory ProbabilityTheory Finset Real |
|
|
| section BernoulliBounds |
| variable {Ω : Type cycleUniverse1} {I : Type cycleUniverse2} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
|
|
| lemma indicator_mem_Icc {Ω : Type cycleUniverse3} [_contextInstance1 : MeasurableSpace Ω] {X : Ω → ℝ} (hX : ∀ ω, X ω = 0 ∨ X ω = 1) (ω : Ω) : |
| X ω ∈ Set.Icc (0 : ℝ) 1 := by rcases hX ω with h | h <;> simp [h] |
|
|
| lemma mgf_indicator {X : Ω → ℝ} (hm : Measurable X) |
| (hX : ∀ ω, X ω = 0 ∨ X ω = 1) (p t : ℝ) (hp : ∫ ω, X ω ∂μ = p) : |
| mgf X μ t = 1 + p * (exp t - 1) := by |
| have hXi : Integrable X μ := Integrable.of_mem_Icc 0 1 hm.aemeasurable |
| (ae_of_all _ (indicator_mem_Icc hX)) |
| have heq : (fun ω => exp (t * X ω)) = fun ω => (1 : ℝ) + (exp t - 1) * X ω := by |
| funext ω |
| rcases hX ω with h | h <;> simp [h] |
| rw [mgf, heq, integral_add (integrable_const _) (hXi.const_mul _), integral_const_mul, |
| hp, integral_const] |
| simp [mul_comm] |
|
|
| lemma mgf_indicator_sum_bound {X : I → Ω → ℝ} (s : Finset I) |
| (hm : ∀ i, Measurable (X i)) (hi : iIndepFun X μ) |
| (hX : ∀ i ω, X i ω = 0 ∨ X i ω = 1) |
| (p t : ℝ) (hp : ∀ i, ∫ ω, X i ω ∂μ = p) : |
| mgf (∑ i ∈ s, X i) μ t ≤ exp ((s.card : ℝ) * p * (exp t - 1)) := by |
| by_cases hs : s.Nonempty |
| swap |
| · have he := Finset.not_nonempty_iff_eq_empty.mp hs |
| simp [he] |
| rw [hi.mgf_sum hm] |
| have heq : (∏ i ∈ s, mgf (X i) μ t) = (1 + p * (exp t - 1)) ^ s.card := by |
| simp_rw [mgf_indicator (hm _) (hX _) p t (hp _)] |
| simp |
| rw [heq] |
| have hn : 0 ≤ 1 + p * (exp t - 1) := by |
| obtain ⟨i, _⟩ := hs |
| rw [← mgf_indicator (hm i) (hX i) p t (hp i)] |
| exact mgf_nonneg |
| calc |
| (1 + p * (exp t - 1)) ^ s.card ≤ exp (p * (exp t - 1)) ^ s.card := |
| pow_le_pow_left₀ hn (by linarith [add_one_le_exp (p * (exp t - 1))]) _ |
| _ = _ := by rw [← exp_nat_mul]; congr 1; ring |
|
|
| lemma exp_neg_half_bound : exp (-(1 / 2 : ℝ)) ≤ 5 / 8 := by |
| have he : exp (1 / 2 : ℝ) * exp (1 / 2 : ℝ) = exp 1 := by |
| rw [← exp_add]; norm_num |
| have hlow : (8 / 5 : ℝ) ≤ exp (1 / 2 : ℝ) := by |
| nlinarith [exp_one_gt_d9, exp_pos (1 / 2 : ℝ)] |
| rw [exp_neg] |
| exact (inv_le_comm₀ (exp_pos _) (by norm_num)).mpr (by norm_num at *; exact hlow) |
|
|
| lemma exp_half_bound : exp (1 / 2 : ℝ) ≤ 5 / 3 := by |
| have he : exp (1 / 2 : ℝ) * exp (1 / 2 : ℝ) = exp 1 := by |
| rw [← exp_add]; norm_num |
| nlinarith [exp_one_lt_d9, exp_pos (1 / 2 : ℝ)] |
|
|
| theorem bernoulli_sum_lower_tail {X : I → Ω → ℝ} (s : Finset I) |
| (hm : ∀ i, Measurable (X i)) (hi : iIndepFun X μ) |
| (hX : ∀ i ω, X i ω = 0 ∨ X i ω = 1) |
| (p : ℝ) (hp0 : 0 ≤ p) (hp : ∀ i, ∫ ω, X i ω ∂μ = p) : |
| μ.real {ω | (∑ i ∈ s, X i ω) ≤ p * s.card / 2} ≤ exp (-p * s.card / 8) := by |
| have hInt : Integrable (fun ω => exp (-(1 / 2 : ℝ) * (∑ i ∈ s, X i) ω)) μ := |
| hi.integrable_exp_mul_sum hm (fun i _ => integrable_exp_mul_of_mem_Icc |
| (hm i).aemeasurable (ae_of_all _ (indicator_mem_Icc (hX i)))) |
