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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | import Mathlib
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Real
open Set Real
open scoped Topology
open scoped BigOperators
open Set
open Set MeasureTheory
open scoped Classical
namespace CriticalPercolation.Priority
noncomputable def uniform01 : Measure ℝ := volume.restrict (Icc 0 1)
instance : IsProbabilityMeasure uniform01 := by
constructor
simp [uniform01, Real.volume_Icc]
variable {I : Type*} [Fintype I]
noncomputable def law : Measure (I → ℝ) := Measure.pi fun _ => uniform01
instance : IsProbabilityMeasure (law (I := I)) := by
unfold law
infer_instance
noncomputable def split (i : I) : (I → ℝ) ≃ᵐ (ℝ × ({j : I // j ≠ i} → ℝ)) where
toEquiv := Equiv.funSplitAt i ℝ
measurable_toFun := (measurable_pi_apply i).prodMk
(Measurable.of_eval (fun coordinate => measurable_pi_apply coordinate.val))
measurable_invFun := by
apply Measurable.of_eval
intro j
by_cases h : j = i
· subst j
simpa [Equiv.funSplitAt, Equiv.piSplitAt] using measurable_fst
· simpa [Equiv.funSplitAt, Equiv.piSplitAt, h, Function.comp_def] using
(measurable_pi_apply (⟨j,h⟩ : {j : I // j ≠ i})).comp measurable_snd
omit [Fintype I] in
lemma split_apply (i : I) (u : I → ℝ) : split i u = (u i, fun j : {j : I // j ≠ i} => u j.val) := rfl
omit [Fintype I] in
lemma split_symm_apply (i j : I) (u : ℝ × ({j : I // j ≠ i} → ℝ)) :
(split i).symm u j = if h : j = i then u.1 else u.2 ⟨j,h⟩ := by
simp only [split, MeasurableEquiv.symm_mk, Equiv.funSplitAt, Equiv.piSplitAt,
MeasurableEquiv.coe_mk, Equiv.coe_fn_symm_mk]
split_ifs with h
· subst j; rfl
· rfl
lemma split_preserving (i : I) : MeasurePreserving (split i) law
(uniform01.prod (law (I := {j : I // j ≠ i}))) := by
apply MeasurePreserving.symm (split i).symm
refine ⟨(split i).symm.measurable, ?_⟩
unfold law
refine (Measure.pi_eq fun s _ => ?_).symm
rw [MeasurableEquiv.map_apply]
have he : (split i).symm ⁻¹' (Set.univ.pi s) =
s i ×ˢ (Set.univ.pi fun j : {j : I // j ≠ i} => s j) := by
ext x
simp only [Set.mem_preimage, Set.mem_prod, Set.mem_pi, Set.mem_univ, forall_true_left]
constructor
· intro h
constructor
· simpa [split_symm_apply] using h i
· intro j; simpa [split_symm_apply, j.property] using h j
· rintro ⟨h₁,h₂⟩ j
by_cases h : j = i
· subst j; simpa [split_symm_apply] using h₁
· simpa [split_symm_apply, h] using h₂ ⟨j,h⟩
rw [he, Measure.prod_prod, Measure.pi_pi]
have hprod := Fintype.prod_subtype_mul_prod_subtype (fun j : I => j = i)
(fun j => uniform01 (s j))
simpa only [Fintype.prod_unique, (default : {j : I // j = i}).property] using hprod
lemma uniform01_Ioi {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) :
uniform01 (Ioi t) = ENNReal.ofReal (1-t) := by
rw [uniform01, Measure.restrict_apply measurableSet_Ioi]
have he : Ioi t ∩ Icc (0:ℝ) 1 = Ioc t 1 := by
ext x
simp only [Set.mem_inter_iff, Set.mem_Ioi, Set.mem_Icc, Set.mem_Ioc]
constructor
· intro h; exact ⟨h.1,h.2.2⟩
· intro h; exact ⟨h.1, ht.1.trans h.1.le, h.2⟩
rw [he, Real.volume_Ioc]
lemma integral_survival (S : Finset I) {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) :
(∫ u : I → ℝ, if ∀ j ∈ S, t < u j then (1:ℝ) else 0 ∂law) = (1-t)^S.card := by
have he : {u : I → ℝ | ∀ j ∈ S, t < u j} =
Set.univ.pi (fun j => if j ∈ S then Ioi t else Set.univ) := by
ext x
simp only [Set.mem_ofPred_eq, Set.mem_pi, Set.mem_univ, forall_true_left]
constructor
· intro h j; split_ifs with hj
· exact h j hj
· trivial
· intro h j hj; simpa [hj] using h j
have hf : (fun u : I → ℝ => if ∀ j ∈ S, t < u j then (1:ℝ) else 0) =
({u : I → ℝ | ∀ j ∈ S, t < u j}).indicator (fun _ => (1:ℝ)) := by
funext u
simp [Set.indicator]
rw [hf]
rw [integral_indicator]
· rw [integral_const, smul_eq_mul, mul_one, measureReal_def, Measure.restrict_apply_univ,
he, law, Measure.pi_pi]
simp only [apply_ite, uniform01_Ioi ht, measure_univ, ENNReal.toReal_prod,
apply_ite, ENNReal.toReal_ofReal (sub_nonneg.mpr ht.2), ENNReal.toReal_one]
rw [Finset.prod_ite_mem Finset.univ S (fun _ => 1-t)]
simp
· rw [he]
exact MeasurableSet.univ_pi fun j => by split_ifs <;> measurability
end CriticalPercolation.Priority
open Set MeasureTheory Real
open scoped Interval
end
end
end OAI
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