| have hb := measure_le_le_exp_mul_mgf (X := ∑ i ∈ s, X i) (p * s.card / 2) |
| (by norm_num : -(1 / 2 : ℝ) ≤ 0) hInt |
| simp only [Finset.sum_apply] at hb |
| apply hb.trans |
| calc |
| _ ≤ exp (-(-(1 / 2 : ℝ)) * (p * s.card / 2)) * |
| exp ((s.card : ℝ) * p * (exp (-(1 / 2 : ℝ)) - 1)) := by |
| exact mul_le_mul_of_nonneg_left (mgf_indicator_sum_bound s hm hi hX p _ hp) (exp_pos _).le |
| _ = exp (p * s.card / 4 + (s.card : ℝ) * p * (exp (-(1 / 2 : ℝ)) - 1)) := by |
| rw [← exp_add]; congr 1; ring |
| _ ≤ _ := by |
| apply exp_le_exp.mpr |
| nlinarith [mul_le_mul_of_nonneg_left exp_neg_half_bound |
| (mul_nonneg (Nat.cast_nonneg s.card) hp0)] |
|
|
| theorem bernoulli_sum_upper_tail {X : I → Ω → ℝ} (s : Finset I) |
| (hm : ∀ i, Measurable (X i)) (hi : iIndepFun X μ) |
| (hX : ∀ i ω, X i ω = 0 ∨ X i ω = 1) |
| (p : ℝ) (hp0 : 0 ≤ p) (hp : ∀ i, ∫ ω, X i ω ∂μ = p) : |
| μ.real {ω | 2 * p * s.card ≤ ∑ i ∈ s, X i ω} ≤ exp (-p * s.card / 3) := by |
| have hInt : Integrable (fun ω => exp ((1 / 2 : ℝ) * (∑ i ∈ s, X i) ω)) μ := |
| hi.integrable_exp_mul_sum hm (fun i _ => integrable_exp_mul_of_mem_Icc |
| (hm i).aemeasurable (ae_of_all _ (indicator_mem_Icc (hX i)))) |
| have hb := measure_ge_le_exp_mul_mgf (X := ∑ i ∈ s, X i) (2 * p * s.card) |
| (by norm_num : (0 : ℝ) ≤ 1 / 2) hInt |
| simp only [Finset.sum_apply] at hb |
| apply hb.trans |
| calc |
| _ ≤ exp (-(1 / 2 : ℝ) * (2 * p * s.card)) * |
| exp ((s.card : ℝ) * p * (exp (1 / 2 : ℝ) - 1)) := by |
| exact mul_le_mul_of_nonneg_left (mgf_indicator_sum_bound s hm hi hX p _ hp) (exp_pos _).le |
| _ = exp (-p * s.card + (s.card : ℝ) * p * (exp (1 / 2 : ℝ) - 1)) := by |
| rw [← exp_add]; congr 1; ring |
| _ ≤ _ := by |
| apply exp_le_exp.mpr |
| nlinarith [mul_le_mul_of_nonneg_left exp_half_bound |
| (mul_nonneg (Nat.cast_nonneg s.card) hp0)] |
|
|
| end BernoulliBounds |
|
|
| section ProductSamples |
| variable {V : Type cycleUniverse4} [Fintype V] [DecidableEq V] |
|
|
| noncomputable def sampleMeasure (V : Type cycleUniverse5) [Fintype V] (p : unitInterval) : Measure (V → Bool) := |
| Measure.pi (fun _ : V => bernoulliMeasure true false p) |
|
|
| instance sampleMeasure_probability (p : unitInterval) : IsProbabilityMeasure (sampleMeasure V p) := by |
| unfold sampleMeasure |
| infer_instance |
|
|
| def sampleVertices (ω : V → Bool) : Finset V := Finset.univ.filter (fun v => ω v = true) |
|
|
| def sampleIndicator (v : V) (ω : V → Bool) : ℝ := if ω v then 1 else 0 |
|
|
| lemma sampleIndicator_measurable {V : Type cycleUniverse6} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] (v : V) : Measurable (sampleIndicator v) := |
| measurable_of_finite _ |
|
|
| lemma sampleIndicator_values {V : Type cycleUniverse7} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] (v : V) (ω : V → Bool) : |
| sampleIndicator v ω = 0 ∨ sampleIndicator v ω = 1 := by |
| simp only [sampleIndicator]; split <;> simp |
|
|
| lemma sampleIndicator_indep {V : Type cycleUniverse8} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] (p : unitInterval) : |
| iIndepFun (sampleIndicator (V := V)) (sampleMeasure V p) := by |
| exact iIndepFun_pi (X := fun _ : V => fun b : Bool => if b then (1 : ℝ) else 0) |
| (fun _ => (measurable_of_finite _).aemeasurable) |
|
|
| lemma sampleIndicator_integral {V : Type cycleUniverse9} [_contextInstance1 : Fintype V] [_contextInstance2 : DecidableEq V] (p : unitInterval) (v : V) : |
| ∫ ω, sampleIndicator v ω ∂sampleMeasure V p = (p : ℝ) := by |
| have hl := (measurePreserving_eval (fun _ : V => bernoulliMeasure true false p) v).hasLaw |
| have hi := hl.integral_comp (f := fun b : Bool => if b then (1 : ℝ) else 0) |
| (measurable_of_finite _).aestronglyMeasurable |
| simpa [sampleMeasure, sampleIndicator, integral_bernoulliMeasure] using hi |
|
|
| lemma sampleIndicator_sum (S : Finset V) (ω : V → Bool) : |
| (∑ v ∈ S, sampleIndicator v ω) = ((S ∩ sampleVertices ω).card : ℝ) := by |
| simp only [sampleIndicator, Finset.sum_boole] |
| congr 2 |
| ext v |
| simp [sampleVertices] |
|
|
| lemma sample_card_lower_tail (p : unitInterval) (S : Finset V) : |
| (sampleMeasure V p).real {ω | ((S ∩ sampleVertices ω).card : ℝ) ≤ (p : ℝ) * S.card / 2} ≤ |
| exp (-(p : ℝ) * S.card / 8) := by |
| simpa only [sampleIndicator_sum] using bernoulli_sum_lower_tail S |
| sampleIndicator_measurable (sampleIndicator_indep p) sampleIndicator_values |
| (p : ℝ) p.property.1 (sampleIndicator_integral p) |
|
|
| lemma sample_card_upper_tail (p : unitInterval) (S : Finset V) : |
| (sampleMeasure V p).real {ω | 2 * (p : ℝ) * S.card ≤ ((S ∩ sampleVertices ω).card : ℝ)} ≤ |
| exp (-(p : ℝ) * S.card / 3) := by |
| simpa only [sampleIndicator_sum] using bernoulli_sum_upper_tail S |
| sampleIndicator_measurable (sampleIndicator_indep p) sampleIndicator_values |
| (p : ℝ) p.property.1 (sampleIndicator_integral p) |
|
|
| lemma sample_miss_probability (p : unitInterval) (S : Finset V) : |
| (sampleMeasure V p).real {ω | Disjoint S (sampleVertices ω)} = (1 - (p : ℝ)) ^ S.card := by |
| have heq : {ω | Disjoint S (sampleVertices ω)} = |
| Set.univ.pi (fun v : V => if v ∈ S then {false} else Set.univ) := by |
| ext ω |
| simp only [Set.mem_ofPred_eq, Set.mem_pi, Set.mem_univ, forall_true_left] |
| simp only [Finset.disjoint_left, sampleVertices, Finset.mem_filter, Finset.mem_univ, true_and] |
| constructor |
| · intro h v |
| by_cases hv : v ∈ S |
| · simp only [hv, ↓reduceIte, Set.mem_singleton_iff] |
| cases hi : ω v <;> simp_all |
| · simp [hv] |
| · intro h v hv ht |
| have := h v |
| simp_all |
| rw [heq, Measure.real, sampleMeasure, Measure.pi_pi, ENNReal.toReal_prod] |
| have hpv (v : V) : |
| ((bernoulliMeasure true false p) (if v ∈ S then {false} else Set.univ)).toReal = |
| if v ∈ S then 1 - (p : ℝ) else 1 := by |
| by_cases hv : v ∈ S |
| · simp only [hv, ↓reduceIte] |
| change (bernoulliMeasure true false p).real {false} = _ |
| exact bernoulliMeasure_real_apply_of_notMem_of_mem p (by simp) (by simp) (by simp) |
| · simp [hv] |
| simp_rw [hpv] |
| simp |
|
|
| lemma sample_miss_bound (p : unitInterval) (S : Finset V) : |
| (sampleMeasure V p).real {ω | Disjoint S (sampleVertices ω)} ≤ exp (-(p : ℝ) * S.card) := by |
| rw [sample_miss_probability] |
| calc |
| (1 - (p : ℝ)) ^ S.card ≤ exp (-(p : ℝ)) ^ S.card := by |
| apply pow_le_pow_left₀ (sub_nonneg.mpr p.property.2) |
| linarith [add_one_le_exp (-(p : ℝ))] |
| _ = _ := by rw [← exp_nat_mul]; congr 1; ring |
|
|
| lemma sample_contains_probability (p : unitInterval) (S : Finset V) : |
| (sampleMeasure V p) {ω | S ⊆ sampleVertices ω} = ENNReal.ofReal (p : ℝ) ^ S.card := by |
| have heq : {ω | S ⊆ sampleVertices ω} = |
| Set.univ.pi (fun v : V => if v ∈ S then {true} else Set.univ) := by |
| ext ω |
| simp only [Set.mem_ofPred_eq, Set.mem_pi, Set.mem_univ, forall_true_left] |
| simp only [Finset.subset_iff, sampleVertices, Finset.mem_filter, Finset.mem_univ, true_and] |
| constructor |
| · intro h v |
| by_cases hv : v ∈ S <;> simp_all |
| · intro h v hv |
| simpa [hv] using h v |
| rw [heq, sampleMeasure, Measure.pi_pi] |
| have hpv (v : V) : |
| bernoulliMeasure true false p (if v ∈ S then {true} else Set.univ) = |
| if v ∈ S then ENNReal.ofReal (p : ℝ) else 1 := by |
| by_cases hv : v ∈ S |
| · simp only [hv, ↓reduceIte] |
| rw [bernoulliMeasure_apply_of_mem_of_notMem p (by simp) (by simp) (by simp)] |
| exact ENNReal.ofReal_coe_nnreal.symm |
| · simp [hv] |
| simp_rw [hpv] |
| simp |
|
|
| lemma sample_contains_real_probability (p : unitInterval) (S : Finset V) : |
| (sampleMeasure V p).real {ω | S ⊆ sampleVertices ω} = (p : ℝ) ^ S.card := by |
| rw [Measure.real, sample_contains_probability, ENNReal.toReal_pow, |
| ENNReal.toReal_ofReal p.property.1] |
|
|
| lemma sample_blocks_independent {I : Type cycleUniverse10} (p : unitInterval) (S : I → Finset V) |
| (hS : Pairwise fun i j => Disjoint (S i) (S j)) : |
| iIndepSet (fun i => {ω | S i ⊆ sampleVertices ω}) (sampleMeasure V p) := by |
| classical |
| rw [iIndepSet_iff_meas_biInter (fun _ => MeasurableSet.of_discrete)] |
| intro t |
| have heq : (⋂ i ∈ t, {ω | S i ⊆ sampleVertices ω}) = |
| {ω | t.biUnion S ⊆ sampleVertices ω} := by |
| ext ω |
| simp only [Set.mem_iInter, Set.mem_ofPred_eq, Finset.biUnion_subset] |
| rw [heq, sample_contains_probability, Finset.card_biUnion] |
| · simp_rw [sample_contains_probability] |
| exact (Finset.prod_pow_eq_pow_sum t (fun i => (S i).card) _).symm |
| · intro i _ j _ hij |
| exact hS hij |
|
|
| noncomputable def blockIndicator (S : Finset V) : (V → Bool) → ℝ := |
| {ω | S ⊆ sampleVertices ω}.indicator (fun _ => 1) |
|
|
| lemma blockIndicator_values (S : Finset V) (ω : V → Bool) : |
| blockIndicator S ω = 0 ∨ blockIndicator S ω = 1 := by |
| classical |
| by_cases h : S ⊆ sampleVertices ω <;> simp [blockIndicator, h] |
|
|
| lemma blockIndicator_integral (p : unitInterval) (S : Finset V) : |
| ∫ ω, blockIndicator S ω ∂sampleMeasure V p = (p : ℝ) ^ S.card := by |
| rw [blockIndicator, integral_indicator_const _ MeasurableSet.of_discrete, |
| sample_contains_real_probability] |
| simp |
|
|
| lemma sample_blocks_lower_tail {I : Type cycleUniverse11} (p : unitInterval) (S : I → Finset V) |
| (hS : Pairwise fun i j => Disjoint (S i) (S j)) (d : ℕ) |
| (hd : ∀ i, (S i).card = d) (t : Finset I) : |
| (sampleMeasure V p).real {ω | (∑ i ∈ t, blockIndicator (S i) ω) ≤ |
| (p : ℝ) ^ d * t.card / 2} ≤ exp (-(p : ℝ) ^ d * t.card / 8) := by |
| apply bernoulli_sum_lower_tail t (fun _ => measurable_of_finite _) |
| (sample_blocks_independent p S hS).iIndepFun_indicator |
| (fun i => blockIndicator_values (S i)) ((p : ℝ) ^ d) (pow_nonneg p.property.1 _) |
| intro i |
| change (∫ ω, blockIndicator (S i) ω ∂sampleMeasure V p) = _ |
| rw [blockIndicator_integral, hd] |
|
|
| lemma sample_weight_sum (A : Finset V) (w : V → ℝ) (ω : V → Bool) : |
| (∑ v ∈ A ∩ sampleVertices ω, w v) = ∑ v ∈ A, w v * sampleIndicator v ω := by |
| rw [← Finset.sum_ite_mem] |
| apply Finset.sum_congr rfl |
| intro v _ |
| simp only [sampleVertices, Finset.mem_filter, Finset.mem_univ, true_and, sampleIndicator] |
| split <;> simp_all |
|
|
| lemma sample_weight_integral (p : unitInterval) (A : Finset V) (w : V → ℝ) : |
| ∫ ω, (∑ v ∈ A ∩ sampleVertices ω, w v) ∂sampleMeasure V p = |
| (p : ℝ) * ∑ v ∈ A, w v := by |
| simp_rw [sample_weight_sum] |
| rw [integral_finsetSum A (fun _ _ => Integrable.of_finite)] |
| simp_rw [integral_const_mul, sampleIndicator_integral] |
| rw [← Finset.sum_mul, mul_comm] |
|
|
| lemma nonnegative_markov_quarter {Ω : Type cycleUniverse12} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {X : Ω → ℝ} |
| (hi : Integrable X μ) (hn : ∀ ω, 0 ≤ X ω) {B : ℝ} (hB : 0 < B) |
| (he : ∫ ω, X ω ∂μ ≤ B) : μ.real {ω | 4 * B < X ω} ≤ 1 / 4 := by |
| have hb := mul_meas_ge_le_integral_of_nonneg (ae_of_all _ hn) hi (4 * B) |
| have hm : μ.real {ω | 4 * B < X ω} ≤ μ.real {ω | 4 * B ≤ X ω} := |
| measureReal_mono (fun ω (h : 4 * B < X ω) => h.le) |
| nlinarith [mul_le_mul_of_nonneg_left hm (by positivity : 0 ≤ 4 * B)] |
|
|
| lemma sample_weight_markov (p : unitInterval) (hp : 0 < (p : ℝ)) |
| (A : Finset V) (w : V → ℝ) (hw : ∀ v ∈ A, 0 ≤ w v) |
| {B : ℝ} (hB : 0 < B) (hs : ∑ v ∈ A, w v ≤ B) : |
| (sampleMeasure V p).real {ω | 4 * (p : ℝ) * B < ∑ v ∈ A ∩ sampleVertices ω, w v} ≤ |
| 1 / 4 := by |
| have ht := nonnegative_markov_quarter (μ := sampleMeasure V p) |
| (X := fun ω => ∑ v ∈ A ∩ sampleVertices ω, w v) Integrable.of_finite |
| (fun ω => Finset.sum_nonneg (fun v hv => hw v (Finset.mem_inter.mp hv).1)) |
| (mul_pos hp hB) (by rw [sample_weight_integral]; exact mul_le_mul_of_nonneg_left hs hp.le) |
| simpa only [mul_assoc] using ht |
|
|
| theorem exists_small_weighted_hitting_set {I : Type cycleUniverse13} (A : Finset V) (hA : A.Nonempty) |
| (good : Finset I) (K : I → Finset V) (hKA : ∀ i ∈ good, K i ⊆ A) |
| (w : V → ℝ) (hw : ∀ v ∈ A, 0 ≤ w v) {B d : ℝ} |
| (hB : 0 < B) (hs : ∑ v ∈ A, w v ≤ B) |
| (hd : ∀ i ∈ good, d ≤ (K i).card) |
| (q : unitInterval) (hq : 0 < (q : ℝ)) |
| (hmiss : (good.card : ℝ) * exp (-(q : ℝ) * d) < 1 / 2) : |
| ∃ S : Finset V, S ⊆ A ∧ (S.card : ℝ) ≤ 4 * (q : ℝ) * A.card ∧ |
| (∑ v ∈ S, w v) ≤ 4 * (q : ℝ) * B ∧ ∀ i ∈ good, ¬Disjoint (K i) S := by |
| classical |
| let μ := sampleMeasure V q |
| let E₁ : Set (V → Bool) := {ω | 4 * (q : ℝ) * A.card < ((A ∩ sampleVertices ω).card : ℝ)} |
| let E₂ : Set (V → Bool) := {ω | 4 * (q : ℝ) * B < ∑ v ∈ A ∩ sampleVertices ω, w v} |
| let E₃ : Set (V → Bool) := ⋃ i ∈ good, {ω | Disjoint (K i) (sampleVertices ω)} |
| have h₁ : μ.real E₁ ≤ 1 / 4 := by |
| have ht := sample_weight_markov q hq A (fun _ => (1 : ℝ)) (by simp) |
| (show (0 : ℝ) < A.card by exact_mod_cast hA.card_pos) (by simp) |
| simpa [μ, E₁] using ht |
| have h₂ : μ.real E₂ ≤ 1 / 4 := sample_weight_markov q hq A w hw hB hs |
| have h₃ : μ.real E₃ < 1 / 2 := by |
| calc |
| μ.real E₃ ≤ ∑ i ∈ good, μ.real {ω | Disjoint (K i) (sampleVertices ω)} := |
| measureReal_biUnion_finset_le good _ |
| _ ≤ ∑ i ∈ good, exp (-(q : ℝ) * d) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| exact (sample_miss_bound q (K i)).trans (exp_le_exp.mpr |
| (mul_le_mul_of_nonpos_left (hd i hi) (neg_nonpos.mpr hq.le))) |
| _ = (good.card : ℝ) * exp (-(q : ℝ) * d) := by simp |
| _ < _ := hmiss |
| have hf : μ.real (E₁ ∪ E₂ ∪ E₃) < 1 := by |
| have ha := measureReal_union_le (μ := μ) (E₁ ∪ E₂) E₃ |
| have hb := measureReal_union_le (μ := μ) E₁ E₂ |
| linarith |
| have hne : E₁ ∪ E₂ ∪ E₃ ≠ Set.univ := by |
| intro h |
| rw [h] at hf |
| have hu : μ.real Set.univ = 1 := by simp [μ, Measure.real] |
| linarith |
| obtain ⟨ω, hω⟩ := (Set.ne_univ_iff_exists_notMem _).mp hne |
| have hn := hω |
| simp only [Set.mem_union, not_or] at hn |
| refine ⟨A ∩ sampleVertices ω, Finset.inter_subset_left, le_of_not_gt hn.1.1, |
| le_of_not_gt hn.1.2, ?_⟩ |
| intro i hi hdj |
| apply hn.2 |
| simp only [E₃, Set.mem_iUnion, Set.mem_ofPred_eq] |
| refine ⟨i, hi, ?_⟩ |
| apply Finset.disjoint_left.mpr |
| intro v hv hvs |
| exact Finset.disjoint_left.mp hdj hv (Finset.mem_inter.mpr ⟨hKA i hi hv, hvs⟩) |
|
|
| end ProductSamples |
|
|
| section FiniteUnion |
| variable {Ω : Type cycleUniverse14} {I : Type cycleUniverse15} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] |
|
|
| lemma exists_avoiding_finite_events (s : Finset I) (E : I → Set Ω) |
| (h : ∑ i ∈ s, μ.real (E i) < 1) : ∃ ω, ∀ i ∈ s, ω ∉ E i := by |
| by_contra hn |
| push Not at hn |
| have cover : (⋃ i ∈ s, E i) = Set.univ := by |
| ext ω |
| simp only [Set.mem_iUnion, Set.mem_univ, iff_true] |
| obtain ⟨i, hi, he⟩ := hn ω |
| exact ⟨i, hi, he⟩ |
| have hu := measureReal_biUnion_finset_le (μ := μ) s E |
| rw [cover] at hu |
| have : μ.real Set.univ = 1 := by simp [Measure.real] |
| rw [this] at hu |
| linarith |
|
|
| end FiniteUnion |
| end ErdosGallai.Sampling |
| end |
|
|
| end OAI |
|
|
